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OSMANIA UNIVERSITY LIBRARY
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No.
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Accession No.
Author
This book should be returned on
of
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marke4
/>eiow,
THE
EINSTEIN
THEORY
OF
RELATIVITY
Boofcs
by L
/?.
ancf H. G. Lieber
NON-EUCLIDEAN GEOMETRY
GALOIS AND THE THEORY OF GROUPS
THE EDUCATION OF
MITS
C.
T.
THE EINSTEIN THEORY OF RELATIVITY
MITS,
WITS
AND LOGIC
INFINITY
Books of drawings by H. G. Lieber
GOODBYE
MR.
MAN, HELLO MR. NEWMAN
(WITH INTRODUCTION BY
L.
R.
LIEBER)
COMEDIE INTERNATIONALE
THE
EINSTEIN
THEORY
OF
RELATIVITY
Text By
LILLIAN
R.
LIEBER
Drawings By
HUGH GRAY
LIEBER
HOLT, RINEHART
New
AND WINSTON
York / Chicago / San Francisco
Copyright, 1936, 1945, by
L.
R.
and
H. G. Lieber
All rights reserved, including the right to repro-
duce
In
of
this
book or portions thereof in any form.
Holt, Rinehart and Winston
Canada,
Canada,
Limited.
First Edition
October 1945
Second Printing, April 1946
Third Printing, November 1946
Fourth Printing, November 1947
Fifth Printing, May 1950
S/xtfi Printing, March 1954
Seventh Printing, November 1 957
Eighth Printing, July 1958
Ninth Printing, July 1959
Tenth Printing, April 1960
Eleventh Printing, April 1961
Twelfth Printing, April 1964
Thirteenth Printing, November 1966
Firsf Printing,
1975
85251-0115
Printed in the United States of America
To
FRANKLIN DELANO ROOSEVELT
who saved
the world from those forces
of evil which sought to destroy
Art and Science and the very
Dignity of
Man.
PREFACE
In
this
book on
the Einstein Theory of Relativity
the attempt is made
to introduce just enough
mathematics
to
HELP
NOT to HINDER
and
the lay reader/
"lay" can of course apply to
various domains of knowledge
perhaps then
we should
say:
the layman in Relativity.
Many
"popular" discussions of
Relativity,
without any mathematics at
have been written.
But we doubt whether
all,
even the best of these
can possibly give to a novice
an adequate idea of
what
What
in
it is
all
about.
very clear when expressed
mathematical language
is
sounds "mystical"
in
ordinary language.
On
the other hand,
there are
many
discussions,
including Einstein's own papers,
which are accessible to the
experts only.
vii
We
believe that
there
is
a class of readers
who
can get very little out of
either of these two kinds of
discussion
readers who
know enough about
mathematics
to follow a simple mathematical presentation
of a domain new to them,
built from the ground up,
with sufficient details to
bridge the gaps that exist
FOR THEM
in
both
the popular and the expert
presentations.
This book is an attempt
to satisfy the needs of
this
kind of reader.
viii
CONTENTS
PREFACE
Part
I.
II.
III.
I
-THE
SPECIAL THEORY
INTRODUCTION
3
The Michelson-Morley Experiment
8
IV. The
V. The
Remedy
31
Solution of the Difficulty
39
VI. The Result of Applying the
VII.
VIII.
20
Re-Examination of the Fundamental Ideas
The
Four-Dimensional
tinuum
Some Consequences
44
Remedy
Space-Time
Con-
57
of the Theory of
69
Relativity
IX.
A
Point of Logic and a
83
Summary
87
The Moral
Part
A
II
-THE GENERAL THEORY
GUIDE TO PART
91
II
95
X. Introduction
101
XI. The Principle of Equivalence
XII.
XIII.
A
107
Non-Euclidean World!
113
The Study of Spaces
XIV. What
XV. The
Is
127
a Tensor?
Effect
on Tensors
of a
Change
Coordinate System
XVI.
A Very Helpful Simplification
ix
in the
1
41
150
160
XVII. Operations with TenseXVIII.
A Physical
167
Illustration
XIX. Mixed Tensors
XX.
Contraction and Differentiation
XXI. The
XXII.
XXIII.
Our
Last Detour
200
at Last
the Curvature Tensor?
206
Einstein's
Law
of Gravitation
21 3
of Einstein's
Law
of Gravitation
Is
The Big G's or
219
with Newton's
XXVII.
How
Can the
Einstein
Law
of Gravitation Be
227
Tested?
XXVIII. Surmounting the
XXIX. "The Proof
Difficulties
237
Pudding"
255
the Path of a Planet
266
of the
XXX. More About
XXXI. The Perihelion
272
of Mercury
XXXII.
Deflection of a
Ray
XXXIII.
Deflection of a
Ray of
XXXIV. The
78
191
The Curvature Tensor
XXVI. Comparison
1
187
Little g's
XXIV. Of What Use
XXV.
173
of Light
Light, cont.
Third of the "Crucial"
Phenomena
XXXV. Summary
283
289
299
303
The Moral
Would You Like
THE ATOMIC
276
to
Know?
BOMB
310
318
Parti
THE SPECIAL THEORY
INTRODUCTION.
I.
In order to appreciate
the fundamental importance
of Relativity,
it is
how
necessary to
it
Whenever
in
know
arose.
11
a "revolution
takes place,
any domain,
always preceded by
it is
some maladjustment producing
a tension,
which ultimately causes a break,
followed by a greater stability
at least for the
time being.
What was
the maladjustment in Physics
the latter part of the 19th century,
which led to the creation of
in
11
the "revolutionary
Let us summarize
It
Relativity
it
Theory?
briefly:
has been assumed that
space is filled with ether,*
through which radio waves and
all
light
waves
are transmitted
any modern child
*This ether
is
talks quite glibly
of course
NOT the
chemical ether
which surgeons use!
ft is
it
not a liquid, solid, or gas,
has never been seen
by anybody,
presence is only conjectured
because of the need for some medium
to transmit radio and light waves.
its
3
11
about "wave-lengths
in connection with the radio.
Now,
there
if
is
an ether,
does it surround the earth
and travel with it,
or does it remain stationary
while the earth travels through it?
Various known (acts* indicate that
the ether does
NOT travel
with the earth.
THROUGH
the earth is moving
If, then,
there must be an "ether wind/'
person riding on a bicycle
just as a
through
feels an
still air,
air
wind blowing
in his face.
And
so an experiment was performed
and Morley (see p. 8)
Michelson
by
in 1887,
to detect this ether wind/and much to the surprise of everyone,
no ether wind was observed.
unexpected result was explained by
Dutch physicist, Lorentz, in 1 895,
in a way which will be described
This
a
in
Chapter
II.
The search for the ether wind
was then resumed
by means of other kinds of experiments.!
*See the
by A.
in
M
article
Aberration of Light",
S.
Eddington,
the Encyclopedia Britannica, 14th ed.
tSec the article "Relativity"
by James Jeans,
also in the Enc. Brit. 14th ed.
4
the ether,
But, again and again,
to the consternation of the physicists,
no ether wind could be detected,'
until
it
seemed
that
1
nature was in a "conspiracy'
to prevent our finding this effect!
At
this
point
Einstein took
and decided
up the problem,
that
11
a natural "conspiracy
must be a natural
LAW
And
operating.
to answer the question
what is
he proposed
as to
his Theory of
two papers,
1905 and the other
in
Relativity,
in
published
one
this law,
in
1915.*
He first found it necessary to
re-examine the fundamental ideas
upon which classical physics was based,
and proposed certain vital changes in them.
He
then
made
A
VERY LIMITED NUMBER OF
MOST REASONABLE ASSUMPTIONS
from which he deduced his theory.
So fruitful did his analysis prove to be
that
by means
of
it
he succeeded
in:
(1) Clearing up the fundamental ideas.
(2) Explaining the Michelson-Morley experiment
in a much more rational way than
had previously been done.
*Both
now
published
in
one volume
including also the papers
by
Lorentz and Minkowski,
to which
see
we
SOME
shall refer later/
INTERESTING READING, page 324.
Doing away with
(3)
other outstanding difficulties
in physics.
(4) Deriving a
NEW LAW OF GRAVITATION
much more adequate than the
Newtonian one
(See Part II.: The General Theory)
and which led to several
important predictions
which could be verified by experiment;
and which have been so verified
since then.
(5) Explaining
QUITE INCIDENTALLY
a famous discrepancy in astronomy
which had worried the astronomers
for
many
(This also
years
is discussed in
The General Theory).
Thus, the Theory of Relativity had
a profound philosophical bearing
of physics,
on
as well as explaining
ALL
many SPECIFIC
that
outstanding difficulties
had seemed to be entirely
UNRELATED,
and
of further increasing our
of the physical world
by suggesting a number of
NEW experiments which
NEW discoveries.
No
knowledge
have led to
other physical theory
been so powerful
though based on so FEW assumptions.
has
As we
shall see.
THE MICHELSON-MORLEY EXPERIMENT*
II.
On
page 4 we referred to
the problem that
Michelson and Morley
Let us
now
set themselves.
see
what experiment they performed
and what was the startling result.
In
order to get the idea of the experiment
very clearly
it
will
in
mind,
be helpful
first
to consider the following simple problem,
which can be solved
by any boy who
has studied
elementary algebra:
Imagine a river
in which there is a current flowing with
velocity v,
in the direction indicated
by the
Now
which would take longer
for a
man
From
A
to
to
'Published
swim
B and back
in
arrow:
to
A
,
the
Philosophical Magazine, vol. 24, (1887).
8
or
from
if
A
C and back
to
the distances
AB
to
A
,
AB and AC are
equal,
being parallel to the current,
AC perpendicular to it?
Let the man's rate of swimming
and
in still
be represented by c /
then, when swimming against the
from
A
his rate
8
to
water
current,
,
would be only
v
c
f
whereas,
when swimming with
from 8 back to A ,
his rate
the current,
+
would, of course, be c
v.
Therefore the time required
to swim from A to fi
would be a/(c
v),
where a represents the distance
and the time required
for the trip
from
would be a/(c
8
+
to
A
v).
Consequently,
the time for the round
ti
or
Now
ti
let us
=
=
trip
-
a/(c
2
2ac/(c
would be
/)
-
+
v
a/(c
+
v)
2
).
see
how
long the round
from
A
to
AB ;
trip
C and back
to
A
would take.
If he headed directly toward C
,
the current would carry him downstream,
and he would land at some point
to the left of C in the figure on p. 8.
Therefore,
in order to arrive at
C
,
9
he should head
just far
for
some point
D
enough upstream
to counteract the effect of the current.
In
other words,
the water could
be kept
until
he swam
own
from
A
if
to
D
at his
still
rate c
,
and then the current
were suddenly allowed to operate,
carrying him at the rate v from D to C
(without his making any further effort),
then the effect would obviously be the same
as his going directly from
A
iojC
2
2
with a velocity equal to Vc'
v/
as is obvious from the right triangle:
ex
\r
Consequently/
the time required
for the journey from
A
to
C
2
would be a/Vc^- v ,
where a is the distance from
Similarly,
in
going back from C to
easy to see that,
A,
it is
10
A
to
C
method
the same
by
of reasoning,
2
_
the time would again be a/Vc
Hence the time for the round trip
from
A
to
C and back
would be
fa
In
let us write ft for
Then we
get:
ti
and
fa
=
c/
ti
2
.
_
A,
to
= 2a/vV -
order to compare
v
and
Vc
f-
2
v
y\
more
easily,
2
.
2
2a/3
/c
2a/3/c.
Assuming that v is less than c ,
2
2
v being obviously less than
and c
Vc
the
2
v
2
is
therefore less than c
and consequently ft is greater than
(since the denominator
is
less than
Therefore
that
IT
t\
the numerator).
is greater than
1
c
2
,
,
^
fa ,
is,
TAKES LONGER TO
SWIM UPSTREAM AND BACK
THAN TO SWIM THE SAME DISTANCE
ACROSS-STREAM
AND
BACK.
But what has all this to do
with the Michelson-Morley experiment?
In that experiment,
to B:
a ray of light was sent from
A
C-r
^-
HB
11
At 8 there was a mirror
which reflected the light back to
so that the ray of light
makes the round trip from
just as the
A,
AioB and
back,
swimmer did
the problem described above.
since the entire apparatus
shares the motion of the earth,
in
Now,
which is moving through space,
supposedly through a stationary ether/
thus creating an ether wind
in the opposite direction,
(namely, the direction indicated above),
this experiment seems entirely analogous
to the problem of the swimmer.
And,
therefore/ as before/
ti
and
ti
= 2a0Yc
= 2a|S/c.
0)
(2)
is now the velocity of light,
the time required for the light
to C and back to
to go from
Where
and
c
*2 is
A
A
(being reflected from another mirror at
If/
ti
Q.
therefore,
and t> are found experimentally/
then
by dividing (1)
the value of /? would
And
since
by (2),
be easily obtained.
= c/V c
2
-T
2
,
c being the known velocity of light,
the value of v could be calculated.
That
is,
THE ABSOLUTE VELOCITY
OF THE EARTH
would thus become known.
Such was the plan of the experiment.
Now
what actually happened?
12
The experimental values of t\ and
were found to be the SAME,
instead of
ti
ti
being greater than ti
was a most disturbing
\
this
Obviously
result,
quite out of harmony
with the reasoning given above.
The Dutch
physicist, Lorentz,
then suggested the following explanation
of Michelson's strange result:
Lorentz suggested that
matter,
owing
and
this
to
its
WHEN
SHRINKS
electrical structure,
IT
IS
MOVING,
contraction occurs
ONLY IN THE DIRECTION OF
of shrinkage
The
he assumes to be in the ratio of 1/ff
MOTION.*
AMOUNT
2
v
(where /3 has the value c/Vc
Thus a sphere of one inch radius
ellipsoid when
shortest semi-axis
becomes an
with
its
(now only
it is
2
,
as before).
moving,
1//3 inches long)
*The two papers by Lorentz on this subject
are included in the volume mentioned in
the footnote on page 5.
In the first of these papers
Lorentz mentions that the explanation proposed here
occurred also to Fitzgerald.
Hence
it
is
often referred to as
the "Fitzgerald contraction" or
the "Lorentz contraction* or
1
11
the "Lorentz-Fitzgerald contraction.
13
in
the direction of motion,
thus:
a erection
Applying this idea
Michelson-Morlcy experiment,
to the
the distance
AB (=
on
a)
becomes a/jS ,
and ti becomes 2a/3/c
p. 8,
/
2
instead of 2a/3 /c ,
so that now ft
t2
=
just as the
One
,
experiment requires.
might ask
how
it is
that Michelson did not
observe the shrinkage?
Why did not his measurements show
that
AB was
shorter than
AC
(See the figure on p. 8)?
The obvious answer is that
the measuring rod itself contracts
when applied to AB,
so that one is not aware of the shrinkage.
To
this
explanation
of the
Michelson-Morley experiment
the natural objection may be raised
that an explanation
for the express
which
is
purpose
14
invented
of smoothing out a certain
and assumes a correction
of
is
JUST
too
And
difficulty,
the right amount,
to be satisfying.
artificial
Poincare, the French mathematician/
raised this very natural objection.
Consequently,
Lorentz undertook to examine
his contraction
hypothesis
other connections,
to see whether it is in harmony also
with facts other than
in
the Michelson-Morley experiment.
He then published a second paper
in
1904,
giving the result of this investigation.
To present
let us
first
this result in a clear
re-state the
form
argument
as follows:
vt
T
Consider a set of axes, X and Y,
supposed to be fixed in the stationary ether,
and another set X' and Y' ,
attached to the earth and moving with it,
15
with velocity v
X
Let
and
7
V"
Now
,
as indicated
move along X,
move parallel to
above
V.
suppose an observer on the
earth,
say Michelson,
is
trying to
the time
measure
takes a ray of light
to B ,
to travel from
it
A
A
and 8 being fixed points on
the moving axis X .
both
r
At
the
moment
when the
ray of light
starts at
A
f
that
Y and Y
suppose
coincide,
and A coincides with D /
and while the light has been traveling to B
the axis
V
moved
has
the distance vt
,
and B has reached the position
shown in the figure on p. 1 5,
t
being the time
it
takes for this to happen.
DB = x and AB =
Then,
we have x'
if
=
x
x',
vt.
(3)
only another way
of expressing what was said on p.
where the time for
This
is
first part of the journey
was said to be equal to a/(c
9
the
And, as we saw there,
this way of thinking of
did
NOT agree
Applying now
the
v).*
phenomenon
with the experimental facts.
the contraction hypothesis
we are now designating a by x',
= t , or x = ct
we have x'/(c
v)
*Since
vf.
But the distance the light has traveled
is x ,
and x
=
ct,
consequently x'
=x-
vt
is
16
equivalent to a/(c
v)
=
t.
proposed by Lorentz,
x should be divided by /3,
so that equation (3) becomes
r
x7/3
or
x'
Now
- vt
- vt).
(x
=
x
=
when Lorentz examined
as stated
on
1
p.
(4)
other fads,
5,
he found that equation (4)
was quite in harmony with ail these
but that he was now obliged
facts,
to introduce a further correction
expressed by the equation
(5)
where /3 , t , v , x , and c
have the same meaning as before
But what is t?!
Surely the time measurements
in the two systems are not different:
Whether the
origin
is
at
D
or at
A
should not affect the
TIME-READINGS.
In
t'
other words, as Lorentz saw it,
11
sort of "artificial
time
was a
introduced only for mathematical reasons,
because
in
it
helped to give
harmony with the
But obviously
NO
As
had
t
results
facts.
for
him
PHYSICAL MEANING.
Jeans, the English physicist, puts
"If the observer
it:
could be persuaded
to measure time in this
artificial way,
wrong to begin with
and then making them gain or lose permanently,
the effect of his supposed artificiality
setting his clocks
17
would
just
counterbalance
the effects of his motion
11
through the ether
!*
Thus,
the equations finally proposed
by Lorentz
are:
Note
x'
=
z'
=
(x
-
vt)
that
since the axes attached to the earth (p.
are moving along the X-axis,
1
5)
obviously the values of y and z
(z
being the third dimension)
are the
same
The equations
are
known
/
and z
just
given
as
,
respectively.
as
THE LORENTZ TRANSFORMATION,
since they show how to transform
a set of values of x , y , z , t
into a set x',
y, z,
t'
coordinate system
moving with constant velocity
in a
v,
along the X-axis,
with respect to the
unprimed coordinate system.
And,
as
we
saw,
whereas the Lorentz transformation
really expressed the facts correctly,
it seemed to have
NO
PHYSICAL MEANING,
*See the
article
on
Relativity in the
Encyclopedia Britannica, 14th edition.
19
and was merely
a set of empirical equations.
Let us
now
see what Einstein did.
RE-EXAMINATION OF THE
III.
FUNDAMENTAL
IDEAS.
As Einstein regarded the situation,
the negative result of
the Michelson-Morley experiment,
as well as of other experiments
which seemed to indicate a "conspiracy"
on the part of nature
against man's efforts to obtain
knowledge of the physical world (see p. 5),
these negative results,
according to Einstein,
did not merely demand
explanations of a certain number
of isolated difficulties,
but the situation was so serious
that a
complete examination
of fundamental ideas
was necessary.
In
he
other words,
felt that there was something
fundamentally and radically wrong
in
physics,
rather than a
And
mere
superficial difficulty.
so he undertook to re-examine
such fundamental notions as
our ideas of
LENGTH
and
TIME and MASS.
His exceedingly reasonable examination
20
is
most illuminating,
as
we
But
shall
first
why
now
let us
see.
remind the reader
and mass
length, time
are fundamental,
Everyone knows
that
VELOCITY depends upon
the distance (or LENGTH)
traversed in a given
TIME,
hence the unit of velocity
DEPENDS UPON
the units of
LENGTH
and TIME.
Similarly,
since acceleration
is
the change in velocity in a unit of time,
hence the unit of acceleration
DEPENDS UPON
the units of velocity and time,
and therefore ultimately upon
the units of
LENGTH
and TIME.
Further,
since force
is
measured
by the product
of
mass and acceleration,
the unit of force
DEPENDS UPON
the units of mass and acceleration,
and hence ultimately
the units of
MASS, LENGTH
And so on.
In
all
upon
and TIME,
other words,
measurements
depend
in
MASS, LENGTH
That
is
physics
primarily on
and TIME.
why
21
the system of units ordinarily used
is called the "C.G.S."
where C stands
for
system,^
"centimeter"
(the unit of length),
stands for "gram" (the unit of mass),
and 5 stands for "second" (the unit of time)/
G
these being the fundamental units
from which all the others are derived.
Let us
now
return to
Einstein's re-examination of
these fundamental units.
Suppose
that
two observers
wish to compare their measurements of time.
If they are near each other
they can, of course/ look
and compare them.
at
each other's watches
If they are far apart/
they can still compare each other's readings
BY
MEANS OF
SIGNALS,
say light signals or radio signals/
1
that is/ any "electromagnetic wave*
which can travel through space.
Let us/ therefore/ imagine that
one observer/ f , is on the earth/
and the other/ 5 / on the sun/
and imagine that signals are sent
as follows:
By his own watch/ 5 sends a message to
which reads "twelve o'clock/"
f receives this message
say/ eight minutes later;*
*Since the sun is about
from the earth,
93 000 000
miles
light travels about 186 000 miles per second,
the time for a light (or radio) signal
to travel from the sun to the earth/
and
is
approximately eight minutes.
22
now,
it
if
when
his
watch agrees with that of S
n
f<
,
12:08
read
will
the message arrives.
then sends back to
5
the message "12:08,"
and, of course,
5
receives this message 8 minutes
namely, at 12:16.
Thus 5 will conclude,
from
later,
this series of signals,
that his
watch and that of f
are in perfect agreement.
But
let us
now imagine
that the entire solar system
is moving through
space,
so that both the sun and the earth
moving in the direction
shown in the figure:
are
without any change
in
the distance between them.
Now
let
the signals again
be
sent
as before:
S sends
his message "1 2 o'clock,"
but sincejf js moving awayjrom jjta
IRelatter wiffnot reaclTE in 8 minutes,
but will take
to overtake
Say,
9
f
some longer time
,
minutes.
23
If
Fs watch
it
will read
when
is in agreement with
12:09
that of
5,
the message reaches him,
and
accordingly sends a return message/
reading "12:09."
Now 5
and
traveling toward this message/
is
will therefore
it
reach him
LESS than 8 minutes,
in
say, in 7 minutes.
Fs message
Thus S receives
/
at 1 2:1 6,
just as before.
Now
5 and F
if
UNAWARE
are both
of their motion
(and, indeed,
we
in
are
undoubtedly moving
ways
so that
that
we
are entirely unaware of,
this
assumption
is far from being an imaginary one.)
5 will not understand
why Fs message reads
"12:09" instead of "12:08,"
and will therefore conclude
that Fs watch
must be fast.
Of
course, this
only
is
Fs
we know,
an apparent error
because, as
it is
really
and not
to
any
due
in
watch,
to the motion,
at all
error in
Fs
watch.
must be noted, however,
that this omniscient "we"
It
who
can see exactly
what
is
"really" going on
does not
and
in
exist,
that all
human observers
24
the universe,
are really in the situation
in
which 5
is,
namely,
that of not knowing
about the motion in question,
and therefore
being OBLIGED to conclude
that 's watch is wrong!
And
therefore,
the message
5 sends
telling him that
if
sets his clock
back one minute,
then their clocks will agree.
In
the same way,
that other observers,
suppose
j4,B,C,ctc.,
all
of
whom
5 and
all
are at rest
WITH RESPECT
TO
,
set their clocks to agree with that of
S
,
by the same method of signals described above.
They would all say then
that all their clocks are in agreement.
Whether this is absolutely true or not,
they cannot tell (see above),
but that is the best they can do.
Now
let us
when
these observers wish
see what will happen
to measure the length of something.
of an object,
To measure the length
you can place it,
say, on a piece of paper,
put a mark on the paper at one end of the object,
and another mark at the other end,
then, with a ruler,
find out
how many
units of length there are
25
between the two marks.
This
is quite simple provided that
the object you are measuring and the paper
are at rest (relatively to you).
But suppose the object
say, a
fish
is
swimming about
To measure
in a
tank?
length while it is in motion,
by placing two marks on the walls of the tank,
one at the head, and the other at the tail,
it
to
its
would obviously be necessary
make these two marks
SIMULTANEOUSLY otherwise,
the mark 6 is
for,
if
C
made
at a certain time,
B
/
V-
then the
in
fish
allowed to swim
the direction indicated
and then the mark
is made
when it
at
some
at the
later
has reached
by
the arrow,
head
time,
C,
then you would say that
the length of the fish
is the distance BC,
which would be a
Now
suppose
after their
fish-story
indeed!
that our observers,
clocks are
all in
agreement (see p. 25),
undertake to measure
the length of a train
26
which
is
moving through
their universe
with a uniform velocity.
They send out orders that
2 o'clock sharp,
whichever observer happens to be
at 1
place where
the front end of the train,
at the
arrives at that
to
A'9
moment,
NOTE THE SPOT;
and some other observer,
who happens to be at the place where
the rear end of the train, B ,
is at that same moment,
to put a mark at THAT spot.
Thus, after the train has gone,
they can, at their leisure,
measure the distance between the two marks,
this distance being equal to
the length of the train,
since the two marks were
SIMULTANEOUSLY,
their clocks
in
being
made
namely
at
12
o'clock,
all
perfect agreement with each other.
Let us
now
talk to the
people on the
train.
Suppose, first,
that they are unaware of their motion,
and that, according to them,
A f B f ^, t etc., are the ones who are moving,
a perfectly reasonable assumption.
And suppose that there are two clocks
one
and
on the
at A', the other at B',
that these clocks
have been set
in
agreement with each other
method
of signals described above.
, B , C, etc.,
Obviously the observers
admit that the clocks at
and B'
wit!
by
the
A
NOT
27
A
train,
arc in agreement with each other,
since they "know" that the train is in motion,
and therefore the method of
signals
used on the moving train
has led to an erroneous setting
of the moving clocks (see p. 25).
Whereas the people on the train,
since they
"know"
A, B , C,
etc./ are the
claim that
it is
that
belonging to A, 8, C,
which were set wrong.
What
is
who
ones
are moving,
the clocks
etc.,
the result of this
difference of opinion?
When
the clocks of
A
and B
,
say,
both read 12 o'clock,
and
at that instant
A
and B
each makes a mark at a certain spot,
then A and B claim, of course,
that these
marks were
made
simultaneously;
but the people on the train
that the clocks of
A
have been properly
do not admit
and 8
set,
and they therefore claim that
the two marks were
NOT
and
made SIMULTANEOUSLY,
that, therefore,
the measurement of the
is
LENGTH
of the train
NOT correct.
Thus,
when the people on the
make the marks
train
simultaneously,
judged by their own clocks,
the distance between the two marks
as
28
will
NOT be the same as
Hence we see
before.
that
MOTION
prevents agreement
in
the
setting of clocks,
and, as a consequence of
this,
prevents agreement in the
measurement of LENGTH!
Similarly,
we
as
shall
see on p. 79,
motion also
affects
the measurement of mass
different observers obtaining
different results
when measuring the mass
And
all
same object.
since,
we mentioned on
as
of the
p.
21,
other physical measurements
depend upon
length, mass,
it
seems
and time,
that
be agreement
any measurements made
therefore there cannot
in
by different observers
who are moving with different
velocities!
Now, of course,
observers on the earth
partake of the various motions
to which the earth
the earth turns on
is
its
subject
axis,
goes around the sun,
and perhaps has other motions as wefl.
it
Hence
it
would seem
observations
that
made by people on
29
the earth
cannot agree with
those taken from
some
other location in the universe,
and are therefore
not really correct
and consequently worthless!
Thus Einstein's careful and reasonable examination
led to the realization that
Physics was suffering from
no mere
single ailment,
evidenced by the
Michelson-Morley experiment alone,
but was sick from head to foot!
as
Did he
HE
find a
remedy?
DID!
IV.
So
far,
then,
we
THE REMEDY.
see that
THE OLD IDEAS REGARDING
THE MEASUREMENT OF
LENGTH, TIME AND MASS
involved an "idealistic" notion of
11
"absolute time
which was supposed to be
the same for
and
all
observers,
that
Einstein introduced
a
more
PRACTICAL
notion of time
based on the actual way of
setting clocks
by means
of
SIGNALS.
This led to the
DISCARDING
of the idea that
31
the
is
LENGTH
and
is
of an object
about the object
a (act
independent of the person
who does the measuring/
since we have shown (Chapter
III.)
measurement of length
that the
DEPENDS UPON
THE STATE OF MOTION OF THE MEASURER.
Thus two observers,
moving
relatively to each other
with uniform velocity/
DO NOT
GET THE SAME VALUE
A GIVEN OBJECT.
FOR THE LENGTH OF
Hence we may
LENGTH
is
say that
NOT a FACT about an
OBJECT,
but rather a
RELATIONSHIP between
OBJECT and the OBSERVER.
And similarly for TIME and MASS
the
In
(Ch.
III.).
other words,
from
it is
this
point of view
NOT CORRECT to
x'
=
x
say:
vt
Michelson did* (see p. 16, equation (3)
since this equation implies that
the value of x'
as
is
a perfectly definite quantity,
*We do
not wish to imply that
a crude error
Michelson made
ANY
CLASSICAL PHYSICIST
would have made the same statement,
for those were the prevailing ideas
thoroughly rooted in everybody's mind,
before Einstein pointed out
the considerations discussed in Ch. III.
32
),
namely,
length of the arm AB of the apparatus
the Michelson-Morley experiment
THE
in
(See the diagram on p. 1 5).
Nor is it correct to assume that
(again as Michelson did)
two different observers,
which would imply that
both observers agree in their
time measurements.
for
These ideas were contradicted by
Michelson's EXPERIMENTS,
which were so ingeniously devised
and so precisely performed.
And
so Einstein said that
instead of starting with such ideas,
and basing our reasoning on them,
let us rather
START WITH THE EXPERIMENTAL DATA,
and see
they
to
what relationships
will lead us,
relationships
between
the length and time measurements
of different observers.
Now
what experimental data
take into account here?
must
we
They
are:
FACT
(1):
It is
impossible
to measure the "ether wind,"
or, in other
words,
impossible to detect our motion
relative to the ether.
it is
This was clearly
33
shown by the
Michelson-Morley experiment,
by all other experiments
devised to
measure this motion (see p. 5).
as well as
Indeed, this is the great
"conspiracy*
that started all the trouble,
or, as Einstein prefers to see
1
and most reasonably
THIS
FACT
(2):
IS
A
FACT.
The velocity of light is the same
no matter whether the source of light
is
moving or
stationary.
Let us examine
more
this
it is
statement
fully,
to see exactly what
To do
it
means.
this,
necessary to remind the reader
of a few well-known facts:
Imagine that we have two trains,
one with a gun on the front end,
the other with a source of sound
on the
front end,
say, a whistle.
Suppose that the velocity, u ,
of a bullet shot from the gun,
happens to be the same
as
the velocity of the sound.
Now
suppose that both trains
moving with the same velocity, v ,
in the same direction.
The question is:
How does the velocity of a bullet
are
(fired
it,
so,
from the
MOVING train)
relatively to the ground,
compare with
34
the velocity of the sound
came from the whistle
that
MOVING
on the other
relatively to the
which
in
is
it
Are they
train,
medium/ the
air,
traveling?
same?
the
No!
The velocity of the
bullet,
RELATIVELY TO THE GROUND,
is
v
+
u
,
since the bullet
now propelled forward
own velocity, u ,
is
not only wiih
its
given to
the force of the gun,
it
by
but, in
addition,
has an inertia! velocity, v ,
which it has acquired from
the motion of the train
and which
is
shared
by
objects on the train.
all
But
in
the case of the sound
wave
a series of pulsations,
(which
alternate condensations and rarefactions of the air
is
in rapid succession),
the first condensation formed
in
the neighborhood of the whistle,
out with the velocity u
travels
relatively to the medium,
regardless as to whether
the train
So
is
that this
has only
and does
its
moving or not.
condensation
own
NOT
velocity
have the
due to the motion of the
inertia!
velocity
train,
the velocity of the sound
depending only upon the
medium
35
(that
is,
whether
and whether
it is
air
or water/ etc.,
hot or cold, etc.),
but not upon the motion of the source
from which the sound started.
it is
The following diagram
shows the relative positions
after one second/
in
both cases:
CASE
Both
Tram
I.
trains at rest.
u
jt.
-ft.
CASE
Both
trains
II.
moving with velocity
v
UfV
u 4*.
in
Case
Thus/
the bullet has
u
+
v feet
in
II.,
moved
one second
36
v.
from the starting point,
whereas the sound has moved only u feet
from the starting point/
in that
Thus
one second.
we
see that
the velocity of sound is u feet per second
relatively to the starting point,
whether the source remains stationary
as in Case
I.,
or follows the sound, as in Case
Expressing
it
II.
algebraically,
x
=
ut
applies equally well for sound
in both Case I. and Case
x being the distance
II.,
FROM THE STARTING
Indeed,
this fact
is
true of
POINT.
ALL, WAVE
MOTION,
and one would therefore expect
that it would apply also to LIGHT.
As a matter
it DOES,
and
that
FACT
Now,
it
is
(2)
of
FACT,
what
on
is
meant by
p. 34.
as a result of this,
appears,
by
referring again to the
diagram on p. 36,
that
relatively to the
we should
for
MOVING
train
then have,
sound
x'
=
(u
-
v)t
x being the distance
from T to the point where
the sound has arrived after time
37
t.
(Case
II.)
From which, by measuring x
we could
then calculate v
the velocity of the train.
And,
'
u
,
,
and t,
,
similarly, for light
using the moving earth
instead of the moving train,
we should
then have,
consequence of
as a
x'
where
c
is
=
(c
FACT
-
(2)
on p 34,
v)t
the velocity of light
(relatively to a stationary observer
out
in
space)
the velocity of the earth
relatively to this stationary observer
and v
is
and hence
ABSOLUTE
the
Thus
velocity of the earth.
we should be
able
to determine v.
But
FACT
this contradicts
(1),
according to which
it is
IMPOSSIBLE
Thus
it
to determine v.
APPEARS
that
FACT
(2) requires
the velocity of light
MOVING EARTH
RELATIVELY TO THE
to
be
v (see diagram on p. 36),
c
whereas
*FACT
FACT
(1)
(1) requires
may be
The velocity of
RELATIVE TO
c,
and
to
be
c.*
re-stated as follows:
light
A MOVING
(For example, an observer
on the moving earth)
must be
it
NOT
c
vt
for otherwise,
he would be able to find v
which is contrary to fact.
,
38
OBSERVER
And
so the two facts
contradict each other!
Or,
If,
stating
it
another way:
one second,
moves from E
in
the earth
to E'
while a ray of light,
goes from the earth to L
,
then
FACT(1)
E'/.
requires that
to c ( = 1
be equal
while
EL be
Now
86,000 miles)
FACT
(2) requires that
to
c /
equal
it is
FACTS
needless to say that
CAN NOT CONTRADICT
EACH OTHER!
Let us therefore see how,
the light of the discussion in Ch. III.
(1) and (2) can be shown to be
in
FACTS
NOT
contradictory.
V. THE
We
SOLUTION OF THE DIFFICULTY
have thus seen that
according to the
facts,
the velocity of light
IS
ALWAYS THE SAME,
39
whether the source of
is
stationary or
light
moving
FACT (2) on p. 34),
and whether the velocity of
is measured
(See
medium
relatively to the
in
light
which
MOVING
or relatively to a
it
travels,
observer
(See p. 37).
Let us express these
(acts algebraically,
two observers, K and K ,
who are moving with uniform velocity
f
for
relatively to each other,
thus:
K writes
and K'
x
=
ct
writes x'
(6)
,
=
(7)
ct',
both using
THE SAME VALUE FOR
THE VELOCITY OF LIGHT,
namely, c ,
and each using
his own measurements of
length, x and x',
and time, t and t', respectively.
It is
assumed
at the instant
that
when
the rays of light start on their path,
and K' are at the
place,
and the rays of light
SAME
K
radiate out from that place
in all directions.
Now according to equation (6),
K who unaware of his motion
,
is
(since he cannot measure it),
may claim that he is at rest,
and that in time, t ,
40
through the ether
K' must have
shown
and that/
as
moved
to the right,
the figure below/
the meantime,
in
in
the light/
which
travels out in all directions from
has reached
all
points at
the distance ct from
K,
and hence
all points on the circumference
of the circle having the radius
W claims that he
But
who
is
ct.
the one
and
has remained stationary/
on the contrary,
that
/
has
moved
K
TO THE
LEFT/
furthermore that the light travels out
f
from K as a center,
instead of from K\
And
this
when he
is
what he means
says
x'
How
=
ct'.
can they both be right?
41
K
,
We
may be
willing
not to take sides
in their
controversy regarding the question as to
which one has moved
K' to the right or
K to
the
left
because either leads to the same
But what about the circles?
result.
,-?
They cannot possibly have both
K and K
'
as their centers!
One
This
the
of
them must be
another
is
way
right
and the other wrong.
of stating
APPARENT CONTRADICTION BETWEEN
FACTS
Now,
(1)
and (2) (see
at last,
we
p. 39).
are ready
for the explanation.
Although
K claims
at the instant
that
when
the light has reached the point C(p. 41),
it has also reached
the point
A
f
on the other
side,
still,
WE MUST REMEMBER THAT
when
K says
two events happen simultaneously
(namely, the
arrival of
the light at
C and
DOES NOT AGREE
THAT THEY ARE SIMULTANEOUS
A),
K'
So
that
(see p. 28).
when
K says that
the arrival of the lisht at
(rather than at C and A)
C and B
ARE SIMULTANEOUS,
his
statement
DOES NOT CONTRADICT THAT OF K,
since K and K'
DO NOT MEAN THE SAME THING
42
WHEN THEY SAY
"SIMULTANEOUS-.
11
for
K's clocks at
C and A
not agree with K'*s clocks at C and A.
Thus when the light arrives at A ,
do
the reading of K*s clock there
is exactly the same as that of K's clock at
(K having set all clocks in his system
by the method of signals described on
while
n
K s clock at
C
p. 25),
Ai
when
the light arrives there/
reads a LATER TIME than his clock at
when
the light arrived at
so that K maintains that
C
C,
A
the light reaches
LATER than it reaches
and
as
C,
NOT at the SAME
instant,
K claims.
Hence we see
that
they are not really contradicting each other,
but ihat they are merely using
two
different systems of clocks,
such that
the clocks in each system
agree with each other alright,
but the clocks
have
NOT
in
been
the
one system
set
in
agreement with the clocks
in
the other system (see p. 28).
That
If
we
is,
take into account
the inevitable necessity of
using signals
in order to set clocks which are
43
each other,
at a distance from
and
that the arrivals of the signals
at their destinations
are influenced
by
our state of motion,
of which we are not aware (p. 24),
it
becomes
THERE
clear that
NO
IS
REAL CONTRADICTION HERE,
but only a difference of description
due
to
INEVITABLE
differences
the setting of
various systems of clocks.
in
We
now
in a
general qualitative way,
see
that the situation
not
as
at all
it
But
is
mysterious or unreasonable,
seemed
to
be
we must now
at
first.
find out
whether these considerations,
when applied
QUANTITATIVELY,
actually agree with the experimental facts.
And now
a pleasant surprise awaits us.
THE RESULT OF APPLYING
THE REMEDY.
VI.
In
the
by
last
chapter
two fundamental
we reached
expressed
that
FACTS
(p. 34),
the conclusion
in
the equations
x
and
we saw
starting with
x'
=
=
ct
(6)
ct'
(7)
44
which are graphically represented on p. 41,
and we realized that these equations
NOT
are
if
contradictory,
they appear to be
(as
we remember
at
first),
that there
is
a difference in the setting of the clocks
the two different systems.
in
We
now, from (6) and (7),
between the measurements
of the two observers, K and K
shall derive,
relationships
f
.
And
is
all
the mathematics
we need
for this
a little simple algebra,
such as any high school
boy knows..
From (6) and (7) we get
ct
=0
x'
-
ct'
=
x'
-
ct'
-
x
and
0.
Therefore
where X
is
X(x
-
(8)
ct)
a constant.
Similarly, in the opposite direction,
+
x'
ct'
=
/x
H being another constant.
By adding and subtracting
we
x'
get:
and
ct'
where a
Let us
in
=
now
(X
(x
+
(9)
ct)
(8) and (9)
= ax - feet
= act-fex
-f-
ju)/2
and b
(10)
(11)
=
find the values of a
(X
and
terms of v
K
and /C
(the relative velocity of
the
of
light.
velocity
,
and c
45
),
fe
ju)/2.
This
done
is
the following
in
ingenious manner:*
From (10)
when x =
,
then x = bet/ a
=
but x'
And
x
(12)
/
at the point K'\
in this
case
that
is
K to
the distance from
K! ,
is,
the distance traversed/ in time
f
by K moving with velocity v
t
relatively to K.
Therefore x
=
Comparing
this
v
Let us
now
vt.
with (1 2),
=
K
(13)
consider the situation
/Cgetsx'
Hence
K and
K'.
first:
For the time
or
get
be/a.
from the points of view of
Take
we
-
t
,
ax (from (10)),
x
K says
=
x'/a.
(14)
that
11
*See Appendix in "Relativity by Einstein,
Pub. by Peter Smith, N. Y. (1931).
I
46
to get the "true" value,
;
K should
x,
by a/
divide his x
in particular,
ifx'-1,
K
says that
is
only 1/a of a "true"
K"s
unit of length
unit.
K,
But
att'
=
0,usins(11)
says
fex=act
(15)
and since from (10),
(1
5)
becomes
6x
ac(ax
62^2
"x
ax
ax
x')/6c
f
f
,
from which
x'
And
since 6/a
=
a(1
=
-
2
6 /a
y/c from
2
(16)
)x.
(1 3),
(16) becomes
x
f
=
a(1
-
vVc-)x.
other words,
K' says:
In order to get the "true" value,
(17)
In
K should
multiply his x
a(1
-
x',
by
2
v /c).
In particular,
ifx- 1,
K says
then
f('s
unit
that
2
is
really a(1
vr/c
Thus
47
) units
long.
each observer considers that
his own measurements
are the "true" ones,
and advises the other fellow
make
to
1
a "correction.*
And indeed,
although the two observers,
K and
/(',
1
may
express this "correction*
in different
forms,
still
the
MAGNITUDE
of the "correction"
recommended by each of them
MUST BE THE SAME,
since it is due in both cases
to the relative motion,
only that each observer attributes this
motion
to the other fellow.
Hence, from (14) and
1/a
Solving
this
-
a(1
equation
a
*Note
=
we may
(1 7)
for a
s
v /c
,
we
write
2
).*
get
= c/V?~-v*.
that this equation
NOT obtained
is
by
ALGEBRAIC SUBSTITUTION
from (14) and (17),
but is obtained by considering that
the CORRECTIONS advised by K and K'
in (14) and (17), respectively,
must be equal in magnitude
as pointed out above.
Thus in (14) K says:
"You must multiply your measurement by 1/a",
whereas in (1 7) K' says:
"You must multiply your measurement by
and since these correction factors
must be equal
48
a
(1
Note
is
that this value of a
the same as that of
/?
on
p. 1
1
.
Substituting in (10)
this value of a
and the value
we
x'
which
the
is
first
ay from (13),
0x - |8vf
/3(x-rt)
=
=
x'
or
=
fee
get
(18)
of the set of equations
on page 19!
of the Lorenti transformation
Furthermore,
from
(1
=
=
x
8) and /
\x'
we
get
=
=
ct'
or
t'
Or, since
t
=
ct
ct'
- vt)
- vt/c).
P(ct
/3(t
x/c,
t'
=
/3(t
-
2
vx/c ),
(19)
another of the equations
of the Lorentz transformation!
which
is
That the remaining two equations
= y and z'
z also hold,
y'
Einstein shows as follows:
=
K and
Let
/C'
each have a cylinder
of radius r, when at rest
relatively to each other,
and whose axes coincide with the
Now, unless / = y and z = z,
K and K would each claim that
X (X ) axis/
7
r
f
own
his
We
cylinder
is
OUTSIDE
the other fellow's!
thus see that
the Lorentz transformation was derived
by
Einstein
(quite independently of Lorentz),
as a set of empirical equations
NOT
49
devoid of physical meaning,
but, on the contrary,
as a result of
most rational change
a
in
our ideas regarding the measurement of
the fundamental quantities
length and time.
And
so, according to him,
of the equations of the
Lorentz transformation,
the
first
namely,
x'
is
-
-
/3(x
vt)
so written
NOT
because of any real shrinkage,
supposed,
but merely an apparent shrinkage,
as Lorentz
due
to the differences
And
in
made by
the measurements
K and
/C
*
(see p. 45).
Einstein writes
t'
NOT because
=
/3(t
it is
-
just a
2
vx/c
)
mathematical
WITHOUT any MEANING
trick
(see p. 19)
but again because
it is
the natural consequence of
in the measurements
the differences
of the
two observers.
And
that
each observer may think
he is right
and the oiher one
and yet
is
wrong/
each one,
by
using his
arrives at the
own measurements,
same form
*This shrinkage,
occurs only
in
it
will
be remembered,
the direction of motion (see p.
50
1 3).
when he expresses
as, for
K says
when
K
and
a physical fact,
example,
x
f
=
=
ct
ct' ,
says x'
they are really agreeing as to
the
of the propagation of light.
LAW
And similarly,
K writes any
if
and
other law of nature,
we apply
if
the Lorentz transformation
to this law,
in order to see
when
what form the law takes
expressed in terms of
7
the measurements made by /C ,
we
it is
find that
the law
is still
the same,
although it is now expressed
in terms of the primed coordinate system.
Hence
Einstein says that
although no one knows
what the "true" measurements should be,
yet,
each observer
may
use his
WITH EQUAL RIGHT
in
own measurements
AND EQUAL
SUCCESS
formulating
THE
LAWS OF NATURE,
or,
formulating the
of the universe,
the
which remain unchanged
quantities
namely,
in spite of the change in measurements
in
INVARIANTS
due
to the relative motion of
Thus,
we
can
now
K and K
appreciate
Einstein's Principle of Relativity:
"The laws by which
52
.
the states of physical systems
undergo change,
are not affected
whether these changes of
one
to the
of
two systems of coordinates
in
uniform translatory motion.
Perhaps some one will ask
is not the principle of
"But
and was
be
state
referred
or the other
"
relativity old/
not known long before Einstein?
it
Thus a person
in a train
into a station
moving
with uniform velocity
looks at another
and imagines
whereas
And
by
train
own
his
which
is
at rest,
that the other train
he cannot
is
is
moving
at rest.
find out his mistake
making observations within his train
since everything there
the same as
is
just
if
his train
Surely
it
would be
were really
at rest.
this fact,
and other
similar ones,
must have been observed
long before Einstein?"
In
other words,
RELATIVELY
to an observer
on the
train
everything seems to proceed in the same way
whether his system (i.e., his train)
is
at rest or in uniform*
motion,
and he would therefore be unable
*0f
course
if
the motion
is
not uniform,
but "jerky",
things on the train would jump around
and the observer on the train
would
certainly
that his
own
know
train
was not
at rest.
53
to detect the motion.
Yes, this certainly was known
long before Einstein.
Let us see what connection it has
with the principle of relativity
by him:
as stated
Referring to the diagram
we
on
36
p.
see that
a bullet fired from a train
has the
same velocity
RELATIVELY TO THE TRAIN
whether the latter is moving or not,
and therefore an observer on the train
could not detect the motion of the train
by making measurements on
the motion of the bullet.
This kind of relativity principle
is
the
in
one involved
the question on page 53,
and
WAS
known long before
Now Einstein
EXTENDED this
so that
it
Einstein.
principle
to
would apply
phenomena
electromagnetic
(light or radio waves).
Thus,
according to
this
extension of
the principle of relativity,
an observer cannot detect
his
motion through space
by making measurements on
the motion of
But
ELECTROMAGNETIC WAVES.
why should
be such
why had
a great
it
this
extension
achievement
not been suggested before?
54
BECAUSE
it
must be remembered that
see p. 39,
according to fact (2)
o
-
whereas,
the above-mentioned extension of
the principle of relativity
requires that EL should be equal to c
(compare the case of the bullet on p. 36).
In other words,
the extension of the principle of relativity
to electromagnetic
phenomena
seems to contradict fact (2)
and therefore could not have been made
before it was shown that
fundamental measurements are merely "local"
and hence the contradiction was
only apparent,
explained on p. 42;
so that the diagram shown above
must be interpreted
in the light of the discussion on p. 42.
as
Thus we see that
whereas the principle of relativity
as applied to MECHANICAL motion
(like that of the bullet)
was accepted long before Einstein,
the
SEEMINGLY IMPOSSIBLE EXTENSION
of the principle
to electromagnetic
phenomena
was accomplished by him.
55
This extension of the principle,
for the case in which
and K' move relatively to each other
with
velocity,
and which has been discussed here/
is called
the SPECIAL theory of relativity.
K
UNIFORM
We
see
shall
how
later
Einstein generalized this principle
STILL FURTHER,
to the case in which
K and
K?
that
is,
move
relatively to each other
ACCELERATION,
with an
CHANGING
a
And, by means
velocity.
of this generalization/
which he called
the
GENERAL
theory of relativity,
he derived
A NEW LAW OF
GRAVITATION/
much more adequate even
the
and
is
a
than
Newtonian
law,
of which the latter
first
approximation.
But before
we must
how the
we
first
can discuss
see
ideas which
this in detail
we have
already presented
were put into a
remarkable mathematical form
by a mathematician named Minkowski.
work
was essential to Einstein
This
in
as
the further development of his ideas/
we shall see.
56
THE FOUR-DIMENSIONAL SPACE-TIME
VII.
CONTINUUM.
We shall now see
how Minkowski* put
Einstein's results
remarkably neat mathematical form,
and how Einstein then utilized this
in a
the further application of his
Principle of Relativity,
in
which led to
The General Theory of Relativity,
resulting in a
NEW LAW OF GRAVITATION
and leading to further important consequences
and
discoveries.
NEW
It is
now
clear
from the Lorentz transformation (D.
1
9)
that
measurement, x ,
one coordinate system
depends upon BOTH x and
and that
a length
in
also
depends upon
Hence^
t'
BOTH
t in
another,
x and
t.
instead of regarding the universe
being made up of
as
Space, on the one hand,
and Time, quite independent of Space,
there
is
a closer connection
between Space and Time
than we had realized.
In
other words,
*See collection of papers mentioned
footnote on p. 5.
57
in
that the universe
is
NOT a
universe of points,
with time flowing along
irrespective of the points,
but rather,
this
is
A
UNIVERSE OF EVENTS,
-
everything that happens,
at a certain
happens
AND at a
place
certain time.
Thus, every event is characterized
by the PLACE and TIME of its occurrence.
Now,
since
its
may be
place
designated
three numbers,
by
namely,
the
By
(using
x, y, and z co-ordinates of the place
any convenient reference system),
and since the time of the event
needs only one number to characterize
we need in all
FOUR NUMBERS
TO CHARACTERIZE
just as
AN
EVENT,
we need
three numbers to characterize
a point in space.
Thus
we
we may
say that
live in a
four-dimensional world.
This
does
that
we
but
is
that
we
NOT
mean
live in four-dimensional
Space,
only another way of saying
live in
A WORLD OF
rather than of
and
it
EVENTS
POINTS only,
takes
58
it,
FOUR
numbers to designate
each significant element,
namely, each event.
Now
if
an event
is
designated
the four numbers
by
x, y, z,
f ,
in a
given coordinate system,
the Lorentz transformation (p. 19)
shows how to
find
the coordinates x',
of the same event,
in
y', z', t',
another coordinate system,
relatively to the first
moving
with uniform velocity.
11
In
studying "graphs
every high school freshman learns
how
to represent a point
coordinates, x and y,
using the Cartesian system of coordinates,
by two
that
is,
two
straight lines
perpendicular to each other.
Now, we may also use
another pair of perpendicular axes,
X' and Y' (in the figure on the next page),
as before,
,
having the same origin,
and designate the same point by x and
in this
When
new
y'
coordinate system.
the high school boy above-mentioned
goes to college,
and studies analytical geometry,
he then learns how to find
60
/ H
fl
.-.
't^-
\
\
\
J
6
\
the relationship between
the primed coordinates
and the original ones/
and
finds this to
x
anc
y
=
=
be expressed
x'cosfl
x'sinfl
-
+
as follows:*
/sine
)
/cos0 /
where 6 is the angle through which
the axes have been revolved,
as shown in the figure above.
The equations (20) remind one somewhat
of the Lorentz transformation (p. 19),
since the equations of
*Seep. 310.
61
(20)
the Lorentz transformation
also
show how
to
go
from one coordinate system to another.
Let us examine the similarity
between (20) and the
Lorentz transformation
a little
more closely,
selecting from the
Lorentz transformation
only those equations involving x and t ,
and disregarding those containing y and z,
since the latter remain unchanged
in going from one coordinate system to the other.
we
Thus
wish to compare (20) with:
/
t
Or,
if,
that
is,
x'
t'
-
|8(x
-
/3(t
for simplicity,
taking
the distance traveled
vt)
2
vx/c
).
we
take c
=
by
light in
one second,
as the unit of distance,
we may
we wish
say that
to
compare (20) with
x'
t'
Let us
first
= /3(x- vt))
= P(t- vx)f
solve (21) for
x and
t ,
so as to get them more nearly
in the form of (20).
By ordinary algebraic operations,*
*And temembering
we are taking c =
and
that
1,
that therefore
62
1,
we
get
x
and
t
Before
=
=
fat
j8(t'
+ vt')
+ vx') }
)
'
we 90 any further,
moment
let us linger a
and consider equations (22):
Whereas (21) represents /(speaking,
and saying to K
"Now you must divide x by /3,
'.
before you can get the relationship
between x and x' that you expect,
namely, equation (3) on p. 16;
in other words, your x' has shrunk
although you don't know it."
In
(22),
it is
K
f
speaking,
tells K the same thing,
namely that K must divide x by
and he
to get the "true"
which is equal to
x
x
1
/? ,
,
+
vt'.
Indeed,
quite in accord
with the discussion in Chapter VI.,
this is
in
which
it
was shown
that
each observer
gives the other one
precisely the same advice!
Note
that the
only difference
between (21) and (22)
+
in
is
that
y becomes
v
going from one to the other.
63
And this is again quite in accord
with our previous discussion
since each observer
believes himself to be at rest,
and the other fellow to be in motion,
only that one says:
"You have moved to the
whereas the other says:
"You have moved to the
right*'
(+
left*' (
v),
v).
Otherwise,
their claims are precisely identical;
and
this is exactly what
equations (21) and (22) show so clearly.
now
Let us
return to the
and (20):
Minkowski pointed out
comparison
of (22)
if,
in
t is
and
that
(22),
by
replaced
t'
by
IT
(where
/
= V
1 ),
IT',
then (22) becomes:
x
IT
=
=
/3(x'
=
-
frt'
/?(//
+ ivr')
+ VX')
or
/ X
\ IT
Or (by
I/V
+
+
multiplying the second equation
x
=
T
=
/3x'
/3r'
+
-
i/3vr'
fivx'.
Finally,
64
by
*-
i):
substituting* cos0 for
=
=
x
(
\ r
EXACTLY
]8
and
sin0 for
i
become
these equations
x'cosfl
x'sinfl
r'sinfl )
+
r'cosfl j
like (20)!
other words,
observes a certain event
In
K
ff
and
finds that
the four numbers necessary
to characterize it (see p. 58)
are
and
x, y, z
K
',
,
r
,
observing the
finds that in his
SAME
event,
system
the four numbers
f
are x',
y, i ', r
,
then the form (23)
of the Lorentz transformation
shows
to
that
go from one observer's coordinate system
to the other
it is
merely necessary
first coordinate system
through an angle 0, in the x , r plane,
without changing the origin,
to rotate the
*Since
is
greater than 1 (see p.
11)
must be an imaginary angle:
See
p.
25 of "Non-Euclidean Geometry/'
book by H. G. and L. R. Lieber.
another
Note
that sin
2
+
cos
=
2
1
holds for imaginary angles
as well as for real ones/
hence the above substitutions are legitimate,
thus
2
+
2
=
(-
2
i/5v)
-
= F -
v ),
1/(1
r being taken equal to
since
/3
V=
2
2
1
(see p. 62).
65
(1
-
v
2
)
-
thus:
=
= y and z
y
took (p. 65)
f
that
(remembering
And
since
we
p
and
ifiv
then
tan
That
z').
= cos e
= sin
=
iv.
is,
the magnitude of the angle
depends upon v
,
f
K and K
the relative velocity of
And
since, from (23),
2
/x
(
.
r
2
=
=
7
(x
2
)
7
(x
2
)
cos
-
2
+
2
sin
2xYsin0
2x r $infl
/
/
+ (rj
costf + (r')
cosfl
2
2
sin
cos
2
fl
then, obviously,
x
or (since
x
2
+
y
2
y
2
+r =
2
2
(x')
+
= /
and z
=
+
+
= (xT
z
2
r
2
(rj
7
z
66
),
+ (y? + (z ) +
7
2
(rj
Now,
it
will
be remembered
from Euclidean plane geometry,
that x*
+
2
y represents
the square of the distance
O
and
between
and similarly,
in
A
,
Euclidean three-dimensional space,
+
2
also represents
the square of the distance between two points.
)C
y
Thus, also,
x
2
+
2
y
+r+
2
r represents
1 *
the square of the "interval between two EVENTS,
in our four-dimensional world (see p. 58).
And,
just as in
plane geometry
the distance between two points
remains the same
whether
we
use
the primed or the unprimed
coordinate systems (see p. 61),
that
is,
*>
+
2
y
= (xj
67
+
(yj
(although x does
and y does
in
So,
NOT
NOT
equal
equal /).
x',
three dimensions,
x2
+
2
y
+
z
2
= (x7
+
(yj
+
(zj
and, similarly,
as we have seen on p. 66,
the "interval" between two events,
in our four-dimensional
space-time world of events,
remains the same,
no matter which of the two observers,
K or
/C',
measures
That
is
it.
to say,
although K and K'
do not agree on some things/
as, for
example/
and time measurements,
agree on other things:
their length
they
(1)
(2)
DO
The statement of their LAWS (see
The "interval" between events,
p,
51)
Etc.
In other words/
although length and time
are
no longer
INVARIANTS,
the Einstein theory,
other quantities,
in
like the
ARE
space-time interval between two events,
invariants
in this
theory.
These
invariants are the quantities
68
which have the
SAME
value
for all observers,*
and may therefore be regarded
as the realities of the universe.
Thus, from this point of view,
the things that we see or measure
NOT
are the realities,
since various observers
do not
of the
get the same measurements
same objects,
but rather
certain mathematical relationships
between the measurements
2
2
2
2
r )
z
(Like x
y
+
+
+
are the realities,
since they are the
same
for all observers.*
We
in
shall see,
discussing
The General Theory of
how
Relativity,
fruitful
Minkowski's view-point of a
four-dimensional Space-Time
World
proved to be.
VIII.
SOME CONSEQUENCES OF THE
THEORY OF
We
if
RELATIVITY.
have seen that
two observers,
K and K , move
f
relatively to each other
*Ail observers moving relatively to each other
with
UNIFORM
velocity (see p. 56).
69
with constant velocity,
measurements of length and time
their
are different;
and, on page 29,
we promised
In this
also to
show
measurements of mass are
that their
chapter
we
different.
shall discuss
mass measurements,
as well as other
measurements which
depend upon
these fundamental ones.
We
already
know
that
if
an object moves
direction parallel to
the relative motion of K and
in a
K
,
then the Lorentz transformation
gives the relationship
between the length and time measurements
of K and K'.
We
also
in a
direction
know
that
PERPENDICULAR
motion of K and K
to
f
the relative
there
is
NO difference
LENGTH
and,
in
the
measurements (See footnote on
p. 50),
in this case,
the relationship between the time measurements
may be found as follows:
For this
PERPENDICULAR
direction
Michelson argued that
the time would be
t2
= 2a/c
(seep. 12).
Now
is
this argument
supposed to be from the point of view
of an observer
DOES take
who
the motion into account,
70
and hence already contains
the "correction" factor /}/
hence,
replacing to by t'f
the expression t' = 2a/3/c
represents the time
in
the perpendicular direction
as
K tells
K SHOULD be written.
it
K
Whereas
,
in his
own
system,
would, of course, write
=
t
2a/c
for his "true" time,
t.
Therefore
t'
=
|8f
gives the relationship sought above,
from the point of view of K.
From
a
this
we
see that
body moving
with velocity u
PERPENDICULAR
appear to K and K to
in this
direction,
will
have
different velocities:
Thus,
Since u
=
where
and
c/
d/t and
=
u'
c/'/f'
represent
the distance traversed by the object
</'
measured by K and
and since J = </'
as
(there being
LENGTH
K
f
f
respectively;
NO difference
measurements
in
in this direction
see p. 70)
f
and
then
t
i/
= j8t , as shown above,
= c///3t = (1//3)u.
Similarly,
since a
=
u/t and a'
=
f
u /t
71
where a and
a' are the
accelerations of the
as
we
body,
K and
measured by
/(',
respectively,
find that
In like
manner
we may
find the relationships
between various quantities in the
primed and un primed systems of co-ordinates,
provided they depend upon
length
and time.
But, since there are
THREE
basic units in Physics
and since the Lorentz transformation
deals with only two of them, length and time,
the question now
how to get the
is
MASS
Einstein found
into the
game.
approach
that the best
to this difficult problem was via the
Conservation Laws of Classical Physics.
Then,
just as the
old concept of
the distance between two points
(three-dimensional)
was "stepped up" to the new one of
the interval between two events
(four-dimensional), (see p. 67)
so also the Conservation Laws
will have to be "stepped up" into
FOUR-DIMENSIONAL SPACE-TIME.
And, when
an amazing
this is
done
vista will
come
into view!
CONSERVATION LAWS OF
CLASSICAL PHYSICS:
this means that
no mass can be created or destroyed,
(1) Conservation of Mass:
72
but only transformed from one kind to another.
Thus, when a piece of wood
its mass is not destroyed, for
if
one weighs
all
is
burned,
the substances into which
transformed/ together with the ash
that remains, this total weight is the
it is
same
as the
weight of the original wood.
We
express this mathematically thus:
where 2 stands for the SUM, so that
ASm =
2m
TOTAL
is the
mass, and A, as usual,
stands for the "change", so that
= o says that the change in
ASm
mass
total
is
zero, which
Mass Conservation Law
is
the
in
very
convenient, brief, exact form!
(2) Conservation of Momentum: this says that
if there is an exchange of momentum
and velocity, mv)
between bodies, say, by collision, the
(the product of mass
TOTAL
is
momentum BEFORE
collision
SAME
the
as the
TOTAL
after collision:
A2mv =
o.
which means that
Energy cannot be created or destroyed, but
only transformed from one kind to another.
Thus, in a motor, electrical energy is
converted to mechanical energy, whereas
(3) Conservation of Energy:
the reverse change takes place.
both cases, we take into account
the part of the energy which is
in a
dynamo
And
if,
in
transformed into heat energy,
then the
energy
by
friction,
TOTAL
AFTER the transformation
SAME, thus: A2E = o.
BEFORE
is
the
Now,
a
and
moving body has
73
o
KINETIC energy,
expressible thus:
two moving, ELASTIC bodies collide,
there is no loss in kinetic energy of
the whole system, so that then we have
When
2
Conservation of Kinetic Energy: AS^mv
(a special case of the more general Law);
=
o
whereas, for inelastic collision, where
some
of the kinetic energy
changed
is
other forms, say heat, then AZ^r/nv
2
into
5^ o.
Are you wondering what is the use of all
Well, by means of these Laws, the most
this?
PRACTICAL problems can be solved,*
hence we must know what happens to them
in Relativity Physics!
You
(a)
will see that
they
will lead to:
NEW Conservation
Laws
for
Momentum and
INVARIANT
Energy, which are
under
the Lorentz transformation,
and which reduce, for small v, J
to the corresponding Classical Laws
(which shows why those Laws worked
so well for so long!)
(b) the
IDENTIFICATION
of
MASS and ENERGY!
Hence mass CAN be destroyed as such
and actually converted into energy!
Witness the
ATOMIC BOMB
(see p. 318).
See, for example, "Mechanics for Students of Physics and
Engineering'* by Crew and Smith, Macmillan Co.,
pp. 238-241
1 *
Remembering
c/V c
2
y
that the "correction
2
,
you
the velocity of!
=
see that,
when v
is
is
i
74
equal to
small relatively to
2
v negligible, then
grit, c, thus making
and hence no "correction* is necessary.
1
1
factor, ft,
Thus the Classical Mass Conservation Law
was only an approximation and becomes
merged
into the Conservation of
Energy Lawl
Even without following the mathematics of
the next few pages/ you can already
appreciate the revolutionary
IMPORTANCE
of
these results, and become imbued with
the greatest respect for the human
MIND
which can create
all this
PREDICT happenings
Here
Some
is
MAGIC
readers
for
may be
may
prefer to
after reading Part
II
previously unknown!
you!
able to understand
the following "stepping
others
and
up" process now,
come back
to
it
of this book:
The components of the velocity vector
in Classical
Physics, are:
cfx/c/t, c/y/c/t, c/z/c/t.
And,
these
if
we
replace x, y, z
become,
in
by xi, X2, xs,
modern compact notation:
0-1,2,3).
JxjJt
Similarly, the
m.c/x,/c/t
momentum components
(/= 1,2,3)
so that, for n objects,
the Classical Momentum Conservation
=o
But (24)
is
are:
NOT an
Law
(/= 1,2,3)
invariant
the Lorentz transformation;
75
under
is:
(24)
the corresponding vector which IS
so invariant is:
=o
0=1,2,3,4)
(25)
where s is the interval between two events.
*
and it can be easily shown that c/s = c/t//3,
being, as you know,
c/s
itself invariant
under the Lorentz transformation.
Thus,
in going from 3-dimensional space
-dimensional absolute time
and
1
(i.e.
from Classical Physics)
to 4-dimensional
SPACE-TIME,
we must
the independent variable
use
instead of
s for
t.
Now let us examine (25) which is so easily
obtained from (24) when we learn to speak the
NEW LANGUAGE OF
Consider
Then
is
only the
first
A{33m.c/xi/c/s}
NEW
the
=
o
(26)
Conservation Law,
holds whereas (24) does
makes
BECOMES
And
= 1,2,3)
(i
Momentum
since, for large v, it
and, for small v, which
(26)
SPACE-TIME!
3 components of (25):
first
(24), as
the
taking
now,
it
=
/3
and
1
c/s
NOT/
= c/f ,
should!
FOURTH
_
component
233)
of (25),
namely, m.cta/cfs or mc.dtjds (see p.
and
we
substituting
c/t//J
get mci8 which
-
mc/Vl
is
for </s,
mc.c/Vc
2
2
v /c or mc(1
2
-
2
v or
2
v
"*
2
(27)
.
/c )
Expanding, by the binomial theorem,
we
/,
Since
by
2
/c/0
,
= cVt2 -
2
c/s
dividing
(c/s
v
1
2
3 v
4
cp + j.^ + j-^ +
get
2
c/t
=
1
,
2
(c/x
+ c/y + c/z
2
and taking
-
c
=
2
v anc/
c/s/c/t
76
\
,
1
,
=
.
.
-j,
2
)
we
get
A/1
-
(see p. 233).
y
2
=
1 //S
2
which/ (or small v(neglecting terms after v ),
vr\
~
(1
+
1
And,
r
2
by c, we get me
multiplying
I.
(28)
2
+
^mv
2
.
Hence, approximately,
2
A{Z(roc
Now,
if
m
then A2/nc
which
is
is
2
+
2
|/nv )}
=
o.
(29)
constant, as for elastic collision,
2
o and therefore also A2(J?mv )
=
Law
the Classical
=
of the
Conservation of Kinetic Energy for
elastic collision (see p. 74);
thus (29) reduces to this Classical
for small v, as it should!
Law
we can also see from (29)
INELASTIC collision, for which
Furthermore,
for
A{2|mv
hence also
A2mc ^
2
c being a constant, c
2
}
^o
that
(see p. 74)
o or
2
A2m ^
o
which says that, for inelastic collision,
even when v is small,
any
loss in kinetic
energy
is
compensated
for
by an increase in mass (albeit small)
a new and startling consequence for
CLASSICAL
Physics itself!
this
from
viewpoint
Thus,
even in Classical Physics
NEW
the
Mass
of a
varies with
body
changes
is
in
NOT a
its
we
realize that
constant but
energy
amount of change in mass being
too small to be directly observed)!
(the
Taking now (27) instead of (28),
not be limited to small v/
77
we
shall
o
by
and, multiplying
c as before,
A{2mc /3j =
2
we
get
o for the
NEW
Conservation Law of Energy,
which/ together with (25), is invariant under the
Lorentz transformation, and which,
as
we saw above,
reduces to
the corresponding Classical Law, For small
Thus the
expression for the
NEW
of a
for
body
v
=
showing
o
is:
,
v.
ENERGY
f
=
2
/nc /3, which,
gives fo
= me
2
(30)
,
that
ENERGY
and
MASS are
one and the same entity
instead of being distinct, as previously thought!
Furthermore,
even a
SMALL MASS^m,
LARGE
amount of
equivalent to a
2
since the multiplying factor is c ,
is
ENERGY,
the square of the enormous velocity of light!
Thus even an atom is equivalent to
tremendous amount of energy.
Indeed, when a method was found (see p. 318)
of splitting an atom into two parts
and since the sum of these two masses is
a
less
than the mass of the original atom,
you can see from (30)
this loss in
a
terrific
that
mass must yield
amount
of energy
(even though this process does not transform
the entire mass of the original atom into energy).
Hence
the
Although
stunned us
ATOMIC BOMB!
this terrible
all
gadget has
into the realization
of the dangers in Science,
let us
not forget that
78
(p.
318)
the
is
POWER
behind
it
human
MIND
itself.
the
Let us therefore pursue our examination of
the consequences of Relativity,
the products of this
POWER!
REAL
In
1901 (before
Relativity),
Kaufman*, experimenting with
fast moving electrons,
found that
the apparent mass of a moving electron
is greater than that of one at rest
a result which seemed
very strange at the time!
Now, however, with the aid of (26)
we can see
that his result
is
perfectly intelligible,
and indeed accounts
Thus the use of
c/s
(where
=
c/s
for it quantitatively!
instead of c/t,
dtj(3) brings in
the necessary correction factor, j3, for large v
not via the mass but is inherent in our
NEW
in
RELATIVITY
which
velocity,
LANGUAGE,
c/x /c/s replaces
r
c/Xj/c/t,
the idea of
and makes
it
unnecessary and undesirable to think
mass depending upon velocity.
Many
c/s
by
writers
c(t/j8 in
correction
*
in
terms of
on Relativity replace
(26) and write it:
A{Sinj8.(/Xj/(/t}
Though
,
=
o,
putting the
on the m.
this of
course gives
Gcscll. Wiss. Gott. Nachr., Math.-Phys., 1901
p.
143, and 1902,
p.
291.
79
K1-2,
the same numerical result,
it is a concession to
CLASSICAL LANGUAGE,
and
He
Einstein himself
does not
rightly prefers that since
learning a
we should
like this.
we
are
NEW language (Relativity)
think directly in that language
translating each term
and not keep
into our old
before
We
we
CLASSICAL
LANGUAGE
11
"feel
meaning.
must learn to "feel" modern and talk modern.
its
Let us next examine
another consequence of
the Theory of Relativity:
When
radio waves are transmitted
1
through an "electromagnetic field/
an observer K may measure
the electric and magnetic forces
any point of the
at
field
at a given instant.
The
relationship
between
these electric and magnetic forces
is
expressed mathematically
by
the well-known
Maxwell equations
(see page 311).
Now,
if
moving
another observer, K'/
relatively to
K
with uniform velocity,
makes his own measurements
on the same phenomenon,
and, according to
the Principle of Relativity,
uses the
in his
same Maxwell equations
primed system,
80
it is quite easy to show* that
the electric force
is
NOT an INVARIANT
for the
and
two observers/
similarly
the magnetic force
is
also
NOT AN INVARIANT
although the relationship between
the electric and magnetic forces
expressed in the
MAXWELL EQUATIONS
has the same form for
both observers;
just as, on p. 68,
though x does
equal x'
f
and y does
equal y
NOT
NOT
still
the formula for
the square of the distance between two points
has the same form
in
both systems of coordinates.
we have seen that
SPECIAL Theory of
Thus
the
which
Relativity,
the subject of Part (see p. 56),
has accomplished the following:
is
I
revised the fundamental physical concepts.
(1)
It
(2)
By the addition
of
ONLY ONE NEW POSTULATE,
namely,
the extension of
the principle of relativity
*
See
the
Einstein's
first
paper (pp. 52
bo>k mentioned
in the
&
53)
in
footnote on p. 5.
81
ELECTROMAGNETIC
to
possible
the above-mentioned revision
by
see p. 55),
of fundamental units
it
phenomena*
made
(which extension was
explained
ISOLATED
many
experimental
results
which baffled the
pre-Einsteinian physicists:
for
As,
example,
the Michelson-Morley experiment,
Kaufman's experiments (p. 79),
and many others (p. 6).
It
(3)
led to the merging into
ONE LAW
of the two, formerly isolated, principles^
of the Conservation of Mass and
the Conservation of Energy.
In Part
we
II
see also how
SPECIAL Theory served
shall
the
the
as a
point for
starting
GENERAL THEORY,
*The reader may ask:
"Why
call this a postulate?
not based on Facts?"
The answer of course is that
(s it
a scientific postulate must
BASED
on
but it must 30
and hold also
facts that are
So
(a
be
facts,
further than the
known
facts
for
still
TO
BE discovered.
an ASSUMPTION
really only
most reasonable one, to be sure
that
since
it
it
is
agrees with facts
now known),
which becomes strengthened
continues to agree with
as they are discovered.
if it
in
the course of time
NEW
82
facts
which, again,
by means
ONE
of only
other assumption,
FURTHER
led to
NEW IMPORTANT
RESULTS,
which make the theory
the widest in scope
results
of any physical theory.
IX.
It is
A
POINT OF LOGIC
AND A SUMMARY
interesting here
to call attention to a logical point
which is made very clear
by
In
the Special Theory of Relativity.
order to do this effectively
let us first list
and number
certain statements, both old
to which
we
shall
then refer
and new,
by
NUMBER:
(1)
impossible for an observer
detect
his motion through space (p. 33).
to
(2)
The velocity of light is
independent of the motion of the source
(3)
It is
The old PRE-EINSTEINIAN postulate
and length measurements
that time
are absolute,
that
is,
are the
same
for all observers.
(4) Einstein's replacement of this postulate
by the operational
fact (see p.
that
time and length measurements
83
31)
(p. 34).
are
NOT absolute,
but relative to each observer.
(5) Einstein's Principle of Relativity (p. 52).
We
have seen that
(1)and(2)
are contradictory IF (3)
NOT
but are
(3)
is
is
retained
contradictory IF
replaced by (4).
(Ch. V.)
Hence
may
it
NOT
be
two statements
true to say that
MUST
be
NOT contradictory,
EITHER
contradictory or
without specifying the
ENVIRONMENT
Thus,
in the presence of (3)
(1) and (2)
ARE
contradictory,
whereas,
in the presence of (4),
the very same statements (1) and (2)
are
NOT contradictory.*
We
may now
briefly
summarize
the Special Theory of Relativity:
and (4)
are the fundamental ideas in
(1), (2)
it,
and,
since (1)
and (4)
are
embodied
in (5) 7
then (2) and (5) constitute
the BASIS of the theory.
Einstein gives these
as
two
POSTULATES
*Similarly
whether two statements are
EQUIVALENT or not
may also depend upon the environment
(ee p. 30 of "Non-Euclidean Geometry"
bv H. G. and L R. Lieber).
84
from which he then deduces
the Lorentz transformation (p. 49)
which gives the relationship
between the length and time measurements]
two observers moving relatively to each other
of
with uniform velocity,
and which shows
there
is
that
an intimate connection
between space and time.
This connection was then
EMPHASIZED
who showed
by Minkowslo,
that
may be regarded
r plane
from one set of rectangular axes to another
the Lorentz transformation
as a rotation in the
in a
x
,
four-dimensional space-time continuum
(see Chapter VII.).
fFor the relationships between
other measurements,
jee Chapter VIII.
86
THE
1.
MORAL
Local, "provincial" measurements
are not universal,
although they may be used
to obtain universal realities
if
compared with other systems
local
of
measurements taken from
a different viewpoint.
By examining
certain
RELATIONSHIPS BETWEEN
LOCAL MEASUREMENTS,
and finding those relationships which
remain unchanged in going from
one local system to another,
one may arrive at
the
2.
INVARIANTS
By emphasizing the
of our universe.
fact that
absolute space and time
are pure mental fictions,
and
that the
that
man can have
only
PRACTICAL
are obtainable only
the Einstein Theory
notions of time
by some method
shows
of signals,
that
'Idealism" alone,
11
is, "a priori
thinking alone,
cannot serve for exploring the universe.
that
On
the other hand,
since actual measurements
are local
and not
universal,
87
and
that
only certain
THEORETICAL RELATIONSHIPS
are universal,
the Einstein Theory shows also that
practical
is
measurement alone
also not sufficient
for
exploring the universe.
In short,
a judicious combination
Of THEORY and PRACTICE,
EACH GUIDING the other
a "dialectical materialism"
is our most effective weapon.
PART
II
THE GENERAL THEORY
A
I.
GUIDE FOR THE READER.
The
first three chapters of Part
the meaning of the term
''General Relativity/*
II
give
what it undertakes to do,
and what are its basic ideas.
These are easy reading and important.
II.
XV
introduce
Chapters XIII, XIV, and
the fundamental mathematical ideas
which
will
be needed
and important.
also easy reading
III.
XVI
Chapters
to
XXII build up
the actual
streamlined mathematical machinery
not difficult, but require
the kind of
care and patience and work
go with learning to
that
NEW
run any
The amazing
machine.
POWER
of this
TENSOR CALCULUS,
EASE with which
and the
new
it is
operated,
are a genuine delight!
IV.
Chapters XXIII to XXVIII show how
machine is used to derive the
this
NEW LAW OF GRAVITATION.
This law,
though
at
first
complicated
91
behind
is
its
REALLY
V.
seeming simplicity,
then
SIMPLIFIED.
Chapters XXIX to XXXIV constitute
THE PROOF OF THE PUDDING!
easy reading again
and show
what the machine has accomplished.
Then there are
a
SUMMARY
and
THE
MORAL
92
INTRODUCTION,
In Part
I,
on the SPECIAL Theory,
it was shown that
two observers who
are
moving
relatively to each other
UNIFORM
with
velocity
can formulate
the laws of the universe
EQUAL RIGHT AND
EQUAL SUCCESS,
"W!TH
11
even though
view
their points of
are different,
and their actual measurements
do not
agree.
The things that appear alike
them both
to
11
'TACTS of
INVARIANTS.
are the
the
the universe,
The mathematical relationships
which both agree on
are the
"LAWS
11
of the universe.
Since man does not
know
1
the "true laws of
God/
should any one human viewpoint
be singled out
as more correct than any other?
why
And
it
therefore
seems most
to call
fitting
THOSE
relationships
95
"THE
laws,"
which are
VALID
from
DIFFERENT
viewpoints,
taking into consideration
all experimental data
known up
Now,
to the present time.
must be emphasized
it
that in the Special
Theory,
only that change of viewpoint
was considered
which was due to
the relative
UNIFORM
velocity
of the different observers.
This was accomplished
by
Einstein
in his first
paper*
published
in
1905.
Subsequently, in 1916*,
he published a second paper
which
in
he
GENERALIZED
the idea
to include observers
moving
with a
(that
and
"the
It
relatively to each other
CHANGING
is,
that
with an
is
why
it is
GENERAL
was shown
velocity
ACCELERATION),
called
11
Theory of
in Part
Relativity.
I
that
to
make
possible
even the SPECIAL case considered there,
was not an easy task,
1
*See "The Principle of Relativity*
by A. Einstein and Others,
published by Methuen
&
Co., London.
96
for
it
required
a fundamental
to
in
change
Physics
remove the
APPARENT CONTRADICTION
between
certain
EXPERIMENTAL FACTS!
Namely,
OLD
the change from the
that TIME is absolute
(that
that
is,
the same (or
it is
to the
time
idea
is
NEW
all
observers)
idea that
measured
RELATIVELY
just as the
to an observer,
ordinary
space coordinates, x , y , z,
are measured relatively to
a particular set of axes.
SINGLE new
This
idea
was SUFFICIENT
to accomplish the task
undertaken
in
the Special Theory.
We
shall
now
see that
again
by the addition of
ONLY ONE
more
idea,
called
"THE PRINCIPLE OF EQUIVALENCE/
Einstein
the
made
possible
GENERAL
Theory.
Perhaps the reader
why
may
ask
the emphasis on the fact that
new idea
ONLY ONE
was added?
Are
not ideas good things?
97
1
And
not desirable
is it
many of them as possible?
To which the answer is that
to have as
the adequateness
of a
is
new
scientific
theory
judged
(a)
By
its
correctness, of course,
its
SIMPLICITY.
and
(b)
By
No doubt everyone appreciates
the need for correctness,
but perhaps
the lay reader may not realize
the great importance of
SIMPLICITY!
he
will say,
iJBut,"
"surely the Einstein Theory
is anything but simple!
Has
it
not the reputation
of being unintelligible
to all but a very few experts?"
Of course
"SIMPLE does
11
not necessarily mean
*
"simple to everyone/*
but only in the sense that
*lndeed,
it
everyone
can even be simple to
WHO
will take the trouble to learn
some
mathematics.
Though this mathematics
was DEVELOPED by experts,
it can be UNDERSTOOD
by
any earnest student.
Perhaps even the lay reader
will appreciate this
after reading this little
book.
98
if
all
known
physical (acts
are taken into consideration,
the Einstein Theory accounts for
number
a large
in
now
Let us
what
of these facts
SIMPLEST known way.
the
is
see
meant by
"The Principle
and what
it
of Equivalence/*
accomplishes.
It is impossible to refrain
from the temptation
to brag about
it
a bit
in anticipation!
And
to say that
by making the General Theory
possible,
Einstein derived
A NEW LAW OF
which
is
GRAVITATION
even more adequate than
the Newtonian one,
since it explains,
QUITE INCIDENTALLY,
experimental
which were
facts
left
unexplained
by the older
theory,
and which had troubled
the astronomers
for a
long time.
And, furthermore,
the General Theory
PREDICTED
NEW
FACTS,
which have since been verified
course
this is of
the supreme test of any theory.
But
to
let us
show
get to work
all this.
99
XL THE PRINCIPLE OF EQUIVALENCE.
Consider the following situation:
Suppose
that a
man, Mr. K,
lives in a
away
spacious box,
from the earth
and from
all
so that there
other bodies,
is no force of gravity
there.
And
suppose that
box and all its contents
are moving (in the direction
indicated in the drawing on
the
p.
100)
with a changing velocity,
32 ft. per second
every second.
Now Mr. K,
who cannot look outside of the box,
does not know all this;
but, being an intelligent man,
increasing
he proceeds to study the behavior
of things around him.
We
watch him from the outside,
but he cannot see
We
us.
notice that
he has
And
a tray in his hands.
of course
we know
that
the tray shares the motion of
everything in the box,
101
and therefore remains
him
relatively at rest to
namely, in his hands.
But he does not think of
this
it
in
way/
to him, everything
is
actually
at rest.
Suddenly he
30 the
lets
Now we know
tray.
that the tray will
move upward
continue to
with
CONSTANT velocity/*
we
and, since
also
know
is
moving upwards with
an
ACCELERATION,
we expect
will catch
and
hit
And,
that the
that very soon the
up with the tray
box
floor
it.
we
of course,
see
this
actually happen.
Mr.
K also
sees
but explains
it
it
happen,
differently,
he says that everything was
until he let go the tray,
still
and then the tray FELL and
hit
the floor;
and
K
"A
force of gravity.
attributes this to
11
Now K begins
He
11
to study this "force.
finds that there
is
an attraction
between every two bodies,
*Any moving
with
object
CONTINUES
CONSTANT speed
STRAIGHT LINE due
unless
some
it is
in a
to inertia,
stopped by
external force,
like friction, (or
example.
102
to
move
and
its
strength
proportional to
is
1
their "gravitational
and
masses/
varies inversely as the
square of the distance between them.
He also makes other experiments,
studying the behavior of bodies
pulled along a smooth table top,
and
bodies offer
finds that different
different degrees of resistance to
this pull,
and he concludes
is
that the resistance
proportional to the
"inertial
And
mass" of
a
body.
then he finds that
ANY
object which he releases
with the
acceleration,
SAME
FALLS
and therefore decides
that
the gravitational mass and
the inertial mass of a body
are proportional to each other.
other words, he explains the fact
fall with the
In
that all bodies
SAME
acceleration,
saying that the force of gravity
such that
by
is
the greater the resistance to motion
which
a
body
has,
the harder gravity pulls it,
and indeed this increased pull
is
supposed to be
JUST BIG
ENOUGH TO OVERCOME
the larger resistance,
and thus produce
THE SAME ACCELERATION
Now,
if
Mr.
K is
a
IN
very intelligent
103
ALL
BODIES!
Newtonian
physicist,
he says,
"How
strange that these
properties of a
two
distinct
body should
always be exactly proportional
to each other.
But experimental (acts show
accident to be true,
this
and experiments cannot be denied."
But
it
continues to worry him.
On the other hand,
K is an Einsteinian
if
relativist,
he reasons entirely differently:
"There is nothing absolute about
my way
Mr.
K
of looking at
phenomena.
1
outside,
,
(he means us),
may see this entire room moving
upward with an acceleration,
and
attribute all these
to this
happenings
motion
rather than to
a force of gravity
am doing.
His explanation and mine
as
I
are equally
good,
from our different viewpoints."
This
is
what Einstein called
the Principle of Equivalence.
Relativist
"let
me
K
continues:
try to
see things from
the viewpoint of
f
my good
though
He
I
neighbor, K ,
have never met him.
would
of course see
104
room come up and
the floor of this
ANY object which
hit
and
it
I
might release,
would therefore seem
ENTIRELY
NATURAL
to
him
for all objects released
from a given height
given time
at a
to reach the floor together,
which of course is actually the case.
Thus, instead of finding out by
EXPERIMENTATION
long and careful
that
the gravitational and inertial masses
are proportional,
my Newtonian
as
he would predict
that this
And
MUST
ancestors did/
A
PRIORI
be the
case.
so,
although the
facts are
explainable
in either
way,
K"s point of view
is
the simpler one,
and throws light on happenings which
could acquire only by
arduous experimentation,
if
were not a relativist and hence
I
I
quite accustomed to give
equal weight to
my
neighbor's viewpoint!
Of course as we have told the story,
we know that K is really right:
f
But remember that
in
the actual world
we do
We
not have this advantage:
cannot "know" which of the two
explanations
is
"really" correct.
105
But, since they are
we may
EQUIVALENT,
select the simpler one,
as Einstein did.
Thus we already see
an advantage in
Einstein's Principle of Equivalence.
And,
we
as
this
(or
is
it
said in Chapter X.
only the beginning,
led to his
new Law
of Gravitation which
RETAINED ALL THE MERITS OF
NEWTON'S LAW,
and
NEW
merits which
Newton's Law did not have.
has additional
As we
shall see,
106
XII.
A
NON-EUCLIDEAN WORLD.
Granting, then,
the Principle of Equivalence,
according to which Mr. K may replace
11
the idea of a "force of gravity by
a "fictitious force
due to motion/
1 *
the next question
"How
A
In
new Law
is:
help us to derive
11
of Gravitation?
this
answer to which
we
a
does
ask the reader to recall
few simple things which
he learned
in
elementary physics
in
high school:
11
*The idea of a "fictitious force
due to motion
is
familiar to
in
the following example:
Any
if
everyone
youngster knows that
a pail full of water
he swings
in a vertical
plane
WITH SUFFICIENT SPEED,
the water will not
even when the
out of the pail,
is
down!
actually upside
And
fall
pail
he knows that
the centrifugal "force"
is due to the motion
only,
since,
if
he slows down the motion,
the water
WILL
and give him
a
fall
out
good dousing.
107
If
at
it
on a moving object
an angle to this motion,
will change the course of the object,
a force acts
and we say
the
body
that
has acquired an
ACCELERATION,
even though its speed may have
remained unchanged!
This can best be seen with the aid of
the following diagram:
If
AB
represents the original velocity
magnitude and direction)
(both
in
and
the next second
if
the object
is
moving with
represented by AC
due to the fact that
some
a velocity
,
force (like the wind)
pulled it out of
then obviously
its
course,
108
BC
must be the velocity which
had to be "added" to
11
to give the "resultant
AB
AC,
as any aviator, or even
any high school boy, knows from
1
'
the "Parallelogram of forces.
BC
Thus
is
the difference between
the two velocities,
And,
>ACand AB.
since
ACCELERATION
defined as
is
the change in velocity, each second,
then BC is the acceleration,
even
if
equal
that
AB
and
AC happen
be
to
in length,
is,
even
if
the speed of the object
has remained unchanged;*
the very fact that
has merely
it
changed in DIRECTION
shows that there is an ACCELERATION!
Thus,
if
an object moves
in a circle/
with uniform speed,
it is moving with
an acceleration since
it is
Now
lives
is
its
always changing
direction.
imagine a physicist
a disc which
who
on
revolving with constant speed!
Being a physicist,
he is naturally curious about the world,
and wishes to study it,
even as you and I.
And, even though
we
tell
him
that
11
*This distinction
is
between "speed
discussed on page
1
28.
109
and "velocity"
he
moving with an acceleration
is
he, being a democrat and a relativist,
insists that
he can formulate
the laws of the universe
EQUAL RIGHT AND
EQUAL SUCCESS
"WITH
:
and therefore claims
he
is
but
is
that
not moving at all
merely in an environment
in
which
11
a "force of gravity
is
acting
(Have you ever been on
and actually
Let us
felt this
a revolving disc
11
"force ?!).
now watch him
tackle a problem:
We
see him
He
wants to
become interested
know whether
in circles:
the circumferences of two circles
same
are in the
He
draws two
a large
ratio as their radii.
circles,
one and
a small
one
(concentric with
the axis of revolution of the disc)
and proceeds to measure
and circumferences.
their radii
When
he measures the
larger circumference,
we know,
from a study of
the Special Theory of Relativity*
that he will get a different value
from the one
WE
should get
(not being on the revolving disc);
but this is not the case with
his
measurements of the
radii,
since the shrinkage in length,
described in the Special Theory,
"See Part
I
of this
book.
110
takes place only
IN THE DIRECTION OF
and not in a direction which
PERPENDICULAR
(as a radius
MOTION,
is
to the direction of motion
is).
Furthermore/ when he measures
the circumference of the small circle,
value
his
not very different from ours
is
since the speed of rotation
around a small
is
small
circle,
and the shrinkage
is
therefore
negligible.
And
he
so, finally,
it
turns out that
finds that the circumferences
NOT
are
Do we
him
that
is
and
he
On
is
wrong?
a fool for trying
is
to study Physics
Not
he
not according to Euclid?
that this
that
the same ratio as the radii!
in
tell
on
a revolving disc?
at all!
the contrary,
being modern
"That is quite
relativists,
all right,
we
say
neighbor,
you are probably no worse than we are,
you don't have to use Euclidean geometry
it
does not work on
for
now
a revolving disc,
there are
non-Euclidean geometries which are
exactly what you need
we would not expect
Plane Trigonometry to work on
Just as
a large portion of the earth's surface
for
which we need
Spherical Trigonometry,
in
which
the angle-sum of a triangle
isNOTlSO
,
111
if
we might
as
demand
naively
after
a high school course in
Euclidean plane geometry,
In short/
instead of considering the disc-world
as an accelerated system,
we
by
can,
the Principle of Equivalence,
regard
which
as a system in
it
11
a "force of gravity
is
acting,
and, from the above considerations,
we see that
in a
space having such a
gravitational field
Non-Euclidean geometry,
rather than Euclidean,
is
applicable.
We shall
how
now
illustrate
the geometry of
a surface or a space
may be
studied.
This will lead to
the mathematical consideration of
Einstein's
and
its
Law
of Gravitation
consequences.
XIII.
THE STUDY OF SPACES.
Let us consider
first
the familiar Euclidean plane.
Everyone knows
that
for a right triangle
on such
a plane
the Pythagorean theorem holds:
Namely,
113
that
2
s
Conversely,
it is
true that
IF the distance
on
is
between two points
a surface
given
by
2
s
(1)
=
*
2
+
y>
THEN
MUST BE
EUCLIDEAN PLANE,
the surface
A
Furthermore,
it is
obvious that
the distance from
ALONG
to
A
THE CURVE:
114
is
no longer
the hypotenuse of a right triangle,
and of course
CANNOT write
we
If,
2
here
s
=
x
2
however,
we
take two points,
A
and 8,
sufficiently near together,
the curve
AB
is
so nearly
a straight line,
that
we may
ABC as
in
actually regard
a little right triangle
which the Pythagorean theorem
does hold.
115
+
2
y
!
Only that here
we shall represent
by
as
c/s
c/x
,
done
is
and
its
three sides
c/y ,
in
the differential calculus,
to show that
the sides are small.
So
that here
we have
2
c/s
(2)
Which
and
is
still
=
2
</x
+
2
c/y
has the form of (1)
characteristic of
the Euclidean plane.
be found convenient
x and y
and
xi
xi
by
, respectively,
so that (2) may be written
It
will
to replace
2
c/s
(3)
Now
on
a
-
2
c/x L
+
c/xA
what is the corresponding situation
non-Euclidean surface,
such as,
the surface of a sphere, for
example?
Let us take
two points on
this surface,
A
and 8
designating the position of each
by
its
latitude
and longitude:
116
,
**"^^ \
TV
Let
Df be
the meridian
from which
longitude
is
measured
the Greenwich meridian.
And
and
let
DK be
a part of the equator,
the north pole.
Then the longitude and
latitude of
A
are, respectively,
the
number
the arcs
of degrees in
^F and AG
,
(or in the
corresponding central angles,
Similarly,
117
a and
0).
the longitude and latitude of
8
are, respectively,
the
number
the arcs
of degrees in
CF and BK.
The problem again
is
to find the distance
between
If
A
and
B.
the triangle /ABC
is
sufficiently small,
we may
en
consider
it
to lie
Euclidean plane which
practically coincides with
a
the surface of the sphere
in
this little
region,
and the sides of the triangle
to be straight lines
(as
on page
1 1
ABC
5).
Then,
since the angle at
is
C
a right angle,
we have
AB*
(4)
And now
what
if
this
let us
=
A? +
BC*-
see
expression becomes
we change
the Cartesian coordinates
in
(4)
tangent Euclidean plane)
to the coordinates known as
(in the
longitude and latitude
on the surface of the sphere.
Obviously
AB
has a perfectly definite length
irrespective of
118
c
which coordinate system we use;
but>4C and BC,
the Cartesian coordinates in
the tangent Euclidean plane
may be transformed into
longitude and latitude on
the surface of the sphere, thus:
let r
be
the radius of the latitude circle
FAC,
and R the radius of the sphere.
Then
Similarly
BC
=/?$.
Therefore, substituting
in (4),
we have
2
c/s
(5)
=
+
tJc?
And, replacing a by
may be written
xi
,
and
by
j8
this
2
c/s
(6)
A
=
i>c/x?
+
tfdxt
.
comparison of (6) and (3)
will
show
that
*any
that
an
school student knows
x represents the length of
is
the number of
and
high
if
arc,
radians in
And
it,
then
therefore
x
120
=
on the sphere,
the expression (or
is
as
2
c/s
not quite so simple
it was on the Euclidean plane.
The question naturally arises,
does this distinction between
a Euclidean and a non-Euclidean
surface
always hold,
and
is
this a
way
to distinguish
That
if
between them?
is,
we know
the algebraic expression which represents
the distance between two points
which actually holds
on a given surface,
we then immediately decide
whether the surface
can
is
Euclidean or not?
Or does
it
perhaps depend upon
the coordinate system used?
To answer
let us
this,
30 back to the Euclidean plane,
and use oblique coordinates
instead of the more familiar
rectangular ones
thus:
121
The coordinates of the point
now
xand y
are
represented
A
by
which are measured
parallel to the
X and Y axes,
and are now
NOT at right angles to
Can we now
each other.
find
the distance between
and
A
using these oblique coordinates?
Of
course
we
can,
for,
by the well-known
Law of Cosines in Trigonometry,
we
can represent
the length of a side of a triangle
122
lying opposite an obtuse angle,
by:
=
2
s
Or,
for a
2
x
+
2
2xy cos
y
a.
very small triangle,
2
</$
=
2
c/x
+
2
c/y
-2c/x</ycosa.
And, if we again
replace x and y
by
xi
this
and X2
respectively,
,
becomes
2
c/s
(7)
Here we see
even on
a
-
if
+
Jxl
that
2
c/s
it
was before.
we had used
polar coordinates
on a Euclidean plane,
we would have obtained
or
(8)
-
Euclidean plane,
the expression for
is not as simple as
And,
2
c/x
Js
2
=
2
c/x
+ x?
*
(See page
1
24)
123
2 c/xrc/X2-cosa.
The reader should
remembering that
verify this,
the polar coordinates of point
P
are
O
its distance, xi , from a fixed point,
,
makes with a fixed line
(2) the angle, x2 , which
(1)
OP
Then (8)
is
obvious from
the following figure:
o
124
OX.
Hence we see
that
the form of the expression for c/r
depends upon
(a)
BOTH
pF SURFACE
the
KIND
we
are dealing with,
and
(b) the particular
COORDINATE SYSTEM.
We
shall
soon see
that
whereas
mere superficial inspection
2
of the expression for </s
a
is
not sufficient
to determine the kind of surface
we are dealing with,
DEEPER examination
a
of this expression
help us to
DOES
For
this
know
this.
deeper examination
we must know
how,
from the expression for
the sVcalled
2
c/s ,
"CURVATURE TENSOR
of the surface.
And this brings us to
the study of tensors:
What
are tensors?
Of
what use are they?
and
are they used?
HOW
Let us see.
125
11
XIV.
The reader
is
WHAT
no doubt
1
scalar
is
A
TENSOR?
familiar
11
*
and "vector.
with the words "scalar
A
IS
which
a quantity
has magnitude only,
whereas
a
vector has
both magnitude and direction.
Thus,
if
we
say that
the temperature at a certain place
is
70
there
Fahrenheit,
is
NO
obviously
DIRECTION
to this temperature,
and hence
TEMPERATURE
is
a
SCALAR.
But
if we
say that
an airplane has gone
one hundred miles
east,
obviously its displacement
from its original position
is
a
VECTOR,
whose
MAGNITUDE
and whose
is
DIRECTION
100 miles,
EAST.
is
Similarly,
a
person's
AGE
is
a
SCALAR,
whereas
127
the
VELOCITY
a
VECTOR,*
is
with which an object
moves
and so on;
the reader can easily
find further
examples
and vectors.
of both scalars
We
shall
some
now
discuss
quantities
which come up in our experience
and which are
neither scalars nor vectors,
but which are called
TENSORS.
And,
when we have
we
a
and defined these,
illustrated
shall find that
SCALAR
is
a
TENSOR
a
TENSOR
whose
RANK
ZERO,
is
and
a
VECTOR
and we
a
is
shall
see what
whose RANK
meant by
is
is
ONE,
TENSOR of RANK TWO, or THREE, etc.
"TENSOR" is a more inclusive term,
Thus
*A
distinction
is
made between
often
""speed" and "velocity
the former
Thus when
is
a
we
SCALAR,
the latter a
are interested
HOW FAST a thing
is
and do not care about
ONLY
moving,
its
DIRECTION of motion,
we must then speak of its SPEED,
but if we are interested ALSO in
DIRECTION,
we must speak of its VELOCITY.
Thus the SPEED of an automobile
would be designated by
"Thirty miles an hour,"
but
its
VELOCITY would be
"Thirty miles an hour
EAST."
128
its
VECTOR.
in
of which
"SCALAR
11
and
"VECTOR
11
are
SPECIAL CASES.
Before
we
discuss
the physical meaning of
a tensor of rank two,
let us
consider
the following facts about vectors.
Suppose
that
we have
any vector, AB , in a plane,
and suppose that
we draw a pair of rectangular axes,
X and y,
thus:
T
B
X'
Drop
to the X-axis.
Then we may say
AC
is
BC
a perpendicular
8
from
and CB
that
X-component oF AB
the /-component of
the
is
,
for,
we know
as
from
the elementary law of
"The parallelogram of forces/*
if a force
acts on a particle
AC
and CB also
acts
on
the resultant effect
as that of a force
And
that
>AC and
AB
the
same
alone.
why
is
CB
it,
is
are called
the
"components"
Of
course
if
of
AB.
we had used
the dotted lines as axes instead,
the components of AB
would now be
AD and
DB.
other words,
the vector AB may be broken up
In
into
components
various ways,
in
depending upon our choice of axes.
Similarly,
if
we
use
THREE
rather than
we
two
axes
in
SPACE
in a
plane,
can break up a vector
into
THREE components
as
shown
in
the diagram
1 31 .
on page
130
D
By dropping the perpendicular BD
from 8 to the XY-plane,
and then drawing
the perpendiculars
X and
to the
we have
Y
DC and DE
axes, respectively,
the three components of
namely,
and, as before,
the components
depend upon
the particular choice of axes.
Let us
now
illustrate
the physical meaning
of a tensor of rank two.
Suppose we have
at
a rod
every point of which
there
due
As
is
to
a certain strain
some
force acting on
a rule the strain
131
it,
AB
L
is not the same at all points,
and, even at any given point,
the strain is not the same in
all
directions.*
Now,
(that
is
if
is,
the
STRESS
the
FORCE
at the point A
causing the strain at
represented
both
in
magnitude and direction
by/B
*When
an
breaks
object finally
under a sufficiently great strain,
it does not fly into bits
as
it
would do
if
the strain were the same
at all points
and
in all directions,
but breaks along certain lines
where, for one reason or another,
the strain is greatest.
132
A)
and
we
if
are interested to
the effect of
know
this force
upon
CDEF (through A} f
the surface
we
are obviously dealing
with a situation which depends
SINGLE
not on a
vector,
TWO vectors:
but on
Namely,
one vector,
AB ,
which represents the force
and another vector
it
(call
whose
question;
/AG),
direction will indicate
ORIENTATION
the
in
and whose magnitude
the AREA of CDEF.
of the surface
In other words,
the effect of a force upon a surface
on the force
depends
but
NOT ONLY
ALSO on the
size
and orientation of the
Now, how
can
we
CDEF,
will represent
itself
surface.
indicate
the orientation of a surface
by
If
a line?
we draw
in the
obviously
in
a line through
CDEF ,
we can draw
A
plane
many
this line
different directions,
and there
is
no way
of choosing one of these
to represent the orientation of this surface.
BUT,
if
we
take a line through A
to the plane
PERPENDICULAR
such a line
is
UNIQUE
133
CDEF,
and
CAN
therefore
be used
to specify the orientation
of the surface
CDEF.
we draw
a vector,
direction perpendicular to
and of such a length that
Hence,
if
in a
CDEF
it represents the magnitude of
the area of CDEF,
then obviously
this vector
AG
indicates clearly
both the SIZE and the
of the surface
ORIENTATION
CDEF.
Thus.
the
STRESS
at
A
upon the surface CDEF
depends upon the
AB and AG ,
and is called
TWO vectors,
a
TENSOR
Let us
now
of
RANK TWO.
convenient way
find a
of representing this tensor.
order to
do
And,
in
let us
consider the
upon
a small surface, c/S ,
represented
Now
if
OG,
in
so,
stress,
F,
the following figure
perpendicular to
ABC
the vector which represents
the size and orientation of ABC
is
then,
134
,
z
c
it is
quite easy to see (page
that the
X-component
of
1
36)
OG
represents in magnitude and direction
the projection
And
the
08C
of
ABC upon
similarly,
7 and
Z components
of
OG
represent the projections
0/C and 0/B
,
respectively.
135
the XZ-plane.
To show
both
in
that
OK represents OBC
magnitude and direction:
T
That
is
it
does so
in direction
obvious,
since
OK
is
perpendicular
As
regards the magnitude
we
must
now show
to
OBC
(see p.
1
34).
that
OK _ OBC
OG ABC'
(a)
Now OBC = ABC
ABC and OBC
x cos
of the dihedral angle
between
(since the area of the projection
of a given surface
is
equal to
ihe area of the given surface multiplied
136
by
the cosine of the dihedral angle
between the two
planes).
dihedral angle equals anale
and
since
are respectively
perpendicular to ABC and OBC ,
But
this
06
GO/C
OK
ZGOKisOK/OG.
and cos
Substitution of this in (a)
gives the required
OBC _ OK
OGABC
the force
Now,
if
which
is itself
acts
we
on
F,
a vector,
ABC ,
can examine
its
total effect
by considering
separately
the effects of its three components
fx i fy i
upon
EACH
and
fz
of the three projections
OBC,OAC*ndOAB.
Let us designate these projections
by dSx
/
dSy end dSz
is
the
,
respectively.
Now,
since fx
(which
acts
upon
X-component of F)
one of the three
EACH
above-mentioned projections,
let us
due
designate the pressure
to this
upon
component alone
the three projections
by
Pxx
/
Pxy
,
Pxz
/
respectively.
We
must emphasize
the significance of this notation:
In the first place,
137
the reader must distinguish between
the "pressure" on a surface
and the "force" acting on the
M
The p ressurel1 k
FORCE PER UNIT AREA.
the
So
that
TOTAL FORCE
MULTIPLYING
the
the
surface.
PRESSURE by
obtained by
is
AREA
the
of the surface.
Thus the product
'
PXX dSx
gives the force acting upon
the projection dS,
due to the action of f,
ALONE.
Note
the
DOUBLE
subscripts in
'
PXX
The
first
i
Pxy
i
PXZ
one obviously
refers to the fact
that
these three pressures
from the component
emanate
all
f.,
alone;
whereas,
the second subscript designates
the particular projection upon which
the pressure acts.
Thus
p/.v
means
the pressure due to fr
upon the projection dSy
,
Etc.
It
follows therefore that
fx
And,
=
pxx'dSx
+ Pxy'dSy + pxz'd
similarly,
~
>y
Pyx'dSx -f Pyy'dSy
+ Pyz'J
and
fx
=
'
PZX dSx
'
~\~
138
fzy dSy
+p
zz
dS
Hence
TOTAL
the
STRESS, F,
on the surface c/S,
F=(x+(y +
?,
or
i
=
+ Pyx
+ PZX
Thus
we
'
*
'
'
PXX cfix ~T Pxy
'
cr jj/
~r PXZ
+
uSx H~ Pyy uSy
p yz
dSx
PZU dSy ~\- p Z2
+
'
'
see that
not
just a vector,
with three components in
stress
is
three-dimensional space (see p.
but has NINE components
1
30)
THREE-dimensional space.
in
Such a quantity
A
is
called
TENSOR OF RANK TWO.
For the present
let this illustration of a
Later
It is
if
we
tensor suffice:
shall give a precise definition.
obvious that
we were
dealing with a plane
instead of with
three-dimensional space,
a tensor of rank
two would then have
FOUR
components instead of nine,
since each of the two vectors involved
has only two components in a plane,
and therefore,
there would now be only
only
2X2 components for the tensor
instead of 3
X
3 as above.
And, in general,
if we are dealing
with
n-dimensional space,
139
'
'
dSz
C/S2
C/Sz
.
two
a tensor of rank
has n components
which are therefore conveniently written
in a
as
SQUARE
array
was done on page
1
39.
Whereas,
in
n-dimensional space/
VECTOR
a
has only n components:
Thus,
VECTOR
a
has
in
in a
PLANE
TWO components/
THREE-dimensional space
THREE
it
has
components/
and so on.
Hence,
the components of a
VECTOR
are therefore written
in a
SINGLE ROW;
instead of in a
SQUARE ARRAY
TENSOR of RANK TWO.
as in the case of a
Similarly,
n-dimensional space
3
of rank THREE has n components,
and so on.
in
a
TENSOR
To sum
In
a
up:
n-dimensional space,
VECTOR has n components,
TENSOR of rank TWO has n components,
TENSOR of rank THREE has n components,
2
a
3
a
and so on.
The importance of
tensors
in Relativity
become clear
we 90 on.
will
as
140
XV. THE EFFECT
CHANGE
IN
ON TENSORS OF
THE COORDINATE
SYSTEM.
In Part
we had
I
of this
book (page 61)
occasion to mention
the fact that
the coordinates of the point
141
A
A
in the unprimed coordinate system
can be expressed in terms of
its coordinates in the
primed coordinate system
the relationships
by
=
=
(y
w
/
Ov
as
is
/
known
x
f
x co$0
y
x sin0 + / cos0
any young student
to
of
elementary analytical geometry.
now
see
what effect
this
Let us
change
in
the coordinate system
has
upon
a vector
and
its
components.
Call the vector cfs,
and
let o'x and c/y represent
components in the UNPRIMED SYSTEM,
and dx and c/y'
its
its
as
components in the PRIMED
shown on page 143.
142
SYSTEM
Obviously
is
c/s itself
not affected by the change
of coordinate system,
COMPONENTS
but the
of
</s
the two systems
in
are
as
DIFFERENT,
we have already pointed out
on page
Now
are
if
1
30.
the coordinates of point
x and y
in
one system
143
A
/
in the other,
x' and
the relationship between
these four quantities
and
is
by equations (9) on p. 142.
now, from these equations,
given
And
we
can,
by
differentiation*,
find the relationships
and c/y
and
7
c/x' and c/y
between
c/x
It
will
in
.
be noticed,
equations (9),
that
x depends upon BOTH x and y',
so that any changes in x' and y'
will
BOTH
Hence
namely
the
affect x.
TOTAL
change
in
x,
c/x ,
will
depend upon
(a)
Partially
namely
TWO causes:
upon the change
in x',
c/x' ,
and
(b) Partially
upon the change
in y',
7
namely
c/y
.
Before writing out these changes,
it will be found more convenient
7
to solve (9) for x and y
in terms of x and y.|
7
*See any book on
Differential Calculus.
fAssuming of course
that the
determinant of the coefficients
is not zero.
in (9)
(See the chapter on "Determinants"
"Higher Algebra" by M. Bocher.)
144
in
In
other words,
to express the
NEW,
in
primed coordinates, x' and /,
terms of the
OLD,
original ones,
rather than the other
x and y ,
way around.
This will of course give us
ax
=
ex
,
y
\
where a,
It
x' ==
<
N
(10)
'
will
J
c,
fc,
be even
+ 6y
+ dy
,
,
are functions of 0.
better
to write (1 0) in the form:
(11)
x
'
x2
=
621X1
using xi and XL instead of x and y ,
(and of course x[ and x'2 instead of
x'
and /);
and putting different subscripts
on the single letter a ,
instead of using
four different letters:
The advantage
not only that
easily
a
b
,
,
c
of this notation
we
,
d
is
can
GENERALIZE
to
n dimensions
from the above
two-dimensional statements,
but,
as
we
this
shall
see
later,
notation lends
a beautifully
itself
to
CONDENSED
writing equations,
which renders them
very
EASY
to
work with.
145
way
of
Let us
now proceed
with
the differentiation of (11):
we
get
=
=
k'
I
\
(1
* 2)
'
C/X2
MEANING
The
ail( 'Xl
"^ ai2C^X2
521C/X1
+
a 2 2C/X?
of the a's in (12)
should be clearly understood:
Thus an is
the change in
A
UNIT
x'\
due
to
CHANGE
in
xi,
so that
when it is multiplied by
the total change in xi , namely
we
get
THE
THE
And
612
c/xi ,
CHANGE
CHANGE
similarly in
IN
IN
DUE TO
*(
* ALONE.
a^cfa,
represents
the change in xi
and therefore
PER UNIT
the product of a\?
the total change
CHANGE
and
x / namely dx
in
,
gives
THE
THE
CHANGE
CHANGE
DUE TO
ALONE.
IN x{
IN x2
Thus
the
is
TOTAL CHANGE
given
in xi
by
just as
the total cost of
number of apples and oianges
would be found
a
by multiplying the
cost of
ONE APPLE
by
the total
number
of apples,
146
in
x2 ,
ADDING
and
to a similar
this result
one
for the oranges.
And
We
similarly for
may
Jx 2
in (1 2).
therefore
replace an
by dx(/dx\
symbol which represents
the partial change in xj
per unit change in xi*,
and is called
a
the "partial derivative of xi
with respect to xi ."
Similarly,
dx{
'
3l2
~
/
cl21
~
9X2
And we may
in
9x2
T~~
OXi
/
therefore rewrite (1 2)
the form
dx{
j
dx\
*^i H
dx{
i
f\
OXi
f\
0X2
(13)
</X2
2
=
9X2 ,
T-'CfXi
+
,
<?X2
T~
0x2
But perhaps the reader
is
getting a
and
is
what
little
tired of all this,
it
wondering
has to
do
with Relativity.
*Note
is
that a
PARTIAL
change
always denoted by the
i
A A
in contrast to
letter
"d"
MJM
d
which designates a
TOTAL
change
147
^22
_
9x5
"
^
0X2
c
To which we may give him
a partial answer
now
and hold out hope
of further information
in
the remaining chapters.
What we can
already say
since General Relativity
is
that
concerned with
is
finding the laws of the physical world
which hold good for ALL observers/
and since various observers
differ
from each other,
as physicists,
only
in that
they
use different coordinate systems,
we see then
that Relativity
is
concerned
with finding out those things
which remain
INVARIANT
under transformations of
coordinate systems.
Now,
as
a vector
we saw on page 143,
is
such an
INVARIANT/
and, similarly,
tensors in general
are such
INVARIANTS,
so that the business of the physicist
really
becomes
to find out
which physical quantities
are tensors,
and are therefore
1
the "facts of the universe/
since they hold good
for all observers.
*See
p.
96.
149
Besides,
we promised on page 125,
as
we
must explain the meaning of
"curvature tensor/
since it is this tensor
1
CHARACTERIZES
which
And
a space.
then
with the aid of the curvature tensor of
our four-dimensional world of events,*
we shall find out
how things move
in this world
what paths the planets take,
and in what path
a ray of light travels
as
passes near the sun,
it
and so on.
And
of course
these are
things which
all
can be
VERIFIED BY EXPERIMENT.
XVI.
A
Before
we go any
VERY HELPFUL SIMPLIFICATION
more
further
equations (13) on page
let us write
briefly
thus:
-**
(,4)
*FOUR-dimensional, since
each event
its
THREE
the
TIME
(see Part
is characterized by
space-coordinates and
of its occurrence
I.
of this book, page 58)
150
147
A careful study of (14) will show
(a) That
(although
it
equations
looks like only one),
we
since, as
its
TWO
(14) really contains
M
give
possible values,
1
and 2
,
we have
c/x[
and
on the
c/Xj
we
just as
did
in
left,
(13);
The symbol 2, means that
when the various values of <J ,
(b)
namely
and 2
1
,
are substituted for
(keeping the
cr
constant in any
the resulting two terms
must be
Thus, for
(1
\i
ADDED
/i
=
together.
and
1
one equation)
<r
=
1
2
,
,
4) becomes
it
dxi
=
dx(
,
-dxi
.
+
j
-dx2,
0x2
dxi
just like
dxi
the FIRST equation in (13),
and, similarly,
by
taking
ju
=
2
,
and again "summing on the <r's ,"
since that is what 2, tells us to do,
we
get
If
dx2
which
Thus
is
we
the
is
still
3X2 I
T-'cfxi
OXi
SECOND
all
of (13).
further abbreviation
introduced
by
5X 2 J
+ -OX
I
equation
see that
(14) includes
A
=
omitting
the symbol 2,
151
-ax2
t
2
in (13).
WITH THE UNDERSTANDING THAT
WHENEVER A SUBSCRIPT OCCURS TWICE
IN A SINGLE TERM
(as, for
in
it
a
member
be understood
will
),
is to be made
SUBSCRIPT.
Hence we may
write (14) as follows:
d
<Jx'
*"
(15);
U
=
dx
'*-<lx
*'
dx.
which
we
know
shall
that the presence of the
in
of (14)
that
SUMMATION
ON THAT
in
<r
example,
the right-hand
TWO </s
the term on the right,
means
And
that
2a
understood.
is
now, finally,
c/xi and cfe
since
are the
components of
UNPRIMED
let us
c/s in
the
system
represent them more briefly
A
]
and
by
A
2
respectively.
The reader must
these
with
NOT confuse
SUPERSCRIPTS
EXPONENTS -
is not the "square of" A
but the superscript serves merely
the same purpose as a
thus >4
2
,
SUBSCRIPT,
namely,
to distinguish the
components
from each other.
why we use
SUPERSCRIPTS instead
Just
will
appear
of subscripts
later (p. 1 72).
152
c
v
And
the components of
PRIMED
in the
now be
will
c/s
coordinate system
written
1
A' and A'\
Thus
(1
And
so,
if
5)
we have
that
1
a certain vector
A"
,
is,
a vector
A
becomes
and
whose components
A
in a certain
and
coordinate system,
we change
if
are
2
to
new coordinate system
a
in
accordance with
the transformation represented
then
(16)
the
in
tells us
what
the
be
same vector
will
components of
by (11) on page 145
this
new (PRIMED)
coordinate system.
Indeed, (1 5) or (16) represents
the change in the components
of a vector
NOT ONLY for the
but for
change given
ANY transformation
of coordinates:*
Thus
suppose x, are the coordinates of
one coordinate system,
a point in
*Except only that
the values of (x a ) and (x/) must
one-to-one correspondence.
154
be
in
in
(11),
,
and suppose
that
=
xj
x!2
ft
(xi
x2/ ....)
,
=
ft
=f>(xa )
etc.
Or, representing
this entire
set of
by
equations
Xp
Ifj.
\Xa)f
where the f s represent
any (unctions whatever,
then, obviously
'
-
^
~
HI
J
OXi
OXi
or, since
=
ft
x{
If
afl
JJ J'CTX2 ~r
T" ^
0X2
=
{
r-~
OXi
C/Xi
+
'
^
C/X2
+
0X2
Hence
gives the
manner of transformation
dxa
to
ANY other coordinate system
(see the only limitation
mentioned
in
the footnote on
54).
page
1
And
in fact
ANY set of quantities which
transforms according to (16)
DEFINED
TO
BE
A
is
VECTOR,
or rather,
A CONTRAVARIANT
the meaning of
.
.
.
-
.
.
.
,
etc.
of the vector
.
VECTOR -
"CONTRAVARIANT"
155
will
appear
later (p. 1 72).
The reader must not
forget that
whereas the separate components
the two coordinate systems are
in
different,
the vector
itself is
INVARIANT
an
under the
transformation of coordinates
(see page 143).
It
should be noted further that
(16) serves not only to represent
a two-dimensional vector,
but
may
represent
a three- or four- or
n-dimensional vector,
since
that
all
is
M and
Thus,
may
(r
if
=
JLI
we have
but
if
M
a
-
is
necessary
number
to indicate the
of values that
take.
1
,
2 and
<r
=
1
2
,
,
two-dimensional vector/
1 , 2 , 3 , and a = 1
2
,
,
3
,
(16) represents a 3-dimensional vector,
and so on.
For the case M
^
1
/
2
,
3 and
(7
= 1,2
(16) obviously represents
THREE EQUATIONS
the right-hand
each have
in
which
members
THREE
terms:
(JX[
3*2
/!
-A
l
+
ux-2
0x3
dXo
fa'
I*
-A-+ -
i
,
r
}
0x3
axi
156
5x 3
Similarly
we may now
give
the mathematical definition of
a tensor of rank two,*
or of
any other
rank.
Thus
a contravariant tensor of rank
two
defined as follows:
is
(17)
Here, since 7 and
in
it
5
TWICE
occur
the term on the right,
is
we
understood that
must
SUM
for these indices
over whatever range of values they have.
Thus if we are speaking of
THREE DIMENSIONAL SPACE,
we have 7 = 1 , 2 , 3 and
ALSO a = 1 , 2 , 3 , and
5
= 1
= 1
2
,
,
2
3.
,
,
3;
But
NO SUMMATION
on the a and
since neither of
TWICE
to
is
be performed
j8
them occurs
in a single
term/
so that
any
particular values of
cv
and p
ANY ONE
must be retained throughout
For example,
for the case
a
=
1 , /3
=
2
,
It will be remembered (see page 128)
that
a
VECTOR
is
a
TENSOR
of
RANK ONE.
157
equation.
(1 7) gives the
_
~
,,
/\
<9x[
'
V~~
equation:
dxs
~ All
'
r\
^
,
"I
dx[
" dxj
'
n
..
12
,
'
9X3 9xi
*.
AV
7 and
that
,
??!.
<5
have each taken on
their
THREE
possible values:
which resulted
NINE
1,2,3,
in
terms on the right/
whereas
a =
and
1
ft
=
2
have been retained throughout.
And now
there will
NINE
Thus
/3
the three values,
1,2,3,
be
such
(1
a and
since
may each have
EQUATIONS
in all.
7) represents
nine equations each containing
nine terms on the right,
we
if
are considering
three-dimensional space.
Obviously
(17)
for
two-dimensional space,
will represent
only four equations each containing
only four terms on the
right.
Whereas,
in
four dimensions,
as
we must have
in
Relativity*
*See the footnote on
p. 1 50.
158
dx
?.
9X3 9X3
9X3 9X2
be observed
.
9x2 9x3
dx dx
l.
3
Jl
__
9x2 9x2
+ ^.^./31 +
dx! 3*2
'
9xi 0x3
.
.
6x2 9xi
will
^
9xi 0x2
9xi 9xi
It
,
'
a
(17)
will represent
sixteen equations each containing
sixteen terms on the right.
And
;
in
general,
n-dimensional space,
in
RANK TWO,
a tensor of
defined by (17),
consists of
n equations, each containing
n terms
the right-hand member.
in
Similarly,
a contravariant tensor of
is
RANK THREE
defined by
A'-to-l^-^-A
vXp ox? dx
(18)
ff
and so on.
As
before,
the
number
of equations represented
and the number of terms on the
by (18)
right in each,
depend upon
the dimensionality of the space in question.
The reader can already appreciate somewhat
the remarkable brevity
of this notation,
But
when he
how
are
will
see
in
the next chapter
easily such sets of equations
MANIPULATED,
be
he
will
we
are sure of that.
really delighted,
159
OPERATIONS WITH TENSORS.
XVII.
For example,
take the vector (or tensor of rank one)
2
1
and
having the two components
A
A
in a
a
Af
plane,
with reference to a given set of axes.
And
a
let
6 be another
such vector.
Then, by adding the corresponding components
Aa
of
we
and B
a
,
obtain a quantity
also having
two components,
namely,
A +
1
l
B and
A +B
2
2
which may be represented by
and C
2
,
respectively.
Let us
now prove
that this quantity
is
also a vector:
Since A*
its
is
a vector,
law of transformation
(19)
/V
x
Similarly, for B"
X
(20)
B'
=
is:
M"(see
:
=
^--B\
ox,
Taking corresponding components,
160
we
get, in full:
lAli
*.
/I
^
AH
f\
i
0X2
OXi
and
B/l
=
^ Xl
.....
TT
Dl
D
~r
1
axi
The sum of these
^ Xl D2
-D .
0x2
gives:
Similarly,
6x2
dxi
Both these
results are
included
in:
Or
c^^-e.
dx
(21)
a
Thus
we
see that
the result
a
is
VECTOR
(see p. 155).
Similarly for tensors of
higher ranks.
Furthermore,
note that (21)
QUITE
may be obtained
MECHANICALLY
by adding (19) and (20)
AS
IF
each of these were
A SINGLE equation
containing only
SINGLE term on the right,
A
161
c
instead of
A SET OF EQUATIONS
EACH CONTAINING
SEVERAL TERMS
ON
THE RIGHT.
Thus the notation
AUTOMATICALLY
takes care that
the corresponding components
shall
be properly added.
is even more impressive
the case of multiplication.
This
in
Thus,
to multiply
A'
(22)
by
=
M'
^
""^^
(23)
we
x
(A, /.,,/*
= 1,2)
write the result immediately:
(24)
0-g.g.C*
(X //t/
To convince the reader
it is quite safe
to write the result so simply,
that
let us examine (24) carefully
and see whether it really represents
correctly
the result of multiplying (22)
by (23).
By "multiplying (22) by (23)"
we mean that
EACH equation of (22) is to be
multiplied
EACH
by
equation of (23)
163
,/3-1,2X
which
in
the
in
ordinary algebra.
way
in
would be done
this
Thus,
we
must
first
multiply
by
f \r
fj
j
We
y
8 et,
(25)
9xi 9xi
9x2 9xi
9xi 9xi
-
,
ox* 0x2
Similarly
we
shall set
more such equations/
whose left-hand members are,
three
respectively,
A'T,
A'-B'\ A"- B'
2
,
and whose right-hand members
resemble that of (25).
Now, we may
by
taking X
=
obtain (25) from (24)
1
=
,
1
/*
,
retaining these values throughout,
since no summation
[that
in
is,
is
neither X nor
indicated on X and
/x
is
repeated
any one term of (24)].
164
But since
a and
ft
OCCUR TWICE
each
the term on the right,
in
they must be allowed to take on
all
possible values, namely,
and
1
and 2
,
SUMMED,
thus obtaining (25),
we
except that
by
the simpler
Similarly,
replace A'B**
symbol C
a^
=
2
and summing on a and
/?
by
we
=
taking A
1 , /*
*.
in
(24),
as before,
obtain another of the equations
mentioned on page 164.
And
X
=
2
, /*
=
1
,
gives the third of these equations/
and finally A = 2 , M - 2
gives the fourth and
last.
Thus (24) actually does represent
COMPLETELY
the product of (22) and (23)!
Of
in
course,
three-dimensional space,
(22) and (23) would each represent
THREE
equations, instead of two,
each containing
THREE
terms on the right, instead of two;
and the product of (22) and (23)
*Note
that either
allows for
A" B
FOUR
or C<*
components:
Namely, AW or C
AW or C",
AW or C
and AW or C
11
,
21
,
22
.
And hence we may use
G* instead of A* V.
165
would then
NINE
consist of
equations, instead of four,
each containing
NINE
But
terms on the right, instead of four.
this result
is still
And,
by
represented
of course,
(24)!
four dimensions
in
(24) would represent
SIXTEEN
equations, and so on.
I
:
Thus the tensor notation enables us
to multiply
WHOLE
SETS
containing
MANY
as
EASILY
as
OF EQUATIONS
we
simple monomials
TERMS IN EACH,
multiply
in
elementary algebra!
Furthermore,
see from (24)
we
that
the
is
PRODUCT of two tensors
TENSOR (see page 157),
also a
and, specifically, that
the product of two tensors
each of
ONE,
RANK
gives a tensor of
RANK TWO.
In general,
if
two tensers of ranks
m
and n
.
respectively,
are multiplied together,
the result
a
is
TENSOR OF RANK m
+
n.
This process of multiplying tensors
is
called
OUTER
multiplication,
166
to distinguish
it
another process
from
known
as
INNER
multiplication
which is also important
in
Tensor Calculus,
and which we
XVIII.
But
first
describe later (page 183).
shall
A
PHYSICAL ILLUSTRATION.
let us discuss
a physical illustration of
ANOTHER KIND OF TENSOR,
A COVARIANT TENSOR:*
Consider an object whose density
is
different in different parts of the object.
B
A
^This
is
to
be distinguished from the
CONTRAVARIANT tensors
discussed on pages
1
55ff.
167
We
may
then speak of
A
the density at a particular point,
Now,
NOT a
.
obviously
is
density
directed quantity,
but a
SCALAR
And
since the density of the given object
(see page 127).
is not uniform throughout,
but varies from point to point,
it will
vary as we go from A to
So
that
if
we
B
.
by ^
designate
A,
the density at
then
^ and ^
.
--
-"--
dxi
dx2
represent, respectively,
the partial variation of
in
Thus, although
a
\l/
the xi and x> directions.
DIRECTED
^
NOT
quantity,
the
CHANGE
the
DIRECTION
and
itself is
in
IS therefore a
$
DOES
depend upon
DIRECTED
whose components
quantity,
are
9xi
Now
let us
see
what happens to
this
quantity
the coordinate system
We
is
when
changed (see page 149).
are seeking to express
-d\[/
r
9x1
Now
if
we have
d\l/
,
,
.
in
f
terms o!
d$
-
,
9xi
3x2
three variables,
say^^andz,
169
d\l/
,
9x2
.
such that y and z depend upon x
it is obvious that
the change
IF IT
in
,
z per unit change in
CANNOT
FOUND
BE
x,
DIRECTLY,
may be found by
multiplying
the change
change
in
x
the change in z per unit change
in
y,
in
y per
unit
by
or,
expressing this in symbols:
c/z
..
f
(26)
In
_
cfz
Jy
cfx~d[ydx-
our problem above,
the following similar situation:
we have
A change in x( will
BOTH
xi
and the
affect
and x2 (see
p.
145),
resulting changes in XL
and
X2
will affect 1/7
hence
Note
that here
on the
we
instead of only
ONE,
since the change
affects
have
TWO
terms
right
BOTH
and these
whereas
in
xi
as in (26),
in x{
and x2
in turn
BOTH
affect ty,
(26),
change in x affects y/
which in turn affects z ,
a
and
that
Note
is all
the curved
since
there was to
it.
also that
all
"d"
is
used throughout
the changes here
170
in
(27)
are
PARTIAL
changes
(see footnote on page 147).
And since ^ is influenced also
by
a
in
change
this influence
x2/
may be
similarly represented
by
And, as before,
we may combine (27) and (28)
by means of the abbreviated notation:
where the occurrence of a TWICE
in the single term on the right
indicates a summation on a ,
as usual.
And,
finally,
writing
A^
4
for the
J'in
represented
W
two components
C/ JC
\
and
>A ff
we may
(30)
for the two components,
v
I
/
write (29) as follows:
0*^ =
Al-jfrA.
CXM
If we now compare
we note a
(30) with (16)
VERY IMPORTANT DIFFERENCE,
namely,
that the coefficient
on the
the reciprocal of
the coefficient on the right
right in
(30)
is
in
171
(16),
NOT satisfy
so that (30) does
the definition of a vector given
But it will be remembered that
(16).
the definition of
is
(16)
in
A CONTRAVARIANT VECTOR
And
we
ONLY.
in
(30)
introduce to the reader
the mathematical definition of
A COVARIANT
Note
VECTOR.
that
to distinguish the two kinds of vectors,
it is customary to write the indices
as SUBscripts in the
and
As
as
one case
SUPERscripts
in
the other.*
before (page
1 56),
represent a vector in
any number of dimensions,
depending upon the range of values
(30)
may
given to
and
for
JJL
and
0"
,
ANY transformation
of coordinates.
Similarly,
A COVARIANT
is
defined
and so on,
TENSOR OF RANK
TWO
by
for higher ranks.
COMPARE
and
and
(31)
(17).
CONTRAST
carefully
^Observe that the SUBscripts are used
for the COvariant vectors,
in
which the
are in the
PRIMES
in
the coefficients
DENOMINATORS
(see (30), p. 171).
To remember this more easily
a young student suggests the slogan
"CO, LOW, PRIMES BELOW/
172
1
XIX.
MIXED TENSORS.
Addition of covariant vectors
performed in the same simple manner
is
as for contravariant vectors (see p.
Thus, the
SUM
1
of
A\
r-~7*.n a
ox\
and
Sx=
"
dx(
is
c=
x
3x
.
Also,
the operation defined on page
as
is
166
OUTER MULTIPLICATION
the same for
covariant tensors:
Thus, the
OUTER PRODUCT
A(
=
^-A a
and
is
dx a
Furthermore,
it is
a
a
also possible to multiply
COVARIANT tensor by
CONTRAVARIANT one,
thus,
173
of
61
)
V
OUTER PRODUCT of
the
A(
=
|J.
A.
and
is
(32)
C/5;:=
^'-ax"'
C'-
Comparison of (32) with (31) and (17)
shows that it is
NEITHER
NOR a
called
It is
A
a covariant
contra variant tensor.
MIXED TENSOR
More
the
of rank
TWO.
generally,
OUTER PRODUCT
of
and
is
rfr
That
if
-
***
*
is,
any two
tensors of ranks
m
and n,
respectively,
are multiplied together
so as to form their
OUTER PRODUCL
is a TENSOR
the result
of rank
175
m+
thus, the rank of (33)
and that of (34) is 2 ,
is
3
,
hence,
the rank of their outer product, (35),
5.
is
Furthermore,
suppose the tensor of rank
a
is
MIXED
m
tensor,
having mi indices of covariance*
and
m
indices of contravariancef
mi
m^ = m),
and suppose the tensor of rank n
+
(such that
m
has
indices of covariance*
and n 2 indices of
contravariance/f"
then,
their outer
product
will
be
MIXED
a
+m
tensor having
indices of covariance*
+
indices of contravariance.f
mi
and
m2
All
in
rt2
this has
already been illustrated
the special case given above:
Thus,
(33) has
ONE
index of covariance (7)
and (34) also has
index of covariance
ONE
(5),
therefore their outer product, (35),
has
indices of covariance (7,
TWO
and
<5);
similarly,
since (33) has
indices of contravariance (a/
and (34) has
TWO
*
SUBscripts.
t SUPERscripts.
176
/?)
ONE
index of contravariance
(K),
their outer product, (35),
has
THREE
We
indices of contravariance
(,
/3,
K) O
hope the reader appreciates
the fact that
although
it
takes
many words
to describe these processes
it
to
is
extremely
EASY
DO them
with the AID of the
TENSOR NOTATION.
Thus the outer product of
A*
is
simply
C$
and B75
!
Let us remind him, however, that
behind this notation,
the processes are really complicated:
Thus (33) represents
a whole SET of equations*
terms on the right
each having
MANY*
And
(34) also represents
a SET of equations!
each having
And
MANYj
terms on the right.
their outer product, (35),
obtained by
multiplying
is
*Namely, EIGHT
for
TWENTY-SEVEN
SIXTY-FOUR
two-dimensional space;
for three-dimensional,
for four-dimensional,
and so on.
tFour for two-dimensional space,
NINE
for three-dimensional space,
for four-dimensional space}
SIXTEEN
and so on.
177
EACH
EACH
equation of (33)
by
one of (34),
resulting in a SET of equations, (35),
containing
THIRTY-TWO
equations for
two-dimensional space,
TWO HUNDRED AND
FORTY-THREE
for
three-dimensional space,
ONE THOUSAND AND TWENTY-FOUR
for
four-dimensional space,
and so on.
And
all
with a
correspondingly large number of terms
on the right of each equation!
And
yet
11
"any child can operate
it
as easily as
pushing a button.
CONTRACTION
XX.
This powerful
AND
DIFFERENTIATION.
and
easily operated machine,
the
TENSOR CALCULUS,
was devised and perfected by
the mathematicians
Ricci
and Levi-Civita
about 1900,
and was known to very few people
in
until Einstein
Since then
it
made use of
become
widely known,
and we hope that
will
make
it
it.
has
this little
book
intelligible
even to laymen.
178
make
But what use did Einstein
What
is its
of it?
connection with Relativity?
We
are nearly ready to Fulfill
the promise made on page 1 25.
When we have explained
two more operations with tensors,
namely,
CONTRACTION and DIFFERENTIATION,
we shall be able to derive
the promised
CURVATURE TENSOR,
from which
Einstein's
is
Law
of Gravitation
obtained.
Consider the mixed tensor (33), p. 175:
suppose we replace in it
7 by a,
obtaining
^~f
f\<x
By
a
-\
dx a dx x dxM
the summation convention (p.
**"
1
52),
member is to be summed on
(36) now represents
the left-hand
so that
TWO
only
equations instead of eight,*
each of which contains
TWO terms
on the
left
instead of one;
furthermore,
on the RIGHT,
a occurs twice
we must sum on a
since
for
here,
each pair of values of v and X
Now,
*Secp. 177.
180
:
when
happens to have a value
v
DIFFERENT
from X,
then
dx, dx^
dx
v
__
f
dxa dxx
__
dxx
BECAUSE
the x's are
NOT functions of each
other
(but only of the x"s)
and therefore
there
NO variation of x
is
p
with respect to
a
DIFFERENT
namely xx
x,
Thus coefficients of
will all
and
be
will
.
^
when A
v
ZERO
make
BUT
When A =
A?
these terms drop out.
v.
then
dx,
dx
dxx dx'a
_
_
.
dx'a 3xx
dxl dx x
Thus (36) becomes
=
A'
(37)
which we must
sum on the right
for X and M
in
g-^
still
-
To make
all this
let us write
clearer,
out explicitly
the two equations represented
l
'\
!?
+ A'?
=
+ A'f =
l
|^
UXi
(A\
+
Al
by
l
)
(37):
^ + A?)
+ ^ (A? + Al
2
+
(Al
OX'2
d
f (A? +
UXi
181
^l
2
1
)
).
may be
Thus (37)
08)
written
more
briefly:
c-g-e
where
C'^/T + ^T,
C = A'\ + X?
2
2
and
=
C2
In
A[
422
2
other words,
by making one upper and one lower index
ALIKE
in
(33),
we have REDUCED
a tensor of rank
a tensor of rank
THREE
ONE.
to
The important thins to note
is
that this process of reduction
or
as
CONTRACTION,
is called/
leads again to
it
A
TENSOR,
and
it is
obvious that
every such contraction
the rank is reduced by TWO,
since for every such contraction
foi
two of the
partial derivatives in
the coefficient
cancel out (see page 181).
We
how
shall
see
important
Now,
we form
if
in
later
the
the
this
process ot contraction
OUTER PRODUCT
way already described
182
(p. 1
of
75)
is.
two
tensors,
and if the result
a mixed tensor,
is
then,
by
as
contracting this
mixed tensor
shown above,
we
get a tensor which
an
INNER PRODUCT
in
is
called
contrast to
OUTER PRODUCT.
their
Thus the
OUTER
is
now,
we
if
product of
A*
and B7
Ck
(see page
1
77);
in this result
replace
7 by
|8 ,
obtaining
CS, or
then
D
A*
Da
an
is
(see pages
INNER
1
and B\
must remind the reader that
if
y
=
uv
where // u, and v are variables,
then
dy
_
c/x
dv
c
j,
c/x
f?f*
c/x
Applying this principle to
the differentiation of
(39)
A* = ^'A',
OXff
*See any book on
differential calculus
183
to
product of
And now we come to
DIFFERENTIATION.
We
80
1
82),
with respect to
we
x,' ,
get:
Or, since
dA
dA* dxr
hence (40) becomes
( 41
^
\
=
dxT
From (41 ) we see that
if the second term on the right
were not present,
then (41) would represent
a
mixed tensor
And,
this
in certain
of rank two.
special cases,
second term does vanish,
so that
in
SUCH
cases,
differentiation of a tensor
leads to another tensor
whose rank
is
one more than
the rank of the given tensor.
Such a special case is the one
in
which the coefficients
dx.
in
(39)
are constants,
as in (13)
on page 147,
since the coefficients in (1 3)
are the same as those in (1 1) or (10);
184
and are therefore (unctions of 6 ,
6 being the angle through which
the axes were rotated (page 141),
and therefore a constant.
other words,
transformation of coordinates
In
when the
is of the simple type
described on page 141
,
then
ordinary differentiation of a tensor
leads to a tensor.
BUT, IN
GENERAL,
these coefficients are
and
IN
NOT constants,
so,
GENERAL
differentiation of a tensor
does
as
is
NOT
give a tensor
evident from (41).
BUT
there
is
a process called
COVARIANT DIFFERENTIATION
ALWAYS
which
and which we
leads to a tensor,
shall presently describe.
We
cannot emphasize too often
the
IMPORTANCE
any process which
leads to a tensor,
since tensors represent
of
the
"FACTS
11
of our universe
(see page 149).
And, besides,
we shall have to employ
COVARIANT DIFFERENTIATION
in deriving
185
t/
-
X
the long-promised
CURVATURE TENSOR
and
EINSTEIN'S
LAW OF GRAVITATION
XXI.
THE
To explain covariant
we must
first
LITTLE
g's.
differentiation
refer the reader
back
to chapter XIII;
which
in
was shown that
it
the distance between two points,
or, rather, the square of this distance,
2
namely, c/s ,
takes on various forms
depending upon
(a) the surface in question
and
(b) the coordinate system used.
But now,
with the aid of the remarkable notation
which
we
ALL
in
we have
since explained,
can include
the
these expressions for
SINGLE
expression
Jf = g^Jx^Jx9ff
(42)
and, indeed,
this holds
NOT ONLY
ANY
2
c/s
for
SURFACE,
but also for
any THREE-dimensional space,
or
FOUR-dimensional,
187
or, in general,
any n-dimensional space!*
Thus, to
show how (42)
represents
equation (3) on page 116,
we take M = 1 , 2 and ^ =
1,2,
obtaining
=
2
c/s
(43)
guc/xi-c/Xi
'
C/Xi
g2lC/X2
since the presence of
ju
and
+
+
gi>c/xrc/x2
'
g22C/X>
C/X2 /
P
TWICE
the term on the right
in
requires
Of
in
SUMMATION
may be
course (43)
2
c/s
(44)
=
(42)
on both
/*
and
p.f
written:
+
giic/x?
g, 2 c/xic/X2
+
and, comparing (44) with (3),
we
find that
the coefficients in (3)
have the particular values:
=
ffll
1
=
=
I ff!2
,
7
jjfel
^Except only at a so-called "singular point"
of a space/
that
is,
a point at
matter
is
which
actually located
In other words,
f42) holdr. for any region
Pee page 1 52.
AROUND
matter.
| Note that in c/xi (as well as in c/x:)
the upper
and
"2"
NOT a
since (44)
is
is
really an
exponent
SUPERSCRIPT
an
ordinary algebraic equation
and is
in the
NOT
ABBREVIATED TENSOR NOTATION.
188
22
=
on page 120,
Similarly, in (6)
=
gu
in
and,
=*
gn
(7) on
1, 912
r\
=
ffi2
=
g2 i
,
, 322
=
/?*
page 123,
^
= -cos a,
- cos
a, g2i
g2 2
=
1,
and so on.
Note
that gi2
and
And
indeed,
in
have
general
=
9?"
in
g?i
SAME value.
the
9w
(42) on page 187.
Of course, if, in (42),
we take /*- 1,2, 3 and ^=1,2, 3,
we shall get the value for c/s2
in a
THREE-dimensional space:
2
c/s
(45)
=
guc/xi
+
gi2C/xrc/x2
Thus,
+
gisc/xrc/xs
+
4- g>2]dx2-dxi
in particular,
Euclidean three-space,
using the common rectangular coordinates,
we now have:
for ordinary
gn
and
all
=
1
,
=
g22
1
/
gas
=
1 /
the other g's are zero,
so that (42) becomes,
for
THIS
PARTICULAR CASE,
the familiar expression
2
c/s
2
c/s
and
=
c/x?
=
c/x
2
+
c/xl
+
c/y
2
similarly for
higher dimensions.
189
+
c/xi
+
c/z
2
/
/
Thus/ for a given space/
two-, three-, four-, or n-dimensional,
for a given set of coordinates,
we get a certain set of g's.
and
easy to show* that
It is
any such
(which
is
set of g's,
represented
by gv)
constitutes
COMPONENTS
the
and;
of a
TENSOR,
in fact, that
COVARIANT TENSOR OF RANK TWO,
and hence
is appropriately
designated with
SUBscriptsf:
TWO
Let us
now
briefly
the story so
sum up
far:
By introducing
the Principle of Equivalence
Einstein replaced the idea of
M
a
force of gravity"
by the concept of
a geometrical space (Chap. XII).
And
is
the
is
since a space
characterized
knowledge
by
its
g's,
of the g's of a space
essential to a study of
how
things
and hence
move
in
the space,
essential
to an understanding of
Einstein's
Law
of Gravitation.
*Seep.313.
tSeep.172.
190
OUR LAST DETOUR
XXII.
As we
said before (page 185),
to derive the
Einstein Law of Gravitation,
we must employ
COVARIANT DIFFERENTIATION.
COVARIANT DERIVATIVE
the
Now,
known
contains certain quantities
as
CHRISTOFFEL SYMBOLS*
which are functions of the tensor g^
discussed
in
chapter XXI,
and also of another
set
g^
(note the SUPERscripts here)
we
which
shall
now
describe:
For simplicity,
ourselves for the
moment
let us limit
to
TWO-dimensional space,
that
is,
let us
take
/x
=
1
,
2 and v
=
1
,
2
;
then gMV will have
FOUR
components,
namely,
the four coefficients on the right
in
(44).
And
in a
let us
arrange these coefficients
SQUARE ARRAY,
thus:
ff21
which
Now
is
called a
since gt2
*Named
for the
=
ff22
I
MATRIX.
921
(see page
1
89)
mathematician, Christoffcl.
191
of a terror
this
is
called a
SYMMETRIC MATRIX,
since
it
is
symmetric with respect to
the principal diagonal
(that
IF
is,
the
one which
starts
the upper left-hand corner).
in
we now
replace the double bars
on each side of the matrix
by SINGLE bars,
shown in the following:
as
we
a
get what
is
known
as
DETERMINANT.*
The reader must
carefully
DISTINGUISH between
*The reader probably knows that
a square array of numbers
with single bars on each side
is
5
6
2
3
called a determinant,
that its value is found thus:
and
5X3-6X2 = 15-12Or, more generally,
at =*J
I
I
c
A determinant does not necessarily
have to have
but
TWO
rows and columns,
n columns,
may have n rows and
and is then said to be of order n
The way to find the VALUE of
a determinant of the nth order
is
described
in
any book on
college algebra.
193
.
a square array with
SINGLE
bars
DOUBLE bars:
FORMER a DETERMINANT
from one with
The
and has
is
a
SINGLE
VALUE
found by combining the "elements"
in a certain
way
mentioned
Whereas
as
the
is
in
the foot-note on p. 193.
DOUBLE-barred
a set of
array
SEPARATE
"elements,"
NOT to
be COMBINED
They may be just
in
any way.
the coefficients of the separate terms
on the right in (44),
which,
as
we mentioned on page
are the separate
1
90,
COMPONENTS
of a tensor.
The determinant on page 193
may be designated more briefly by
!
or,
still
g.
I
better, simply
And now
let us
Oi
,
by
form a
=
1
,
2
/
v
=
1
,
2)
g.
new
the following manner:
DIVIDE the
square array
in
COFACTOR*
of
EACH ELEMENT
on page 193
by the value of the whole determinant,
of the determinant
namely, by g,
thus obtaining the corresponding element of
the
NEW array.
*For readers unfamiliar with determinants
this term is explained on p. 195.
194
The
COFACTOR
of a given element
of a determinant
is
found by
out
striking
the row and column containing the given element,
and evaluating the
determinant which
is left
over,
or
prefixing the sign
according to a certain rule:
+
Thus,
in
the determinant
523
4
6
1
8
/
the cofactor of the element 5,
1X7-8X0 = 7-0 = 7.
=
1
8
7
Similarly, the cofactor of
2
3
8
7
is:
4
is:
= -(14 -24) = 10;
and so on.
Note
that in the
first
case
+
we
prefixed the sign
,
while in the second case
we
prefixed a
The
rule
prefix a
the
.
is:
+
or
according as
NUMBER
from the
(that
is,
go
element
first
the
of steps required to
one
in
the upper left-hand corner)
to the given element,
is
EVEN
ODD,
or
respectively/
go from "5" to "4"
it takes one
step,
hence the cofactor of "4" must have
thus to
a
MINUS
But
prefixed before
2
3
8
7
all this is
explained
in
'
more thoroughly
any book on
college algebra.
195
Let us
now 90 back
to the array described
at the
bottom of
p.
194.
new array,
we shall designate by
This
p
which
g"
can also be shown to be
TENSOR,
a
and,
this
time,
A CONTRAVARIANT
TENSOR
OF RANK TWO.
That
it is
also
SYMMETRIC
can easily be shown by the reader.
We
can
now
give the definition
of the Christoffel
which
is
It
and
In
designated
is
symbol
we need.
a
symbol
by
{^v, \]
for:
other words,
the above-mentioned Christoffel symbol
involves partial derivatives of
the coefficients in (44),
combined
as
shown
in
(46)
and multiplied by
the
components
Thus,
in
of the tensor
g^
.
two-dimensional space,
*There arc other Christoffel symbols,
but we promised the reader
to introduce only the
barest minimum of mathematics
necessary for our purpose!
196
:
since
we
M
v
,
X
,
a each have
,
the values
1 ,
2
have, for example,
(11
*
I
I
-
1){
a 11
I
/
2
<J
I
9n
4-
1-1
dxi
i
\dxi
-
9n
^T
dxi
12
2
and
g
similarly (or the remaining
SEVEN
values of
obtained by allowing M
to take
Note
on
their
,
v
and X
two values
for each.
that in evaluating
above,
{11 ; 1
on the a,
take on BOTH values, 1,2,
}
SUMMED
we
allowing
a
to
BECAUSE
if
(46) were multiplied out,
would contain
EACH TERM
and
a TWICE,
this calls for
SUMMATION
Now
that
on the a (see page
we know
1
the meaning of
the 3-index Christoffel symbol
X},
we
are ready to define
the covariant derivative of a tensor,
from which it is only a step to
the
If
its
new Law
Ar
is
of Gravitation.
a covariant tensor of rank
with respect to
is
one,
COVARIANT DERIVATIVE
DEFINED
(47)
xr
as:
dA.
.
.
}
^-{r,aM..
197
52).
,
can be shown to be a
It
in fact/
it is
TENSOR
a
COVARIANT TENSOR OF RANK TWO*
and may therefore be designated by
A,T
.
Similarly,
we have
if
a contravariant tensor of rank one,
represented
its
by
Aa ,
COVARIANT DERIVATIVE
with respect to x,
is
<
the
TENSOR:
48 >
K.
+ln
.
r]
*.
Or,
starting with tensors of rank
we have
TWO,
the following three cases:
(a) starting with the
CONTRAVARIANT tensor, 4",
we get the COVARIANT DERIVATIVE:
(b) from the MIXED tensor,
we get the
A:,
COVARIANT
',*See p.
60
DERIVATIVE:
of
1
"The Mathematical Theory of Relativity/ by
A.
S.
the
1930
Eddington,
Edition.
198
COVARIANT tensor, A. ,
COVARIANT DERIVATIVE:
(c) from the
we
=
Arrp
And
T
set the
AA
-
{*P/
-jT^
UXp
Ar
*}
-
\Tp / *}
A,
.
similarly (or the
COVARIANT DERIVATIVES
of tensors of higher ranks.
Note that IN ALL CASES
COVARIANT DIFFERENTIATION
OF A TENSOR
leads to a
TENSOR
having
ONE MORE UNIT OF
COVARIANT CHARACTER
than the given tensor.
Of
course since
the covariant derivative of a tensor
is itself
a tensor,
we may
find
ITS covariant derivative
which
is
then the
SECOND COVARIANT DERIVATIVE
the original tensor,
and so on for
higher covariant derivatives,
Note also that
when the g's happen
as, for
in
to
be
constants,
example,
the case of a Euclidean plane,
using rectangular coordinates,
in
which case
we have
(see p.
1
88)
so that
=
flfll
I
/
l2
=
, ff21
199
=
,
22
of
all
constants,
then obviously
the Christoffel symbols here
are
ZERO,
all
since the derivative of a constant
is
zero,
and every term of the
Christoffel
symbol
has such a derivative as a factor/
so that (47)
That
is,
becomes simply
~~
.
in this case,
the covariant derivative
becomes
simply the ordinary derivative.
But of course
this
is
NOT so
XXIII.
IN
GENERAL
THE CURVATURE TENSOR AT LAST.
Having now built up the necessary machinery,
the reader will have no trouble
in following the derivation of
the new Law of Gravitation.
Starting with the tensor,
form
its
with respect to x r
(49)
A,
,
covariant derivative
:
A =
d
- K!A.(seep.197).
*See page 196.
200
Now form the covariant
AT (see page 1 99)
derivative
of
with respect to
A
(50)
xp
:
=
d
!<rp, e)
rp
A, -
[rp,
ej
obtaining
a
SECOND
which
is
covariant derivative of
Aa
,
a
COVARIANT TENSOR
OF RANK THREE.
Substituting (49) in (50),
we
get
A
A,,,
=
l/
^ff
T.-*-
~
)
(
(
ffr'/
9A
i
^^a
^7"
UJ\p
- AA
^
i
<
{"/
)
"I
v/JCp
,
dA,
or
o
202
4
A,.
If
we had
the
in
taken these derivatives
REVERSE
order,
namely,
FIRST with respect to xp
and THEN with respect to xr
we would
,
of course have obtained
the following result instead:
|f
M<rT
+
{<7T,e}
-
f
)
a
is
}
A
ax~
{tp,a} A.
Vf\a
{pT'tlfc
which
3
)
'
+,f!PT /
)
(
)
l
(T6 /
a
)
l
l
A,
again
COVARIANT TENSOR OF RANK THREE
Now,
comparing (51) with (52)
we
shall find that
they are
NOT alike THROUGHOUT:
Only SOME of the terms are the
SAME in both,
but the remaining terms are different.
Let us see:
the FIRST term
2
3 Ar
in
d~A
each
j
,- ,
V- ^-~ and T
dx dx/
dxTdxp
is:
ff
,
respectively.
p
These,
by ordinary
calculus,
203
ARE the same.*
SECOND term
The
is
of (51)
the same as
the
FOURTH
term of (52)
since the occurrence of
TWICE
implies a
a
(or e)
the same term
in
SUMMATION
and it is therefore immaterial
what letter is used (a or e) f
!
Similarly for
the
FOURTH
term of (51)
and the SECOND of (52).
The SIXTH term (and the SEVENTH)
is the same in both
since the reversal of r and p in
*For, suppose that z
as, for
Then
=
.
2x
+
=
2
a (unction of
x
example, z
v
v"
dx-dy
and
is
2y
2
+
,
2xy,
(treating
x
(treating
x and /
y as constant)
as constant).
if we reverse the order of differentiation,
finding FIRST the derivative with respect to y
with respect to x ,
and
And,
THEN
we would
get
= 2x
Y
(treating
x
as constant)
\0
and
7,
2
dydx
the
SAME FINAL
And
t
An
(treating
this is true
IN
index which
y as constant)
result.
GENERAL.
is
thus easily replaceable
"dummy"!
204
is
called a
does not
alter the value of
this Christoffel
symbol:
This can easily be seen by referring
to the definition of this symbol;*
and remembering that the tensor gM ,
is
SYMMETRIC,
that
is,
ft,
g,M (see
page
Similarly the last term
in
is
1
89).
the same
both (51) and (52).
But the
are
THIRD and FIFTH
NOT equal
Hence by
to
terms of (51)
any of the terms
we
subtraction
in
(52).
get
\
}
{^
f
}
A*
Aa
oxp
{
or
(53)
And
since addition (or subtraction)
of tensors
gives a result which is itself a tensor (see page 161)
the left-hand member of (53) is
A COVARIANT
TENSOR OF RANK THREE,
hence of course the right-hand member
is
also such a tensor.
But,
now,
since
An
is
an arbitrary covariant vector,
*Seepa 9 e196.
205
its
coefficient;
namely; the quantity
must also be a tensor
in
square brackets,
according to the theorem on p. 31 2.
Furthermore,
this
bracketed expression
must be a
it
MIXED
tensor of
RANK FOUR,
on inner multiplication by A.
since
must give a result which
of rank
is
THREE;
and indeed
must be of the form
it
(see page 31 3).
This
AT LAST
is
the long-promised
CURVATURE TENSOR
and
known
is
( P a 8 e187),
as
THE RIEMANN-CHRISTOFFEL TENSOR.
Let us examine
we may
so that
its
it
carefully
appreciate
meaning and value.
XXIV.
OF WHAT USE
IS
THE CURVATURE
TENSOR?
In
the
first
place
we must remember
it
is
the expression
in
in
if
that
an abbreviated notation for
in
square brackets
(53) on page 205;
which,
we
substitute for the Christoffel symbols,
206
and so on,
accordance with
the definition on page 196,
(crp, e}
their values in
we
find that
we have
an expression containing
First
and second
of the g's
partial derivatives
,
which are themselves the coefficients
the expression for
in
How many
2
c/s
(see p.
1
87)
components does
the Riemann-Christoffel tensor have?
Obviously
that
depends upon the
dimensionality of the space
under consideration.
if we are
studying
two-dimensional surface,
Thus,
a
then each of the indices,
will
have two possible values,
would then have
so that B*TP
sixteen components.
Similarly,
three-dimensional space
in
4
would have 3 or 81 components,
and so on.
it
For the purposes of Relativity,
which we have to deal with
in
FOUR-dimensional continuum
a
this tensor has
44
We
add
hasten to
or
256 components!
that
not quite so bad as that,
as we can easily see:
it is
In
the
if,
in this tensor,*
*That
in
first
place,
is, in the expression
(53) on page 205.
in
square brackets
207
we interchange r and p ,
the result is merely to change
its
sign.J
Hence,
of the possible 16 combinations of r and p
only 6 are independent:
This
is
in itself
we
that
so interesting
here for a moment:
shall linger
Suppose we have 16 quantities,
1 , 2, 3 7 4, and
a,,;j ,
- 1,2,3,4),
(wherea=
which we may arrange
And
(that
suppose
is,
,
as follows:
a-2t
322
a23
324
ast
332
a;j;}
331
641
^42
^43
44
that a a)3
=
a reversal of the
a fta
two
subscripts
change of sign of the term),
=
an implies that an =
then, since a n
,
and similarly for the remaining terms
results
in
only
in a
the principal diagonal,
hence,
the above array becomes:
324
323
324
Thus there are only
SIX
distinct quantities
instead of sixteen.
Such an array
is
called
ANTISYMMETRIC.
tThc reader would do well to compare this expression
with the one obtained from it by an interchange of r and
p throughout.
208
Compare
with the definition of
this
SYMMETRIC
And so,
a
come back
we now have
to
on page 193.J
on page 208,
to the discussion
combinations of r and p
six
to
array
be used with
<r and
a,
96 components
sixteen combinations of
giving
X
6
16
or
instead of 256.
Furthermore,
it can be shown
that
we
can further reduce
this
number
to 20.*
Thus our curvature tensor,
for the situation in Relativity,
20 components and
has only
Now
let us
consider for a
IMPORTANCE
the great
in the study of spaces.
fThus
in
an
whereas,
Note
were
it
in a
that
if
a/3
of this tensor
matrix
we have
,
SYMMETRIC
the
256!
moment
ANTISYMMETRIC
ap =
NOT
first
matrix
matrix
on
p.
we have
208
SYMMETRIC,
would reduce to
TEN
distinct elements,
since the elements
would
NOT
in
be zero
the principal diagonal
in that case.
*See A. S. Fddington's
The Mathematical Theory of Relativity,
page 72 of the 1930 edition.
209
Suppose we have
Euclidean space
a
more dimensions,
and suppose we use
of two, three, or
ordinary rectangular coordinates.
Here the g*s are all constants.*
Hence,
since the derivative of a constant
zero
is
the Christoffel symbols will
also be zero (see page 200);
and, therefore,
the components of the
all
curvature tensor
will be zero too,
because every term contains
a Christoffel
symbol (see page 205).
BUT,
if
in
the components of a tensor
any given coordinate system
are
all
zero,
obviously
its
components
in
any other coordinate system
would also be zero
(consider this
And
in
the simple case on page 129) c
so,
whereas from a mere superficial inspection
of the expression for
we cannot
2
cfe
whether
Euclidean or not,!
an examination of the curvature tensor
(which of course is obtained
from the coefficients
the space
in
tell
is
the expression for
*Sec page
JScc page
1
1
2
c/s )
89.
25.
210
can definitely give this information,
no matter what coordinate system
2
used
is
in setting
up
cfs
.
Thus,
we
whether
all
116
use (3) on page
or (7) or (8) on
page 123,
of which represent
the square of the distance
between two points
ON A
EUCLIDEAN PLANE,
using various coordinate systems,
we
shall find that
the components of B"Tp
three cases
in all
ARE ALL ZERO.*
The same
for all
is
true
coordinate systems
for any number of dimensions,
provided that we remain in
Euclidean geometry.
and
*To have
a clear idea of
the meaning of the symbolism,
the reader should try the simple exercise
of showing that
He
must bear
911
=
in
for (8) on
B?TP =
mind that here
1,
312
=
=
g-2i
o,
p.
123.
gfc^xj,
and use these values in the bracketed expression
in (53) on page 205,
remembering of course
is
given
also that
by
all
that the
meaning of [ap 1
the definition on page 196;
indices,
<r
,
have the possible values
since the space here
p
1
,
c
,
etc.
and 2,
is
two-dimensional;
and he must not forget to
SUM
whenever an index appears
TWICE IN
ONE TERM.
ANY
211
1]
,
etc.
Thus
is
=
B; rp
(54)
NECESSARY
a
condition
be
that a space shall
EUCLIDEAN.
It
can be shown that
this
is
also a
SUFFICIENT
condition.
other words,
given a Euclidean space,
In
this tensor will
be zero,
whatever coordinate system
AND
is
used,.
CONVERSELY,
given this tensor equal to zero,
then we know that
the space must be Euclidean.
We
shall
how
the
is
now see
new Law
EASILY
from
of Gravitation
derived
this tensor.
XXV. THE BIG G'S OR EINSTEIN'S
OF GRAVITATION.
In
(54) replace p
by a ,
obtaining
B
(55)
Since
in
a
a appears
=
twice
the term on the
we
must,
0.
left,
,
according to the usual convention,
sum on a,
213
LAW
t\
V
so that (55) represents
SIXTEEN equations
corresponding to the
values of <r and r
only
4X4
four-dimensional continuum.
in a
=
when er = r
(55) becomes
Thus,
+
Bin
Similarly for
we
=
<r
1
7
Bin
1 ,
+
=
r
Bin
2
+
B?u
+
Bi24
=
0.
,
get
0121
+
Bl22
+
Bi23
=
and so on,
16
for the
We
possible combinations of
<r
and r
therefore write (55) in the form
may
G =
(56)
ffr
where each G consists of 4
shown above.
B's
as
In
other words,
by
CONTRACTING
which
we
is
B,' rp ,
get a tensor of the
namely,
FOURTH rank,
SECOND rank,
a tensor of the
G
ffT
,
as explained
on page 182.
The QUITE INNOCENT-LOOKING
EQUATION (56) IS
EINSTEIN'S LAW OF GRAVITATION.
Perhaps tht reader
by
But
this
is
startled
sudden announcement.
let us
look into (56)
carefully,
and see what
is
behind
its
innocent simplicity,
215
and why
the
In
it
Law
it
deserves to be called
of Gravitation.
the first place
must be remembered
that before contraction,
B*rt>
represented the quantity in brackets
in the right-hand member
of equation (53)
on page 205.
Hence,
when we contracted
by replacing p by a
we
it
,
can see from (53) that
Gar represents
the following expression:
\
(57)
{era, e}
which,
by
(er, a}
-
^
r\
{<"",
-
+
{aa, a]
{err,*}
{,<*},
a]
in turn,
the definition of
the Christoffel symbol (page 196)
represents
an expression containing
first
and second
of the
little g's.
And,
of course,
partial derivatives
(57) takes 16 different values
as <r and r each take on
their
4
different values,
while the other Creek
letters in
(57),
namely, a and e,
are mere dummies (see page 204)
and are to be summed
(since each occurs twice in each term),
as usual.
216
To get
clearly in
mind
what (57) means,
just
the reader
is
advised
to replace each Christoffe!
symbol
accordance with the definition on page 196,
and to write out in particular
in
one
by
of the
16 expressions represented by (57)
putting, say
=
<r
and allowing a and
in succession,
the values 1,2, 3
It
and r
1
6
=
2
,
to assume,
,4.
can easily be shown
that (56) actually represents
NOT
16 DIFFERENT equations
but only 10,
6 are independent.*
new Law of Gravitation
and, of these, only
So
is
as
that the
not quite so complicated
it
appears at
first.
why do we call it a
Law of Gravitation at all?
But
It
be remembered
will
that a space,
of any number of dimensions,
is
characterized
by
2
its
expression for
cfs
(see page 187).
Thus
(56)
is
completely determined by
the nature of the space which,
by the Principle of Equivalence
determines the path
of a freely moving object
in the space.
*Se
p. 242.
217
even granting the
But,
Principle of Equivalence,
that
is,
granting the idea
that the nature of the space,
11
rather than a "force
how
determines
move
in that
of gravity,
objects (or light)
space
other words,
in
granting that the g's alone
determine the
one may
Why
is
still
Law
of Gravitation
ask:
this particular
expression (56)
taken to be the
Law
of Gravitation?
To which the answer
it is
the
SIMPLEST
is
that
expression which
is
ANALOGOUS to
Newton's Law of Gravitation.
Perhaps the reader
unpleasantly surprised
is
at this reply,
and thinks
made
that the choice has
rather
May we
therefore suggest to him
to read through the rest of this
in
been
ARBITRARILY!
book
order to find out
the
CONSEQUENCES
of the
We
WISDOM
will
choice
who
is
be convinced
of this choice,*
appreciate that
part of Einstein's
*The reader
*
of Gravitation.
predict that he will
of the
and
Law
1
of Einstein
this
is
GENIUS!
particularly
interested in this point
wish to look up a book called
"The Law of Gravitation in Relativity"
by Levinson and Zeisler, 1931.
may
218
He
example, on page 271
will see, for
,
that the equations giving
the path of a planet,
derived
are the
by Newton,
SAME,
to a
first
approximation,
as the Einstein equations,
so that the latter can
ALL
and
that the
do
Newtonian equations do,
FURTHERMORE,
ADDITIONAL
term in (84)
accounts for the "unusual" path
of the planet Mercury,
the
which the Newtonian equation (85)
did not account for at all.
But
we
are anticipating the story!
Let us
now
a form
which
XXVI.
express Newton's Law in
will show the analogy clearly.
COMPARISON OF
EINSTEIN'S
LAW
OF GRAVITATION WITH NEWTON'S.
Everyone knows
that,
according to Newton,*
two bodies attract each other
with a force which
is
proportional
to the product of their masses,
and inversely proportional to the
square of the distance between them,
thus:
*See the chapter on the
"Theory of Attractive Forces*' in
Ziwet and Field's
Introduction to Analytical Mechanics,
219
formula
In this
we
regard the two bodies,
of masses mi and m ,
>
as each concentrated at a single point
*
11
(its
and
"center of gravity
r is then precisely
),
the distance between these two points.
Mow we may consider that mi
5
surrounded by a "gravitational field*
which the gravitational force at A
is
in
(see the diagram on page 221)
is given by the above equation.
If we divide
we get
both sides by m%
fc/m
_F_~ __
1
o~~
r"
ni2
according to Newton,
And,
=
a
the acceleration with which
,
/3l2
m> would move due to
F acting on it.
the force
We
may
therefore write
a
(58)
=
where the constant C now includes
since
we
are speaking of
the gravitational field around mi
*Thus
it is
a fact that
to support a
it is
hold
body
not necessary to
it
up
all
over,
but one needs only support it
right under its center of gravity,
as
if its
entire mass
were concentrated
at that point.
220
.
Now,
acceleration
and
may be
it
is
split
a vector quantity/*
up
into
components:!
Thus take the origin to be at mi
and the mass 012 at A i
OA =
then
and
r;
AB represent the acceleration
m is being pulled toward mi)
let
(since
in
,
2
both magnitude and direction.
Now
it is
if
X
is
the x-component of a
obvious that
X
a
x
r'
Therefore
Or,
better,
*See page 127.
fSee page 129.
221
,
at
A
to
show
that the direction of
X
is
to the
Substituting in this equation
the value of a from (58)
we
get:
^*
Ay - And,
y=
-
similarly,
--
-
3-dimensional space,
in
and,
3
we would have
By
7
also
we
differentiation,
--p
Z=
get:
6
dx
=
2
But, since r
(as
is
r
x
2
+
2
y
z
2
obvious from the diagram
on page 131
i
,
if
AB =
dr
= -x
dx
r
then
r),
'
Substituting this in the
it
+
above equation,
becomes
dx
And,
r
6
similarly,
From these we
(59)
get:
3X,ay,az
_
TV"
[-
ox
ay
az
222
left.
This equation
may be
dV
(60)
ax
where
and
is
is
,
2 "*"
written:
^
*
,
2 "*"
a/
dz
2
a (unction such that
called the
*
1
"gravitational potential
obviously (60)
is
*/
merely another way
of expressing the field equation (59)
obtained from
Newton's Law of Gravitation.
This form of the law,
namely (60),
generally 'known as
the Laplace equation
is
and
is
more
briefly
denoted by
v ^ 2
where the symbol
V
2
merely denotesf
that
the second partial derivatives
with respect to x , y , and z ,
respectively,
be taken and added together,
shown in (60).
are to
as
We see
from (60), then,
that the gravitational field equation
obtained from
Newton's Law of Gravitation
an equation containing
the second partial derivatives
of the gravitational potential.
is
*
See footnote on page 21 9.
is read "nabla",
fThe symbol
2
and
is read *'nabla square".
V
V
223
Whereas (56)
is
a set of equations
which also contain
nothing higher than
the second partial derivatives
of the g's,
which,
by the Principle of Equivalence,
replace the notion of
a gravitational potential
derived from the idea of
11
a "force
by
of gravity,
the idea of
the characteristic property of
the SPACE in question (see Ch. XII).
It is therefore reasonable
to accept (56) as the
gravitational field equations
which follow from the idea of
the Principle of Equivalence.
HOW
REASONABLE
it
is
be evident
when we test it by
will
EXPERIMENT!
It
has
that
been
each
said (on
G
page 21 5)
consists of four B'S O
Hence,
if
the B's are
all
then the G's will
zero,
all
be zero;
but the converse
is
obviously
NOT true:
Namely,
even
it
if
the G's are
all
zero,
does not necessarily follow
that the B's are zero.
925
we know
But
to have the
that,
fi's all
zero
implies that
the space
Euclidean (see p. 213).
is
Thus,
if the condition for Euclidean
space
is
fulfilled,
namely,
= V/
Ba ~
Off Tp
then
=
G<rT
automatically follows/
thus
is
true in the special case of
Euclidean space.
But, more than
since
this,
NOT NECESSARILY
does
that the
imply
are zero,
fi's
hence
can be true
EVEN
IF
THE SPACE
IS
NOT EUCLIDEAN,
namely,
in the space around
a
body which
creates a gravitational field.
Now
but
all this
still
"How
Law
be
one
can
sounds very reasonable,
naturally asks:
this
new
of Gravitation
tested
EXPERIMENTALLY?"
226
Einstein suggested several ways
in which it might be tested.
and,
as every child
when
now knows/
the experiments were
actually carried out,
were
his predictions
and caused
all fulfilled,
a great stir
not only in the scientific world,
but penetrated even into
the daily news
the world over.
But doubtless the reader
would
like to
know
the details of these experiments/
and just how the above-mentioned
Law
is
of Gravitation
applied to them.
Thai
is
what
XXVII.
we
shall
show
HOW CAN
next.
THE EINSTEIN
GRAVITATION
We
have seen that
G, r
=
represents Einstein's
Law of Gravitation/
new
and consists of 6 equations
containing partial derivatives of
the little g's.*
*See pages 21 5 to 21
7.
227
LAW OF
BE TESTED?
,u
In
order to test
law
this
we must obviously
substitute in
it
the values of the g's which
actually apply in our ohysical world;
in
other words,
we must know
what
first
the expression for d*
which applies to our world
is
(see Chapter XIII).
Now,
if
we
use
the customary polar coordinates,
we know
in
that
EUCLIDEAN
two-dimensional
space
we have
2
c/s
=
2
c/r
+ rW.*
Similarly,
for three-dimensional
EUCLIDEAN
we have
space
the well-known:
2
c/s
The reader can
2
c/s
-
=
2
c/r
+ rW +
2
r
2
sinW</>
easily derive this from
c/x?
+
Jxl
+ Jx]
(on pa g c 189),
to polar coordinates
with the aid of the diagram
by changing
on page 230.
*See page 123
229
where
xi
X2
xs
=
=
=
=
x
r
y
z
= 01 = OMcos
cos
/
POM
= LM =
= PM =
cos
=
OM sin
r
cos
0.
And,
for
<t>
</>
4-dimensional space-time
230
<
r sin ^
=
r sin
cos
<
sin
we have
2
fc/s
(61 a) or
2
= -
(where xi
and c
as
we
Note
-
c/x?
- xl^ -
2
= -
[c/s
=
Jr
=
r, x2
rW - fWe/fl-cfy +
2
x?sinVc/xi
= 0, x =
9, x3
4
t
-f c/xi
7
taken equal to 1),
can readily see:
is
that the general form for
four-dimensional space
in
Cartesian coordinates,
analogous to the 3-dimensional one on p. 189,
=
2
c/s
in
2
+
2
dy
+
2
</z
67
But 7 on page
we showed
Jx
that
order to get
the square of an "interval"
in
space-time
in this
with
form/
all
we had
four plus signs,
to take r
the time, t,
BUT to take T
i
c
=
=
NOT
= -
equal to
ict* where
the velocity of light;
from which
2
c/r
= - cW,
and the above expression becomes:
*As
a matter of fact,
11
in
we
"Special Relativity,
took r
=
it,
but that was because
we
also took c
otherwise,
=
1/
we must
take r
=
231
ict.
+
2
c/r
.
V
And,
furthermore,
since in actual fact,
2
c
2
is
c/t
always found to be
Sreater than (</x
2
+
2
c/y
+
2
c/z ),
therefore,
to
make c/s come out reel instead
more reasonable to write
of imaginary,
it is
=
2
c/s
which
in
-c/x
-Jy
2
-c/z
2
+cW,
polar coordinates,
becomes (61
a).
The reader must
this
2
formula
still
clearly realize that
applies to
EUCLIDEAN
which
the
is
space-time,
involved in
SPECIAL
theory of Relativity*
where we considered only
observers moving with
UNIFORM
velocity relatively to each other.
But now,
in the GENERAL theory (page 96)
where we are considering
accelerated motion (page 102),
and therefore have a
NON-EUCLIDEAN
space-time
(see Chapter XII),
what expression
shall
In
we
2
for
c/s
use?
the
it is
first place
reasonable to assume that
2
(61 b)
Js
= -
e* </x?
-
e* (xldxl
+
(where Xi
*See Part
,
X2 / Xa / X4 represent
I
of this book.
233
+
e'c/x
the polar coordinates r, 0,
$, and
t,
respectively,
and A
,
IJL
,
and
v are (unctions of xi only),
BECAUSE:
(A) we do not include product
of the form
cfa
cfxi
terms
,
more generally,
of the form dx c/xr where
or,
ff
(which
ARE
included
in
cr
^
r
,
(42), p. 187)
since
from astronomical evidence
it
seems
that
our universe
is
(a)
ISOTROPIC and
(b)
HOMOGENEOUS:
That
is,
the distribution of matter
(the nebulae)
is
the
SAME
ALL DIRECTIONS
and
(a)
IN
(b)
FROM WHICHEVER POINT WE LOOK.
Now,
how does
the omission of terms like
dx JxT where
ff
<r
^
r
represent this mathematically?
Well, obviously,
a term like c/rc/0
(or C/0-C/0 or c/rc/(/>)
would be
for 6 (or
different
</>
or r) positive or negative,
and, consequently,
the expression for
would be
in
different
2
c/s
if
we
turn
opposite directions
234
which would contradict the
experimental evidence that
the universe is ISOTROPIC
And
of course the use of
2
the same expression for
c/s
ANY point
from
reflects the idea of
And
so
we
HOMOGENEITY.
see that
it is
reasonable
to have in (61 b)
2
2
2
only terms involving c[0 , c[</> , c/r
in which it makes no difference
whether
we
substitute
+c/0 or
,
c/0, etc.
Similarly,
2
since in getting a measure for
c/s /
we are considering
STATIC condition,
a
and not one which
from
moment
we must
to
is
changing
moment,
therefore not include
terms which will have different values
for
+dtand
-c/f/
other words,
we must not include
in
product terms like c/rc/t,
etc,
In short
\we must not have any terms involving
cfx^-c/Xr
where
or
^T,
but only terms involving
dx -dxr where
ff
(B)
The
in
factors
e
x
,
e
M
the coefficients
, e",
(7
=
T.
are inserted
*
to allow for the fact
*Cf.(61a)and(61b).
235
that our space
is
now
NON-EUCLIDEAN.
Hence they are so chosen as to
allow freedom to adjust them
to the actual physical world
/since they are variables),
jnd yet
iheir
FORM
is
be easy
such that
in
them
making the necessary adjustment-
as
we
it
will
to manipulate
shall see.*
Now,
(61 b) can
by
be somewhat
simplified
replacing
2
e*x?
and taking
by
x( as a
(x() ,
new
coordinate,
thus getting rid of e" entirely/
and we may even drop the prime,
since any change in c/xi which arises
from the above substitution
can be taken care of
by taking X correspondingly different.
Thus (61 b) becomes, more simply,
2
(62)
c/s
= -
e\/x?
And we now
-
xldxl
- xrsinWx2 +
have to find
the values of the coefficients
x
e and e"
in
terms of xi.j
^Further justification for (61 b)
may be found
in
R. C. Tolman's
Relativity
p.
239
Thermodynamics
& Cosmology,
ff.
ISee page 934.
236
.
2
e'-c/x4
We warn the reader that this a
COLOSSAL UNDERTAKING,
is
but, in spite of this bad news,
hasten to console him by
we
telling
him
that
terms will reduce to zero,
and the whole complicated structure
many
will
we
melt down to almost nothing;
can then apply the result
to the physical data
with the greatest ease.
To any reader who "can't take
we
1
'
it
suggest that
he omit the next chapter
and merely use the result
to follow
the experimental tests of the
Law of Gravitation
Einstein
given from page
255
on.
A
BUT HE WILL MISS
SURMOUNTING THE
XXVIII.
So
far,
LOT OF FUN!
DIFFICULTIES.
then,
the following values:
we have
SLI
=
and gar
*
=
x
/ ff22
when
Furthermore,
the determinant
is
= cr
g33
x?
,
^
r.
= -
2
x?sin
=
(see (62) on p. 236.)
g (see page 194)
simply equal to
the product of the four elements in
its
x2 , g44
principal diagonal,
237
e*
since
ail
the other elements are zero:
Hence
Also 7
in this case,
and
T
g*
We
in
=
when a
^
r.
f
need these relationships
determining e* and e in (62).
shall
Now we
shall
see
how
the big G's will help us to
Find the little g's
and how the little g's will help us
to reduce the number of big G's 10
ONLY
THREE!
First let us
show
that
the set of quantities
is
SYMMETRIC,}:
and therefore
*See the chapter on determinants
in any college algebra,
to Find out
a
how
to evaluate
determinant of the fourth order.
"
M
tSee the definition of g on page 196.
193.
{Seepage
239
Gar =
reduces to
instead of sixteen/
v and r each take
as
their values
To show
1
*
TEN
equations
on
,2/3,4.
this,
we must remember
that
G
really represents (57)
By
definition (page 196),
on
and let us examine {act, a}
which occurs in (57):
ffT
p.
216;
But,
that
remembering
the presence of
in
term
a and
c
TWICE
EACH
(after
multiplying out)
SUM
on
implies that we must
the reader will easily see that
many
and
a and
e
of the terms will cancel out
that
we
shall qet
Furthermore,
v
the definition of g" on page 196,
the reader may also verify the fact that
by
i
2ff
where g
And,
is
dg
=
1
dg
the determinant of p. 239.
from elementary calculus,
*See pase 193.
240
Similarly,
/} =
:->-
v-gr.
log
Substituting these values in (57),
we
get:
<\
G =
(63)
We
can
{er, a}
ffT
{era, e}
now
easily see
-
jar, aj
+
that (63) represents
10 equations and not
sixteen/
(or the following reasons:
the
In
first
place/
{er, a]
=
{re, a}
(see pp. 204, 205).
Hence,
by
the
its
interchanging a and r ,
first term of (63) remains unchanged,
two
*Notc
factors
merely change places
_
that
we might
also have obtained
but since g
is
V +
always negative
g,
= -y,
(we shall show on p. 252 that X
4
2
and therefore g on p. 239 becomes
xi -sin X2)
it is more reasonable to select v
g f which
will
make
the Christoffel symbols,
and hence also the terms
the
new Law
REAL
in
of Gravitation,
rather than imaginary.
241
and a are mere dummies,
(since
explained on page 204).
as
And,
the second, third and fourth terms of (63)
are also unchanged by
the interchange of a and r.
in other words,
Thus, if we arrange
the 16 quantities in
in a
We
have
this
is
JT
a
just
Gn
Gi2
Gis
vJ21
tf22
VJ23
Gsi
Gs2
633
G41
642
Gl3
shown
we
We
G
\3
G
6
that
SYMMETRIC
Hence (63) reduces
instead of 1 6 ,
as
G
square array:
to
matrix.*
10 equations
said before.
shall
not burden the reader
with the details of
how
is further reduced
SIX equations.!
(63)
to only
But perhaps the reader
that
is
thinking
"only six" equations
no great consolation,
particularly if he realizes
are
still
*See page 239.
he is interested,
he may look this up on page 115 of
M
1
The Mathematical Theory of Relativity/
flf
by A. S. Eddington,
the 1930 edition.
24*
how long each of these equations
But does he realize this?
he would do well to take
cr and r
= 1 , and r = 1 ,
say <r
in order to see just what
particular values of
ONE
is
of the equations
,
(63)
sum on the dummies
the reader wondering
what we are trying to
just
Is
!
really like!
(don't forget to
Is
in
is
this a subtle
do
to
I)
him?
mental torture
by which we
alternately frighten
The fact is that
we do want
to
make him
and console him?
to frighten him sufficiently
realize
the colossal amount of computation
that is involved here,
and yet to keep up
by the knowledge
it
his
courage too
that
does eventually boil down
to a really simple form.
He
the
if
might not appreciate
simple form
final
he did not know
the labor that produced
With
it.
this
we shall
how the
apology,
now proceed to indicate
further simplification
takes place.
In
we
each Christoffel symbol in (63),
must substitute specific values
for the
It is
Greek
letters.
obvious then
243
that there will
all
three
Thus:
be
four possible types:
which the values of
(a) those in
Greek
ffo-
g
letters are alike:
a}
(b) those of the form
0-<r ,
r
(c) those of the form
or
r
,
and
(d) those of the form
{err ,
p}.
Note that it is unnecessary to consider
the form {rer, r}
since this is the same as (or, r} (see p. 204).
Now, by
definition (page 196),
and, as usual,
we must sum on a.
But since the only g's which are not zero
are those in which
the irrdices are alike (see p.
and,
237)
in that case,
flT*=1/ff~
(P- 239).
Hence
f
_
(jcr,
a }}
dS
_ jL( v~
^'l
Adx
2g
(T
d
5x
<y
ff
and therefore
which,
(a)
by elementary
{<r<r,<r}
calculus, gives
=
^
^
244
log
---d 9~-
9~
4-r ~^-
gro.
\
I
3x,/
Similarly,
+
*"-' = Mi!*-"
2
Vox,
Here the only values
keep the outside
are those for which
and since r
^
(for otherwise
we
a
of
a =
T
ra
g
from being zero
,
a
we
should have case (a)
)
get
or
/L\
(b)
Likewise
{CTT.T}
(c)
=
^
g
log grr
and
(d)
Let us
{<rr
now
/P }=0.
evaluate these various forms
for specific values:
Thus, take,
in
Then
case (a),
{11/1}
x
Butgn= -c
hence {11
which,
f
=
cr
=
1
:
1
r)
g
^7
lo
33n
(Seep. 239).
i}=l.|og(-O
by elementary
-Hdx
that
factor
will
1dx,
calculus, gives
245
a
/
L
V
4
A represents
where
=
since xi
r (see
/
or
-
dr
6x1
page 233),
Similarly,
i4
in,
if
*-J
But, since
in taking a PARTIAL derivative
with respect to one variable,
all the other variables are held constant,
hence
and therefore
(22,2} =0.
And,
likewise,
{33 ,3)
Now,
take
for case (b),
first (r
=
1
,
=
r
= {44,41=0.
2
/
then
{11
,
2}
= -
^
x2
= -
gn
But, since X is a function of x\
and is therefore held constant
~
2flL> 2
only/
while the partial derivative
with respect to x>
hence {11,2}
and so on.
Let us see
we
shall
*Se
taken,
how many
have
Obviously
is
=0,
specific values
in all.
(a) has
4
specific cases,
page 234.
246
~3x2
=
1 ,2,3,4,
namely, <r
which have already been evaluated above.
(b) will have 12 specific cases,
since for each value of a = 1 , 2
r can have 3 of
here a
(for
^
(c) will also
its
r)
possible 4
,
3
,
4
,
values
/
have 12 cases,
and
X
(d) will have 4
but since {or , p}
reduces to
this
Hence
3
=
X
2
=
[TV , p\
24
cases,
(see p. 204),
1 2.
in all
there are
40
cases.
The reader should verify the fact that
31 of the 40 reduce to zero,
the
9 remaining ones being
(64)
'Remember
that
x2
=
Note
that
~=%
d
/=
oxi
Now,
.
or
in (63),
when we
give to the various
Greek
letters
their possible values,
we find that,
since so many of the ChristofFel symbols
are equal to zero,
a great meny (over 200) terms drop out !
And there remain now
only FIVE equations,
each with a much smaller number of terms.
These are written out in full below,
and,
lest
this
the reader think that
is the promised
final
simplified result,
hasten to add that
the BEST is yet to come!
we
Just
how
Gii is obtained,
the
reader how
showing
to
on a. and e
and which terms drop out
(because they contain zero factors}
SUM
will be found
on page 31 7.
in
V,
And,
similarly (or the other G's.
Here we give
the equations which result
after the zero terms have been
eliminated.
248
Gu=
11,1} {11,1}+
13, 3} {31,3}
=
+
12, 2} {21, 2}
14,4)141,4}
0.
Similarly,
,3] (83,3)
=
0.
= 2{33 1}{13,3}
/
+ 2{33,2}{23,3}
-ST {33,11-ir
133, 2}
,
=
0.
,1}
{14,41-(44,1}
/
=
0.
=
0.
249
we now
If
substitute
these equations
the values given in (64),
in
we
get
=
424
=
0*
v
"
+
""
-
xv - -
Similarly
G =
33
2
sin
0-e-
=
0.
=
and
\"
=
v"
=
x
l
+
0.
d X
j
250
r
(/
-
X')
-
2
sin
and 612 becomes:
-
cotfl
-
-
cotfl
=
r
r
which is identically zero
and therefore drops out,
thus reducing the number of equations
FOUR.
to
Note
also that
633 includes 622
,
so that these two equations
are not
independent
hence now the equations are
THREE.
And
now, dividing
and adding the
we
x
G
44
by
e*~
Gn
result to
,
get
A'
(65)
d\
or
T"
or
Therefore,
by
where
/ is a
dv
T"
dr
= ~
'
integration,
A
(66)
= - /
= -
v
+
I
constant of integration.
But, since
at
an
infinite distance
from matter,
our universe would be Euclidean,*
*See page 226.
251
and then, (or Cartesian coordinates,
we would have:
that
is,
the coefficient of c/x? and of dx\
must be 1 under these conditions/
hence,
if
(61 b)
is
to hold also for
this special case,
it must
do,
should then have A =
as of course
we
,
v
=
0.
other words,
In
since,
when
v
=
,
X also equals
,
/ , too, must be zero.
then, from (66),
Hence
\=-v.
(67)
Usins (65) and (67),
622
on page 250 becomes
(68)
If
we
and
we
+n/) =
e'(1
put
7
=
e
1.
,
differentiate with respect to r
get
dy
^~
_
e
v
or
"
dv
dr
or
Hence (68) becomes
(69)
7
+ fy =
1.
*See pages 189 and 231.
252
,
This equation
may now be
easily integrated/
obtaining
=
T
(70)
-
1
~
where 2m is a constant of integration.
The constant m will later be shown
to have an important physical meaning.
Thus
we have succeeded
in finding
x
and
e an
e'
*From elementary theory of
differential equations,
we
write (69):
or
or
T_ = -Jr
cf
1-7
'
r
Having separated the
we
can
now
variables,
integrate both sides
thus:
7)
log (1
= -
log r
+ constant;
or
log r (1
7)
=
constant,
and therefore
r (1
We
may
=
constant.
then write
r (1
from which
7
7)
=
we
A
1
7)
= 2m
,
get
2m
.
253
in
terms of xi
e"
:
=
=7 = 1- 2m/r = -
x
1/e
1
2/n/x,
and (62) becomes:
= - 7-i</r2 - rV0 2
ctf
(71 )
2
2
r sin
c/</
where, as before (p. 233),
r
And
=
=
xi /
x2
=
,
x3
t
,
=
x4
.
hence
new Law of Gravitation,
consisting now of only
the
the
THREE
remaining equations:
Gn =
are
now
the
little
We
can
fully
g
s
,
G =
33
,
and
G =
44
,
determined by
of (71).
now proceed
to test this result
to see whether
it
really applies
to the physical world
XXIX. "THE
The
first
test
is
we
live in.
PROOF OF THE PUDDING.
naturally
to see what
the
new Law
of Gravitation
has to say about
the path of a planet.
It
was assumed by Newton that
a
body
1 *
"naturally
moves
along a straight line
255
11
not pulled out of
if it is
by some
for
As,
a
its
on
force acting
course
it:
example,
body moving on
a frictionless Euclidean plane.
Similarly,
according to Einstein
if it
moves "freely
the surface of a
it
11
on
SPHERE
would 30 along the
1
"nearest thing to a straight line/
that
is,
along the
GEODESIC
namely,
along a great
And,
for this surface,
circle.
for other surfaces,
or spaces of higher dimensions,
it
would move along
the corresponding geodesic for
the particular surface or space.
Now
our problem
is
to find out
what
is
the geodesic
in
our non-Euclidean physical world,
since a planet must
move
along such a geodesic.
In
order to find
the equation of a geodesic
it is
necessary to know
something about the
1
"Calculus of Variations/
so that we cannot go info details here.
But
we
shall give the reader
a rough idea of the plan,
together with references where
257
he may look up this matter
Suppose, For example, that
we have given
two points, A and B
,
on
further.*
a
Euclidean plane,it is obviously possible to
them by various paths,
join
thus:
B
Now,
which of
all possible paths
the geodesic here?
Of course the reader knows the answer:
is
the straight line path.
It is
But
how do we
set
up
the problem mathematically
so that we may solve
similar
problems
in
other cases?
*(1) For fundamental methods see
"Calculus of Variations/' by
G. A.
Bliss.
problem see
"The Mathematical Theory of
(2) For this specific
1
Relativity/ by A. S. Eddington,
p. 59 of the 1930 edition.
(3)
Or
see pages 128-1 34 of
1
"The Absolute
by
Differential Calculus/
T. Levi-Civita.
258
Well,
we know
from ordinary calculus
that
if
on
a short arc
any
of these paths
by
is represented
then
f.
c/s ,
B
f
JA
A
represents the total length of
that entire path.
And
this of
any one
How
course applies to
of the paths from
do we now
A
select from
to B.
among
these
the geodesic?
This
problem is similar to
one with which the reader
undoubtedly
is
familiar,
namely,
if
y
=
f(x),
y for which
16
an "extremum" or a "stationary.
find the values of
y
is
Such values of y are shown
in
the diagram on the next page
at a , 6 , and c:
259
a
and/ (or
all
we must have
these/
=
dy/dx
or
<fr=ffo)-c/x
(72)
where x,
is
a
,
6
or
,
=
c.
Similarly/
go back to our problem on pp. 258 and 259,
to
the geodesic
we
are looking for
would make
rB
f
JA
A
a stationary.
This
is
expressed
in
the calculus of variations
(73)
f
c/s
by
-
JA
analogously to (72).
To
find the
equation of the geodesic/
satisfying (73),
is
not as simple as finding
260
maximum
a
minimum
or
in
ordinary calculus,
and we
shall give here
the
result:*
only
+
(74)
W .,
Let us consider (74):
In
the
first
place,
for ordinary three-dimensional
Euclidean space
a would have
three possible values:
since we have here
1,2,3,
three coordinates xi, X2,
x3 /
furthermore,
by choosing
Cartesian coordinates,
we would have
(see page
==
ffll
522
=
1
89):
ff33
=
1
/i
^
and
& =
for
v
and therefore
which involves derivatives of the g'sf
would be equal to zero,
so that (74) would become
(75)
~
*For details, look up the references
the footnote on p. 258.
fSee (46) on p. 196.
261
in
Now, if in (75)
we replace c/s by
it
*
dt
becomes
=
(76)
which
-0
a short
is
way
(*
of writing
the three equations:
/ 77 \
(77)
But what
the
*lf
is
PHYSICAL MEANING
we
of (77)?
who
consider an observer
has chosen his coordinates
in
such a
that
way
=
C/Xl
C/X2
other words,
an observer who
=
C/X3
=
9
in
is
traveling with
a
moving object,
and for whom the object
standing
is
therefore
with reference to
still
his ordinary space-coordinates,
so that only time is changing for him,
then, for him (61 a)
2
=
2
=
c/s
or
c/s
or
c/s
That
c/s
is
=
becomes
2
c/x4
2
c/t
c/t.
to say,
of the nature of "time/"
becomes
for this reason
c/s is
often called
"the proper time"
since
it is
for the
a
"time"
moving object
itself.
262
= 1,2,3)
Why, everyone knows
(or uniform
that,
motion,
v=s/t,
where v
a
it
t
If
is
the velocity with which
body moves when
goes
a distance of 5 feet in
seconds.
the motion
we
NOT
is
uniform,
can, by means of
elementary differential calculus,
express the velocity
by
=
v
Or,
if
AT AN INSTANT,
ds/dt
x, /, and z
.
are
on the
and
Z
X, Y,
axes, respectively,
and vx , vy , and vz are
the projections of v
on the three axes,
the projections of
s
then
_
in the
_
~e/x
dy
__
~
di
abridged notation,
v,
where we use
=
^
xi, X2 , Xs
instead of
x, y, z,
and
, va
vi ,
v2
('=1,2,
instead of
vx ,vv ,Vz'
Furthermore,
since acceleration
is
the change in velocity per unit of time,
263
3)
we have
Jv
or
(78)
a
Thus (77)
=^
<T
tr-1
V*
*
1.2
I
2
*
/
3)
*/
states that
the components of the acceleration
must be zero,
and hence the acceleration
must be zero, thus:
itself
or
From
this
we
by
set,
v
=
integrating,
a constant,
or
-r
=
a constant,
and therefore
by
integrating again,
s
which
A
In
is
= at+
b,
the equation of
STRAIGHT
LINE.
other words,
when the equations
for a geodesic,
namely, (74),
are applied to the special case of
THREE-DIMENSIONAL EUCLIDEAN SPACE,
they lead to the
(act that
264
in this special
case
THE GEODESIC
A
IS
STRAIGHT
We
hope (he reader
and
NOT DISAPPOINTED
is
to get a result which
LINE!
DELIGHTED
is
so familiar to him;
and we hope
gives him
it
a friendly feeling of
confidence
in (74)!
And
of course
he must
realize
(74) will work also
for any non-Euclidean space,
that
since
the
it
contains
little g's
which characterize the space/*
and
for
since
cr
any dimensionality,
may be
any number of
given
values.
In particular,
in
our four-dimensional
non-Euclidean world,
(74) represents
the path of an object moving
in the presence of matter
(which merely makes the space
non-Euclidean),
with no external force acting upon the object;
and hence (74)
is
THE PATH OF
A
which we are looking
PLANET
for!
*Seep. 190.
265
XXX.
MORE ABOUT
THE PATH OF
PLANET.
A
Of
a
course (74)
only
is
GENERAL
expression,
and does not yet apply to
our particular physical world,
since the Christoffel symbol
involves the g's,
is therefore not specific until
we substitute the values of the g's
and
which apply
in
specific case
in a
the physical world.
Now
in
(64)
we have
in
the values of \a/3
terms of X , v , r and 0.
-
And, by (67), A hence we know {&$ ,
v
, r
Further, since e"
and 7
we
is
known
and
<?}
,
terms of
in
6.
= 7
in
(see page 252)
terms of r from (70),
therefore have [ct$
r
v
a}
,
and
f
a}
in
terms of
0.
The reader must bear
in
mind
that
whereas (76), in Newtonian physics,
represents only three equations,
on the other hand,
(74) in Einsteinian physics
is an abridged notation for
FOUR
equations,
266
as
<J
its
takes on
FOUR
Taking
possible values: 1,2,3,4.
the value a = 2 ,
first
=
and, remembering that x2
(see page 233),
we
have,
(or one of the equations of (74),
the following:
a
/-rn\
(79)
And
since
now,
a and
/?
+{^} dx
.
2
f
^
a
dxa
=
^
-
each occur
TWICE
the second term,
must sum on these as usual,
so that we must consider terms
in
we
containing, respectively,
{11,2}, {12,2}, [13,2}, {14,2},
{21,2}, {22,2}, {23,2}, {24, 2}, etc
which a always equals 2 ,
and a and /3 each runs its course
in
from
1
to 4.
But, from (64), we see that
most of these are zero,
the only ones remaining being
and
{33,2}
= -
sin0-co$0.
Also, by page 204,
{21,2}
=
{12,2}.
267
Thus (79) becomes
<PO
fftfYk
(80)
If
in
1 dr dd
+ -.._.-
we now choose our
such a way
tm fl. eoi
..
,/</</>
Y
=0.
coordinates
that
an object begins moving
6
=
7T/2
in
the plane
,
then
-j-
=
and cos0
=
di
and hence
If we now substitute all these values in
we see that this equation is satisfied,
and hence
=
thus
that
showing
ir/2
is
a solution of the equation,
the path of the planet
is
in a
plane.
Thus from (80)
we have found out that
a planet,
according to Einstein,
must move in a plane,
just as in
Newtonian physics.
Let us now examine (74)
and see what
further,
the 3 remaining equations in
tell
us
(80),
it
about planetary motion:
263
For
=
<T
1 ,
(79) becomes
Or
But since
we have chosen
6
=
7r/2
then
In
~-r
hence
And
for
a-
this
=
and
sin0
=
1 ,
equation becomes
similarly,
=
3
,
(79) sives
( 82 )
as
and
we
for
cr
=
2
r
as as
4,
get
(83)
+F f**
^
or
as as
269
o.
,
And now
from (81), (82), (83), and (71),
tosether with (70),
we
get*
2
c/
u
___
,
W
m
A
5
(84)
==
n
where c and h are
constants of integration,
and u
1/r.
Thus (84) represents
the path of an object moving freely,
that
is,
not constrained
and
is
in a
sense,
by any
external force,
therefore,
analogous to a
straight line in
Newtonian physics.
But it must be remembered
that in Einsteinian physics,
owing to the
Principle of Equivalence (Chapter XI),
an object is
constrained by any external force
NOT
even when
it
is
moving
in
the presence of matter,
as, for
example,
a planet
moving
around the sun.
And hence (84)
would represent the path of
*For details see page
86
in
"The Mathematical Theory of
A.
S.
a planet.
Relativity/'
Eddington (the 1930 edition).
270
by
From
we
point of view
this
are not interested in
comparing (84) with
the straight line motion
in
Newtonian physics,
as mentioned on page 270,
but rather with
the equations representing
the path of a planet
in
Newtonian physics,
which, of course,
the planet is supposed to
under the
in
move
GRAVITATIONAL FORCE
of the sun.
it
in
been shown
Newtonian physics
has
body
that a
moving under a "central force/
(like a planet moving
1
under the influence of the sun)
moves
in
an ellipse,
with the central force (the sun)
located at one of the foci.*
And
the equations of this path are:
2
e/
u
,
jLTsi
~r u
*
r
is
m
TO
*
(85)
where
__
-
"
*
the distance
from the sun to the planet,
m is the mass of the sun,
*Sec Ziwct and Field: "Mechanics/*
or
any other book on mechanics.
271
a
the semi-major-axis of the ellipse,
the angle swept out by the planet
is
4> is
in
time
We
t.
notice at once
the remarkable resemblance between
(84) and (85).
are indeed
They
IDENTICAL EXCEPT
for
3mu2,
the presence of the term
and of course the use of
instead of dt in (84).*
</s
Thus
the
we
see that
Newtonian equations (85)
are really
a first approximation to
the Einstein equations (84);
that
why
is
they worked so
satisfactorily
for so long.
Let us
how
now
see
the situation
is
affected
by
the additional term 3mir.
XXXI. THE PERIHELION
Owing
to the presence of the term
3mu2
(84) is no longer an ellipse
but a kind of spiral
in
is
which the path
NOT
retraced
each time the planet
*See
p.
OF MERCURY.
262.
272
makes
but
is
a
complete revolution,
shifted as
shown
in
the following diagram:
in
which
the "perihelion/* that
the point in the path
is,
nearest the sun, S, at the focus/
is at
the first time around/
A
at
at
6
C
the next time/
the next/
and so on.
273
In
other words,
does not 30
a planet
round and round
in the same path,
but there is a slight shift
in the entire path,
each time around.
And the shift of the perihelion
can be calculated
by means
of (84).*
This shift can also
be
MEASURED
experimentally,
and therefore can serve
as a
method
of
TESTING
the Einstein theory
in actual fact.
Now
it is
when
a planetary orbit
is
obvious that
very nearly
this shift in
CIRCULAR
the perihelion
not observable,
this is unfortunately
the situation with
most of the planets.
There is one, however,
is
and
in
which
this shift
IS
measurable,
namely,
the planet
MERCURY.
Lest the reader think
that the astronomers
*For details see aqain
Eddington's book referred to
footnote of page 270.
in the
274
can make only
crude measurements,
let us say in their defense,
that the discrepancy
even in the case of Mercury
is an arc of
ONLY ABOUT
43
SECONDS PER CENTURY!
Let us make clear what we mean by
19
"the discrepancy:
when we say that
the Newtonian theory
requires the path of a planet to be
an ellipse,
it must be understood that
this would hold only if
there were a SINGLE planet;
the presence of other planets
causes so-called "perturbations,"
so that
even according to Newton
there
would be
some
shift in
the
perihelion.
But the amount of shift
due to this cause
has long been known to be
531 seconds of arc per century,
whereas observation shows
that the actual shift is
574 seconds,
thus leaving a shift of
43 seconds per century
UNACCOUNTED FOR
in
the
Newtonian theory.
Think of the
DELICACY
275
of the measurements
and the patient persistence
over a long period of years
generations of astronomers
by
that
by
is
represented
the
And
above
figure!
this figure
was known
to astronomers
long before Einstein.
worried them deeply
It
since they could not account
for the presence of this shift.
And
then
the Einstein theory,
which originated in the attempt
to explain
the Michelson-Morley experiment,*
with the intention
and
NOT AT ALL
of explaining the shift
in the perihelion of Mercury,
QUITE INCIDENTALLY EXPLAINED
THIS DIFFICULTY
for the
ALSO,
presence of the term 3mu'
in
(84)
leads to the additional shift of perihelion
of 42.9" |
!
XXXII.
DEFLECTION OF
We
in
saw
A RAY OF
LIGHT.
the previous chapter
that the experimental
evidence
*See Part
I.:
the Special Theory of Relativity.
fFor the details of the calculation which leads
from (84) to this correction of perihelion shift,
see p. 88 of the
1930
edition of
"The Mathematical Theory of Relativity/
A.
S.
Eddington.
276
1
by
in
connection with
the
shift of
the perihelion of
was already
at
Mercury
hand
when Einstein's theory was proposed,
and immediately served
as a check of the theory.
Let us
now
consider
further experimental verification
of the theory,
but
this
time
the evidence did not precede
but was
PREDICTED BY
the theory.
This
was
in
connection with
the path of a ray of light
as it passes near a large mass
like the sun.
It
will
be remembered
that
according to the Einstein theory
the presence of matter in space
makes the space non-Euclidean
and that the path of anything moving freely
(whether it be a planet
or a ray of light)
will
be along
in that
will
a geodesic
space, and therefore
be affected by the presence
of these obstacles in space.
Whereas,
according to classical physics,
the force of gravitation
could be exerted
only by one mass (say the sun)
upon another mass (say a planet),
but
NOT
upon
a ray of light.
277
Here then was
a definite difference in viewpoint
between the two theories,
and the facts should
decide between them.
For this it was necessary
to observe
what happens to a ray of light
coming from a distant star
as
passes near the sun
it
bent toward the sun,
as predicted by Einstein,
or does it continue on
is
it
in a straight line,
*
as required by classical physics?
Now
to
it is
make
obviously impossible
observation
this
under ordinary circumstances,
we cannot look at a star
since
whose
rays are passing near the sun,
on account of the brightness of the sun
Not only would the star be invisible,
but the glare of the sun
would make
to
look
And
it
in that
so
it
impossible
direction at
all.
was necessary
to wait for a total eclipse,
however,
*lf,
were considered
light
be
a stream of incandescent particles
instead of waves,
to
the sun
WOUL6
have
a gravitational effect upon
a ray of light, even by classical theory,
the
AMOUNT of deflection
calculated even
on
this basis,
DOES NOT AGREE
as
we
shall
show
with experiment,
later (see p.
287).
278
itself:
when
but
the sun
its
glare
so that the
is
up in the sky
hidden by the moon,
is
stars
become
distinctly visible during the day.
Therefore, at the next total eclipse
astronomical expeditions were sent out
to those parts of the world
where the eclipse could be
advantageously observed/
and,
since such an eclipse
lasts only a few seconds,
they had to be prepared
to take photographs of the stars
rapidly and clearly,
so that afterwards,
upon developing the
the positions of the
plates/
stars
could be compared
with their positions in the sky
when the sun is
present.
NOT
279
The following diagram shows
/A
B
F
the path of a ray of light,
from a star, v4 ,
when the sun is
in that part of the sky.
AOE ,
NOT
And,
also,
when the sun
IS present,
and the ray is deflected
and becomes ACF ,
280
so that,
when viewed from F,
the
star
APPEARS to
be
at B.
Thus,
if such photographs
could be successfully obtained,
AND
IF
they showed
that all the stars
in the part of the sky near the sun
were really displaced (as from A to B)
AND
the
IF
MAGNITUDE
of the displacements
agreed with the values
calculated by the theory,
then of course
this
would
constitute
very strong evidence
of
in favor
the Einstein theory.
Let us
now determine
the magnitude of this displacement
as predicted by the Einstein theory:
We
have seen (on page 233)
that
1
in
the "Special Theory of Relativity/
which applies
2
c/s
we now
we get
if
-
in
EUCLIDEAN
-
cW
divide
2
(c/x
this
+
c/y
expression
but
Jx c/y </z
"~T i ~~r i ~~r
space-time,
2
are
281
+
by
2
c/z )/
2
cfc ,
the components of the velocity, v
moving thing (see p. 263),
then obviously the
above quantity in brackets is v2,
and the above equation becomes:
,
of
a
= c
Now
when
1*
the "moving thing
happens to be
a light-ray,
=
then v
and we
c
,
get,
FOR LIGHT,
c/$ = 0.
But what about our
NON-EUCLIDEAN
world,
containing matter?
It
will
be remembered (see
p.
118)
that in studying a
non-Euclidean two-dimensional space
(namely, the surface of a sphere)
in a certain
small region,
we were aided by
the Euclidean plane which
practically coincided with
the given surface in
that small region.
Using the same device for
space of higher dimensions,
we
can,
studying a small region of
NON-Euclidean four-dimensional space-time,
such as our world is,
also utilize the
EUCLIDEAN 4-dimensionaI space-time which
in
282
practically coincides with
in that small region.
And
will
hence
FOR LIGHT even
NON-EUCLIDEAN world.
apply
our
in
it
And
now,
using this result in (71),
together with the condition (or
we
A
on page 261 ,
a geodesic,
shall obtain
A RAY OF
THE PATH OF
LIGHT.
DEFLECTION OF A
LIGHT
(Continued)
XXXIII.
XXX
In chapters XXIX and
we
the condition for a geodesic
given on page
led to (74),
showed
260
which, together with
little g's of (71)
gave us the path of a planet, (84).
the
And
in
now,
order to find
the path of
A RAY OF
we must add
as
we
LIGHT,
the further requirement:
c/s
= 0,
pointed out
in
Chapter XXXII.
in
Substituting c/s
the second equation of (84),
283
RAY OF
that
we
get
which changes the
equation of (84) to
first
which
is
the required
PATH OF
And
by
A RAY OF
LIGHT.
this,
integration*
gives, in rectangular coordinates,
+
for the equation of the curve
2
y
on
page 280.
Now,
since
a (page 280)
is
a very small angle,
the asymptotes of the curve
may be
found by taking y very large by
comparison with x ,
and
so,
neglecting the x terms on the right
in the above formula,
it
becomes
And,
*For details see page
A.
S.
the
1930
90
of
Eddinston's
"The Mathematical Theory of Relativity/'
edition.
284
using the familiar formula for
the angle between two lines
(see any
book on
tan
Analytics):
mi
~~~
__
OL
iii2
f
where a is the desired angle,
and mi and m> are
the slopes of the two lines,
we
get
from which
it is
4/?m
=
tana
4m 2 -fl2
easy to find
.
sin
a
4m
R
+
4m2/R
And, a being
its
value
equal to
so that
in
sin
small,
radian measure
a
is
*
we now have
/O/A
(86)
a
4m
=
R
+ 4mVR
-
Now,
what
is
the actual value of
a
in
the case under discussion,
in
which
R =
the radius of the sun
and
m
is its
mass?
*For the proof of this see
any book on calculus,
or look up a table of
trigonometric functions.
285
Since
and
R = 697,000
m=
kilometers,
1.47 kilometers
4m 2 may be
f
neglected by
comparison with
/?,
so that (86) reduces to the
very simple equation:
4m
from which
we
easily get
a =
In
it
1
.75 seconds.
other words,
was predicted by
the Einstein theory
that,
a ray of light passing near the sun
would be bent into a curve (ACF),
as shown
and that,
in
the figure on p. 280,
consequently
A would
APPEARtobeatB,
a star at
a displacement of
an angle of 1.75 seconds!
If
the reader will stop a
moment
to consider
how small is an angle of
even one DEGREE,
and then consider that
one-sixtieth of that
an angle of one
is
MINUTE,
and again
one-sixtieth of that
is
tSee page 315.
286
an angle of one
he
will realize
SECOND,
how
displacement of
a
small
1
is
.75 seconds!
Furthermore,
according to the Newtonian theory,*
the displacement would be
only half of
And
that!
this
it is
TINY
difference
that must distinguish
between the two
After
all
theories.
the trouble that
the reader has been put to,
to find out the issue,
perhaps he
how
small
is
is
disappointed to learn
the difference
between the predictions of
Newton and Einstein.
And perhaps he thinks that
a decision based on
so small a difference
can scarcely be relied upon!
But we wish to point out to him,
that,
far
from losing his respect and
in scientific
faith
method,
he should,
ON
be
all
THE CONTRARY,
the
more
with
filled
ADMIRATION AND WONDER
to think that
experimental work
IS
in
astronomy
SO ACCURATE
that
*See the footnote on
p.
278.
287
these small quantities* are measured
WIFH PERFECT CONFIDENCE,
and
that
they
DO distinguish
between the two theories and
decide in favor of the
DO
Einstein theory,
as
shown by the
is
following figures:
The
British
to Sobral
gave
expeditions, in 1919,
and Principe,
for this displacement:
1.98"
0.1
2"
and
1.61 "0.30",
respectively;
values which have since
been
confirmed at other eclipses,
as, for example,
the results of Campbell and Trumpler,
who
using
obtained,
two
different cameras,
1.72"
0.11" and 1.82"
the 1922 expedition of the
Lick Observatory.
in
So
that
by now
physicists agree that
the conclusions are
all
beyond
question.
*See also the discrepancy
in
the shift of the perihelion
of Mercury,
on page 275.
288
0.1 5",
We
cannot
refrain,
closing this chapter,
from reminding the reader that
in
191 9 was right after World War
and that
Einstein was then classified dS
a
GERMAN
I,
scientist,
and yet,
the British scientists,
without any of the
stupid racial prejudices then
(and
alas! still)
rampant
in
the world,
went to a great deal of trouble
to equip and send out expeditions
to test a theory
11
an "enemy.
by
OF THE "CRUCIAL
PHENOMENA.
XXXIV. THE THIRD
We
have already seen that
two of the consequences from
the Einstein theory
were completely
verified
by
experiment:
(1)
One, concerning
the shift of
the perihelion of Mercury,
the experimental data for which
was known long before Einstein
BUT NEVER BEFORE EXPLAINED.
And
it
must be remembered
that the Einstein theory
was
289
11
NOT expressly designed
explain
but did
to
this shift,
it
QUITE INCIDENTALLY!
(2) The other, concerning the bending
of a ray of light as it
passes near the sun.
It was never suspected
before Einstein that
a ray of light
when
passing
near the sun
would be bent.*
It
was
for the
first
PREDICTED by
time
this
theory,
and, to everyone's surprise,
was actually
verified
by experiment,
QUANTITATIVELY as well as
QUALITATIVELY (see Chap. XXXIII).
Now there was still another
consequence of this theory which
could be tested experimentally,
according to Einstein.
In order to appreciate
we must
it
say something about spectra.
Everyone probably knows that
if you hang a triangular glass prism
in
a
the sunlight,
band of
different colors,
like a rainbow,
will appear on the wall
where
the light strikes after it has
come through the prism.
The explanation of
this
*But, see the footnote
on
phenomenon
p.
278.
290
is
quite simple,
as
may be seen
from the diagram:
A
When
a
beam
of white light,
strikes the prism
it
SD,
ABC,
NOT continue in
SAME direction, D,
does
the
but
is
bent.*
Furthermore,
if it is
"composite"
like sunlight or
which
is
light,
any other white
composed
light,
of
light of different colors
(or different wave-lengths),
*This
is
bending of
a light ray
called "refraction,"
and has nothing
to
do
with
the bending discussed in Ch. XXXIII.
The reader may look up "refraction"
in any book on elementary physics.
291
each constituent
bends a different amount;
and when these constituents
reach the other side,
BC,
of the prism,
they are bent again,
as shown in the diagram on p. 291,
so that,
by the time they reach the wall,
the colors are all separated out,
XY
as
,
shown,
the light of longest wave-length,
namely, red,
being deflected least.
Hence the rainbow-colored spectrum.
Now,
if
obviously,
the light from 5
is
1
"monochromatic/
that
is,
light of a
SINGLE
wave-length only,
instead of "composite,"
like sunlight,
we have
instead of a "rainbow/*
a single bright line
on
XY
,
having a DEFINITE position,
since the amount of bending,
as
we
said above,
depends upon the color or wave-length
of the light in question.
Now
such monochromatic light
from
may be obtained
the incandescent vapor
of a chemical element
thus sodium, when heated,
burns with a light of
a certain definite wave-length,
characteristic of
sodium.
292
And
This
similarly (or other elements.
is
explained as follows:
The atoms of each element
vibrate with a certain
DEFINITE period
of vibration,
characteristic of that substance,
and, in vibrating, cause
a disturbance in the medium around
this
it,
disturbance being
a light-wave of definite
wave-length
corresponding to
the period of vibration,
thus giving rise to
a
DEFINITE color
which
i
is
visible in
DEFINITE
And
position in the spectrum.
so,
if you look at a
spectrum
you can tell from the bright
just
lines in
it
what substances
are present at 5.
Now
then,
according to Einstein,
since each atom has
a definite period of vibration,
it is
a sort of natural clock
and should serve
a
measure
for
as
an "interval"
c/s.
be
the interval between
the beginning and end of one
and eft the time this takes,
Thus take
c/s
to
vibration,
or the "period" of vibration;
then, using space coordinates
such that
dr=
S=
d<t>
= 0,
293
that
is,
the coordinates of an observer
for
whom
the atom
is
vibrating at
the origin of his space coordinates
(in other words,
an observer traveling with the atom),
equation (71) becomes
<tf
= yd?
7 =
where
1
c/s=
or
2/n
---
Vydt,
(see p. 253).
Now,
if
is
an atom of, say, sodium
vibrating near the sun,
we
should have to substitute
for
m
and
r
the mass and radius of the sun/
and, similarly,
if an atom of the substance
is
vibrating near the earth,
m
and
r
would then have
to
be
the mass and radius of the earth,
and so on:
Thus
7 DEPENDS upon
the location of the atom.
But since ds
is
the space-time interval between
the beginning and end of a vibration,
as
judged by an observer
traveling with the atom,
c/s is
consequently
INDEPENDENT
of the location
of the atom/
then, since
294
obviously
eft
would have
to
be
DEPENDENT UPON THE LOCATION.
Thus,
though sodium from a source
in a laboratory
gives rise to a line in
a definite part of the spectrum,
on the other hand,
sodium on the sun, which,
according to the above reasoning,
would have a
DIFFERENT
period of vibration,
and hence would emit
DIFFERENT
wave-length,
would then give
DIFFERENT
light of a
a bright line in a
part of the spectrum
from that ordinarily due to sodium.
And now
let us
see
HOW MUCH of
a
change
in
the period of vibration
is predicted by the Einstein theory
and whether
by
the
it is
borne out
facts:
If Jt and dt' represent
the periods of vibration near
the sun and the earth,
respectively,
then
or
295
Now
Tearth is
very nearly
1
;
hence
cfc
1
&
V^un
_
~
_
<(
2m
R
J,
m*
_
.
T
.
+
f
!l
m.*
7?
Or, using the values of
and R given on page 286,
m
we
qet
get
This result implies that
an atom of a given substance
should have a
LONGER
slightly
when
when
it
is
it
is
period of vibration
near the sun than
near the earth,
and hence a
LONGER
slightly
wave-length
and therefore
its
lines
should be
a little toward the
SHIFTED
RED end
of the spectrum (see p. 292).
This was a most unexpected result!
and since the amount of shift
was so
slight,
^Neglecting higher powers of A
since
p
is
very small
(see the values of
m and R
on
296
p.
286).
it
made
the experimental verification
difficult.
very
For several years after
Einstein
announced
this result
(1917)
experimental observations on this point
were doubtful,
and this caused many physicists
to doubt the validity of the
Einstein theory,
in spite of its other triumphs,
which we have already discussed.
BUT FIN ALLY,
in
1927,
the very careful measurements
made by Evershed
definitely settled the issue
FAVOR OF THE
IN
EINSTEIN THEORY.
Furthermore,
experiments were performed
similar
Adams
by W.
S.
on the
star
known
as
the companion to Sirius,
which has a relatively
LARGE MASS
thus
and
SMALL
RADIUS,
making the ratio
S7eft
much
1
+r
larger than
the case of the sun (see p. 296)
and therefore easier to observe
in
experimentally.
Here too
the verdict was definitely
IN
So
FAVOR OF THE
that
EINSTEIN THEORY!
to-day
297
physicists are agreed
all
that the Einstein theory
marks a definite step forward
for:
EXPLAINED
PREVIOUSLY KNOWN FACTS
(1) IT
MORE ADEQUATELY THAN
PREVIOUS THEORIES DID
(2) IT
(seep. 103).
EXPLAINED FACTS
NOT EXPLAINED AT ALL
BY PREVIOUS THEORIES
such
(a)
as:
The Michelson-Morley experiment,*
(b) the shift in the perihelion
of Mercury (see Ch. XXXI),
(c) the increase in mass of
an electron
(3) IT
when
in
motion. f
PREDICTED FACTS
NOT PREVIOUSLY KNOWN AT ALLThe bending of a
light ray
passing near the sun (see Ch. XXXII).
(b) The shift of lines in
the spectrum (see p. 296).
(a)
when
(c)
The identity of mass and energy,
which/
in turn,
led to the
ATOMIC BOMB!
(Seep. 318
And
by
ff.)
all this
using
VERY FEW
and
*See Part
I,
"The Special Theory.
tSee Chap. VIII.
298
11
VERY REASONABLE
hypotheses (see p. 97),
in the slightest degree
"far-fetched" or "forced."
not
And
what greater service
can any physical theory
render
than this
!
We
has
trust that
the reader
been led by
book
this little
to have a sufficient insight
into the issues involved,
and to appreciate
the great breadth and
fundamental importance of
THE EINSTEIN THEORY OF RELATIVITY!
XXXV. SUMMARY.
I.
In
the
SPECIAL
Relativity
Theory
was shown that
two different observers,
may, under certain
it
SPECIAL
conditions,
study the universe from their
different points of
view
and yet obtain
SAME LAWS and the SAME
the GENERAL Theory,
the
II.
In
democratic
hold also for
this
result
was found to
ANY two observers,
without regard to the
special conditions mentioned
299
in
I.
FACTS.
III.
To accomplish
this
Einstein introduced the
PRINCIPLE
OF EQUIVALENCE,
by which
the idea of a
FORCE OF GRAVITY
was replaced by
the idea of the
A
CURVATURE OF
SPACE.
IV. The study of this curvature
required the machinery of
the
TENSOR CALCULUS,
by means of which
the
V.
CURVATURE TENSOR
was derived.
This led immediately to
the
NEW LAW OF GRAVITATION
which was tested by
the
THREE CRUCIAL
and found to work
VI.
PHENOMENA
beautifully!
And READ AGAIN
pages 298 and 299
1
300
THE
MORAL
.
-
THE
MORAL
Since man has been
so successful
can
we
in science,
not learn from
THE SCIENTIFIC
WAY OF
what the human mind
and
HOW
There
is
it
achieves
is
THINKING,
capable
of,
SUCCESS:
NOTHING ABSOLUTE
Absolute space and absolute time
have been shown to be myths.
We
must replace these old ideas
by more human,
OBSERVATIONAL
concepts.
303
in science.
c
II.
But what
we observe
is
profoundly influenced by
the state of the observer,
and therefore
various observers get
widely different results
even
in their
measurements of
time and length!
Ill
However,
in spite of these differences,
various observers may still
study the universe
WITH EQUAL RIGHT
AND EQUAL
SUCCESS,
and CAN AGREE on
what are to be called
the
LAWS
of the universe.
305
IV. To accomplish
this
we need
MORE MATHEMATICS
THAN EVER BEFORE,
MODERN, STREAMLINED, POWERFUL
MATHEMATICS.
307
V. Thus
a combination of
PRACTICAL REALISM
(OBSERVATIONALISM)
and
IDEALISM (MATHEMATICS),
TOGETHER
have achieved SUCCESS.
VI.
And
knowing
that the laws are
MAN-MADE,
we know
that
they are subject to change
and we are thus
PREPARED FOR CHANGE,
But these changes
are
NOT
made
in
science
WANTONLY,
BUT CAREFULLY
AND CAUTIOUSLY
the
by
BEST
MINDS and HONEST HEARTS,
and not by any casual child who
thinks that
the world
may be changed
as rolling off a log.
309
as easily
WOULD YOU
LIKE
HOW THE EQUATIONS
TO KNOW?
ON PAGE
(20)
ARE DERIVED:
x
.'.
x
=
=
=
=
a cosfl
-
(x'
x'
-
(x'
fc)
y' tanfl)
cos?
cos0
cos0
y' sinf?.
\f
^+
y^c-fc/^
a
sinfl
COSC^
f
=
y
cosO
+
/
= -+
i
/.
/
=
x' sinfl
(*
/
~
}
a
x smO
-
+ / cos0.
'
y
co$0
61
HOW
THE
FAMOUS MAXWELL
EQUATIONS LOOK:
X Y Z represent the
f
f
of the
components
ELECTRIC FORCE
at a point x , y , z in
an electromagnetic field/
at a
L
,
given instant,
t,
M N represent the components
,
of the
MAGNETIC FORCE
same point and
the same instant.
at the
at
311
III.
HOW TO JUDGE
WHETHER A SET OF QUANTITIES
A
IS
We
TENSOR 9R NOT:
various criteria:
may apply
(1) See if it satisfies any of
the definitions of tensors of
various character
and rank
in (1 6), (1 7), (1 8), etc.,
given
or in (30), (31), etc.
or in (32), etc.
Or
(2) See
if it is
the
sum, difference, or product
two
of
tensors.
Or
(3) See if it satisfies
the following theorem:
A QUANTITY WHICH
ON INNER MULTIPLICATION
BY ANY COVARIANT VECTOR
(OR ANY CONTRAVARIANT VECTOR)
ALWAYS GIVES A TENSOR,
IS ITSELF A TENSOR.
theorem may be
quite easily proved
This
as follows:
Given
that
XA a
is
known
to
be
a contravariant vector,
for
any choice
of
the covariant vector
To prove
Now
X
XA a
that
since
is
Aa /
a tensor:
is
a contravariant vector,
it
must obey (16), thus:
(X Aft) =
(XA a );
312
X
y
=
A's
but
hence, by substitution,
Ayr Af _
/A/9
dx,;
dXfl
'
\
^
dx y ox a
Ary
/i/sA
or
j/
S\0
But
/A^
3x5 3x^
Ay\I
"^
6xT 3xa
/y/
VA
TV~~
_ ^
v,
does not have to be zero,
hence
XI _
which
C'X^
~
'
C/X0
^
ax T oxa
y
A
satisfies (1 7),
thus proving that
X must be a
CONTRAVARIANT TENSOR
OF RANK TWO.
And
similarly for other cases:
Thus if XA = B, y
then X must be a tensor of
the form
/
C^
and
if
then
XA* =
X must
the form
C
be
rp /
a tensor of
a
Barp
,
and so on.
Now
let us
the set of
is
show
that
little g's in
(42)
a tensor:
313
Knowing
that
SCALAR i.e.
A
2
is
c/s
a
TENSOR OF RANK ZERO
(see p. 128),
then
the right-hand
member
of (42)
is
also
A
TENSOR OF RANK ZERO/
but
c/x, is,
by
(1
5) on p.
1
A CONTRAVARIANT
52,
VECTOR,
hence,
by the theorem on page 312,
gMI dx^ must be
,
A COVARIANT
TENSOR
OF RANK ONE.
And, again,
since C/XM is
a contravariant vector,
then,
by the same theorem,
g^ must be
A COVARIANT
TENSOR
OF RANK TWO,
and therefore
it is
with
as
appropriate to write
TWO SUBscripts
we have been doing
in anticipation
of
this proof.
314
it
IV.
WHY MASS CAN
BE
EXPRESSED IN
KILOMETERS:
The reader may be
surprised to
see the mass expressed in kilometers!
But it may seem more reasonable"
from the following considerations:
In order to decide in what units
a quantity
is expressed
14
consider its "dimensionality
terms of the fundamental units of
we must
in
Mass, Length, and Time:
Thus the "dimensionality" of
a velocity
is
L/T;
the "dimensionality" of
2
an acceleration is i/7 ;
and so on.
Now, in Newtonian physics,
the force of attraction which
the sun exerts
being
F
where
m
=
upon the
earth
2
is
fcmm'/f (see p. 219),
the mass of the sun,
m
the mass of the earth,
and r the distance between them;
and
also,
m
F
j,
being the centripetal acceleration
of the earth toward the sun
j
(another one of the fundamental
laws of Newtonian mechanics);
hence
fcmm'
2
or
_ m ,.
j
m = v r2-
i.
Therefore,
315
the "dimensionality" of
,,
L
L
= V
'r
p
m
is
since a constant, like V,
has no "dimensionality/*
And now
we
if
take as a unit of time/
it takes light to go
the time
a distance of
and
M
kilometer
a
one kilometer/
call this unit
n
of time
300/000 kilometers would
equal one second/ since
(thus
light goes 300/000 kilometers
one second)/
we may express
the "dimensionality" of
3
2
L /T or simply /
in
then
thus
we may
mass also
So
far as
in
m
thus:
express
kilometers.
considerations of
"dimensionality" are concerned/
the same result holds true also for
Einsteinian physics.
If
the reader has never before
encountered
this
idea of
"dimensionality"
(which,
by
the way,
important tool
is
a very
in scientific
thinking),
enjoy reading a paper on
"Dimensional Analysis" by
he
Dr.
will
A. N. Lowan,
published by
the Galois Institute of Mathematics
of
Long
Island University in
Brooklyn, N. Y.
316
V,
HOW
Gu LOOKS
IN FULL:
13,3)131,31
r
If
this
mathematics
BE SURE
PAGES
+
BORES you
TO READ
318-323!
317
THE ATOMIC
We
saw on
p.
78
that the
which a body has when
fo
BOMB
energy
at rest,
is:
= me
Thus, the Theory of Relativity
not only that
tells us
mass and energy are one and the same
but that, even though they are the same
still/ what we consider to be
even a SMALL MASS
ENORMOUS when translated
is
into
ENERGY
so that a mass as tiny as an atom
has a tremendous amount of energy
2
the multiplying factor being c ,
the square of the velocity of light!
How
to get at this great storehouse of
energy locked up in atoms
and use it to heat our homes,
to drive our cars and planes,
and so on and so on?
Now,
so long as
m
is
constant,
as for elastic collision,
fo will also remain unchanged.
But, for inelastic collision,
and therefore Fo , will change;
m,
and
this
is
the situation
AN ATOM
when
SPLIT UP, for then
sum of the masses of the parts
is LESS than the mass of the
original atom.
Thus, if one could split atoms,
IS
the
the resulting loss of mass would
release a tremendous amount of energy!
And so various methods were devised
by
scientists like
Meitner, Frisch,
Fermi, and others
318
terms,
"bombard
to
It
was
Finally
11
atoms.
shown
when
that
Uranium atoms were bombarded with neutrons*
these atoms split up ("fission")
into two nearly equal parts,
whose combined mass is less than
the mass of the uranium atom itself,
mass being equivalent,
this loss in
as the Einstein formula shows,
to a
tremendous amount of energy,
thus released
When
by
Einstein
the
fission!
warned President Roosevelt
that such experiments might lead to
the acquisition of
by
the
new
terrific
ENEMY of the
sources of
human
power
race,
the President naturally saw the importance
of having these experiments conducted
where there was some hope that
they would be used to END the war
and to
PREVENT
instead of
by
future wars
those
who
set out to
take over the earth for themselves alone!
Thus the
ATOMIC BOMB
was born
in
And now
the U.S.A.
that a practical
method
of releasing this energy
has
been developed,
the
MORAL
We MUST
is
obvious:
realize that
it
has
become
too dangerous to fool around with
scientific
GADGETS,
WITHOUT UNDERSTANDING
in
the MORALITY which
is
*
Read about these amazing experiments in
"Why Smash Atoms?" by A. K. Solomon,
Harvard University Press, 1940.
319
Science, Art, Mathematics
SAM,
for short.
These are
NOT
mere
idle words.
We
ROOT OUT the
must
AND DANGEROUS
that SAM
amoral
FALSE
DOCTRINE
is
and
is
Good
We
indifferent to
and
Evil.
must
SERIOUSLY EXAMINE SAM
FROM THIS VIEWPOINT.*
Religion has offered us
a Morality,
u
lc
have
wise guys
refused to take it
but
many
seriously,
and have distorted
its
meaning!
And
now, we are
getting
ANOTHER CHANCE-
SAM
is
now
also
warning us that
MUST
UNDERSTAND
we
MORALITY
the
HE now offering us.
And he will not stand
is
our failure to accept
for
it,
by regarding him merely
as
a source of gadgets!
Even using the atomic energy
"peaceful" pursuits,
for
*See our book
"The Education of
IC
Mits"
for a further discussion
of this vital point.
321
which
like heating the furnaces
our homes/
IS
and
in
NOT ENOUGH,
will NOT satisfy SAM.
For he
is
desperately trying
to prevent us from
merely picking
his
pockets
to get at the gadgets in them/
and is begging us to see
the
the True, and
Good/
the Beautiful
which are
in his
mind and
heart,
And, moreover,
he
is
giving
new and
meanings to
clear
these fine old ideas
1
which even the sceptical
11
"wise guys
will find irresistible.
So
DO NOT
BE
AN
ANTI-SAMITE,
or
SAM
will get
you
with his
atomic bombs,
his cyclotrons/
and
all his
new
whatnots.
He
if
is
so anxious to
only
we would
BEFORE
IT
IS
HELP
us
listen
TOO
LATE!
323
SOME
INTERESTING READING:
(1) "The Principle of Relativity" by Albert Einstein and
Published by Methuen and Company,
Others.
London.
(2) The
original
ment:
paper
on the Michelson-Morley experi24 (1887).
Philosophical Magazine, Vol.
(3) "The Theory of Relativity" by R.
Pub. by John Wiley & Sons., N. Y.
D.
Carmichael.
(4) "The Mathematical Theory of Relativity" by A.
Eddington, Cambridge University Press (1930).
(5) "Relativity" by Albert Einstein.
Smith, N. Y. (1931).
(6)
"An
Introduction
L. Bolton.
to
Pub. by
Published by Peter
the Theory
E. P.
S.
Dutton
of
&
Relativity"
Co., N. Y.
by
(7) Articles in the Enc. Brit., 14" ed., on: "Aberration of
Light" by A. S. Eddington, and "Relativity" by
J.
Jeans.
(8) "Relativity Thermodynamics and Cosmology" by
Pub. by Clarendon Press, Oxford.
R, C. Tolman.
(9) "The
Civita.
Absolute
Calculus" by
Son, London.
Differential
Pub. by Blackie
&
(10) "Calculus of Variations" by G.
A.
T.
Bliss.
Levi-
Open
Court Pub. Co., Chicago.
(11) "The
Meaning of Relativity"
Princeton University Press, 1945
by Albert
Einstein.
(12) "The Law of Gravitation in Relativity" by Levinson
and Zeisler. Pub. by Univ. of Chicago Press.
324