Philosophy
and Einstein's
Discovery of the
Theories of Relativity
John D. Norton
Center for Philosophy of Science and
Department of History and Philosophy of Science
University of Pittsburgh
CARL FRIEDRICH VON WEIZSÄCKER LECTURES
UNIVERSITY OF HAMBURG
June 2010
1
Carl Friedrich von Weizsäcker, The
Structure of Phyics (Aufbau der Physik).
From Preface, 1985.
“…the apparent distractions in my life due to politics and philosophy
only slightly slowed the pace of this work.
Philosophy was indispensable for a philosophically oriented analysis of
physics; attempting to understand Plato, Aristotle, Descartes, Kant,
Frege, Heidegger was no distraction at all from the main topic and hence
entailed no loss of time…”
2
Carl Friedrich von Weizsäcker, The
Structure of Phyics (Aufbau der Physik).
From Preface, 1985.
“When I was nineteen years old, Bohr revealed to me
the philosophical dimensions of physics. He gave me
what I had been looking for in physics. From him I
learned to understand the influence that Socrates must
have exerted of his followers…”
3
Carl Friedrich von Weizsäcker, The
Structure of Phyics (Aufbau der Physik).
From Preface, 1985.
“I have placed the three names Albert Einstein, Niels
Bohr, Werner Heisenberg at the head of of the book.
Einstein was the genius of the century. The theory of
relativity is his work, and it was on his account that
quantum got under way. All younger workers remain
under the spell cast by his insights…”
4
Carl Friedrich von Weizsäcker, The
Structure of Phyics (Aufbau der Physik).
From Preface, 1985.
“For me, the mention of these three names also carries
the personal significance of admiring and affectionate
remembrance. I unfortunately never met Einstein, but
his name was familiar to me by time if was a
schoolboy, and from decade to decade I learned better
to understand his greatness.
5
“…I unfortunately never met Einstein…”
1936
This moment brought to you by P.
Shop.
6
To
Einstein
7
Overview
Mach
8
This Talk
I
II
Einstein’s discovery of special
relativity is decisively advanced by
his reading of philosophy.
Einstein’s discovery of general
relativity converts him to an advocate
of an ancient epistemology.
1905
1915+
9
I
Einstein’s
Discovery of
Special Relativity
1905
10
11
The Final Crisis:
Reconcile the “Apparently Incompatible” …
Principle of Relativity
From ether drift
experiments, observables
in electrodynamics,
classical mechanics.
Equivalence of all inertial
states of motion.
with
Constancy of speed of light
for all inertial observers.
Light Postulate
From Maxwell’s
electrodynamics. All efforts
to alter it had failed.
12
A later recollection…
“Today everyone knows, of course, that all attempts to clarify this paradox
[of chasing the beam of light] satisfactorily were condemned to failure as
long as the axiom of the absolute character of time, or of simultaneity, was
rooted unrecognized in the unconscious. To recognize clearly this axiom and
its arbitrary character already implies the essentials of the solution of the
problem. The type of critical reasoning required for the discovery of
this central point was decisively furthered, in my case, especially by
the reading of David Hume’s and Ernst Mach’s philosophical
writings.”
Albert Einstein, Autobiographical Notes
! Einstein did not mean Hume and
Mach’s analysis of the notions of
space and time specifically…
13
Einstein’s analysis in his 1905 “On the Electrodynamics of Moving Bodies”
(simplified):
The platform observer judges the two flashes to be
simultaneous and the two clocks to be properly synchronized.
The moving observer judges the A flash to happen earlier
and the two clocks not to be properly synchronized.
14
Inertial Observers Find the Same Speed for Light
An observer in relative motion finds
clocks A and B NOT to be properly
synchronized. After correcting,
finds light moves at c.
An observer measures the speed
of a light signal with a rod and
two synchronized clocks, A and
B. Finds light moves at c.
15
The central insight
A view about how concepts should be used
in physical theories.
16
Concepts must be properly grounded in experience...
“After seven years of reflection in vain (1898-1905), the solution
came to me suddenly with the thought that our concepts and laws of
space and time can only claim validity insofar as they stand in a clear
relation to experiences; and that experience could very well lead to
the alteration of the concepts and laws. By a revision of the concept
of simultaneity into a more malleable form, I thus arrived at the
special theory of relativity.”
From a 1924 recording transcribed
by Herneck in 1966.
“The concept [of simultaneity] does not exist for the physicist until he has the
possibility of discovering whether or not it is fulfilled in an actual case.”
A. Einstein, Relativity, §8
“…an illustration which Einstein offered in discussion. Suppose somebody
uses the word ‘hunchback.’ If this concept is to have any clear meaning,
there must be some way of finding out whether or not a man has a hunched
back. If I could conceive of no possibility of reaching such a decision, the
word would have no real meaning for me.”
To Wertheimer in 1916 interview.
17
…so we may purge a priori (absoluteness of simultaneity)
from our concepts.
“The illusion which prevailed prior to the enunciation of the theory of relativity-that, from the point of view of experience the meaning of simultaneity in relation
to spatially distant events and, consequently, that the meaning of physical time is
a priori clear--this illusion had its origin in the fact that in our everyday
experience we can neglect the time of propagation of light. We are accustomed
on this account to fail to differentiate between "simultaneously seen" and
"simultaneously happening"; and, as a result, the difference between time and
local time is blurred.
The lack of definiteness which, from the point of view of its empirical
significance, adheres to the notion of time in classical mechanics was veiled by
the axiomatic representation of space and time as given independently of our
sense experiences. Such a use of notions--independent of the empirical basis
to which they owe their existence--does not necessarily damage science. One
may, however, easily be led into the error of believing that these
notions, whose origin is forgotten, are logically necessary and therefore
unalterable, and this error may constitute a serious danger to the
progress of science.”
Einstein, “Physics and Reality,” 1936.
18
…and Einstein credits Hume and Mach
“Today everyone knows, of course, that all attempts to clarify this paradox
[of chasing the beam of light] satisfactorily were condemned to failure as
long as the axiom of the absolute character of time, or of simultaneity, was
rooted unrecognized in the unconscious. To recognize clearly this axiom and
its arbitrary character already implies the essentials of the solution of the
problem. The type of critical reasoning required for the discovery of
this central point was decisively furthered, in my case, especially by
the reading of David Hume’s and Ernst Mach’s philosophical
writings.”
Albert Einstein, Autobiographical Notes
19
Mach
20
Einstein attributes this view of concepts to Mach
from Einstein’s obituary
for Mach, 1916
Empirical
grounding
of concepts
Dangers
of a priori
“Science is, according to Mach, nothing but the comparison and orderly
arrangement of factually given contents of consciousness, in accord with
certain gradually acquired points of view and methods….
…concepts have meaning only in so far as they can be found in things, just as
they are the points of view according to which these things are organized.
(Analysis of concepts)”
“Concepts that have proven useful in ordering things can easily gain authority
over us such that we forget their worldly origin and take them as immutably
given. They are then rather rubber-stamped as a ‘necessity of thought’ and an
‘a priori given,’ etc. Such errors often make the path of scientific progress
impassable for a long time…”
21
Einstein attributes this view of concepts to Mach
from Einstein’s obituary
for Mach, 1916
Quotes of Mach’s reanalysis of judgments of time
Illustration
(as expressions of dependence upon pendulum oscillations or the Earth’s position) ;
of motion; Newton’s bucket.
Einstein sees
link to
simultaneity
It is not improbable that Mach would have hit upon relativity theory if, in the
time that he was of young and fresh spirit, physicists would already have
been moved by the question of the meaning of the constancy of the speed of
light. In this absence of this stimulation, which follows from MaxwellLorentz electrodynamics, even Mach’s critical urge did not suffice to arouse
a feeling for the necessity of a definition of simultaneity for spatially distant
events.
22
Hume
23
“…and still much more Hume…”
“Your exposition is also quite right that positivism suggested rel. theory,
without requiring it. Also you have correctly seen that this line of
thought was of great influence on my efforts and indeed E. Mach and
still much more Hume, whose treatise on understanding I studied with
eagerness and admiration shortly before finding relativity theory.”
Einstein to Moritz Schlick, Dec 14 1915
on the known reading
list of Einstein’s
Olympia Academy
“treatise on understanding” =
“A Treatise of Human Nature”?
or “An Enquiry concerning Human Understanding”?
24
Hume, A Treatise of Human Nature
Dependence of
concepts on
experience
“…all our simple ideas proceed either mediately or immediately, from
their correspondent impressions.
This then is the first principle I establish in the science of human
nature…”
Book 1, Part 1, Section 1.
Application
to time
“As ‘tis from the disposition of visible and tangible objects we receive
the idea of space, so from the succession of ideas and impressions we
form the idea of time, nor is it possible for time alone ever to make its
appearance, or be taken notice of by the mind.
…time cannot make its appearance to the mind either alone, or attended
with a steady unchangeable object, but is always discover’d by some
perceivable succession of changeable objects.”
Book 1, Part II, Section III.
25
Hume, A Treatise of Human Nature
Inapplicability of
concept without
corresponding
experience
“I know there are some who pretend, that the idea of duration is
applicable in a proper sense to objects, which are perfectly
unchangeable…But to be convinced of its falsehood we need but
reflect on the foregoing conclusion, that the idea of duration is
always deriv’d from a succession of changeable objects, and can
never be convey’d to the mind by any thing stedfast and
unchangeable…
…Ideas always represent the objects or impressions from
which they are deriv’d, and can never without a fiction
represent or be appl’d to any other…”
Book 1, Part II, Section III.
This mode of
analysis is applied
throughout the
Treatise.
We have no idea of substance beyond the collection of
particular qualities. We have no idea of causation beyond
contiguity and succession--no necessary connection.
26
Why Hume more than Mach?
A conjecture: Einstein thought that Mach (but not
Hume) denied the freedom of creation of concepts
exercised by Einstein when he introduced a new
definition of distant simultaneity in 1905?
“I see [Mach’s] weakness in this, that he more or less believed
science to consist in a mere “ordering” of empirical “material”; that
is to say, he did not recognize the freely constructive element in the
formation of concepts. In a way he thought that theories arose
through discoveries and not through inventions. He even went so far
that he regarded “sensations” not only as material which has to be
investigated, but, as it were, as the building blocks of the real
world…”
Einstein to Besso on 6. Jan. 1948
“Hume saw clearly that certain concepts, as for
example that of causality, cannot be deduced
from the material of experience by logical
methods.”
Einstein, Autobiographical Notes
27
Einstein on…
How to Use Concepts in Physical Theories
Concepts must be properly grounded in experience, else they fail to
represent the physically real and are fictional. (From Hume and Mach)
Concepts without proper physical grounding need not be abjured (contrary to
Mach and Hume). They can be retained in a physical theory as long as their
arbitrary character is recognized and in a way that does not unwittingly
introduce false presumptions.
The breakthrough in Einstein’s discovery of special relativity came when he
applied this view to the traditional concept of the simultaneity of distant
events.
28
II
Einstein’s Discovery
of General Relativity:
1907-1915
29
Physical
approach
Based on physical principles with
evident empirical support.
Principle of relativity. Conservation of
energy.
Special weight to secure cases of
clear physical meaning.
Newtonian limit. Static gravitational
fields in GR.
Physical naturalness.
Extreme case: thought
experiments direct
theory choice.
versus
Formal
approach
Exploit formal (usually mathematical)
properties of emerging theory.
Covariance principles. Group structure.
Theory construction via mathematical
theorems.
Geometrical methods assure automatic
covariance.
Formal naturalness.
Extreme case: choose
mathematically simplest
law.
Considerable overlap. Often both are the
same inferences in different guises.
30
31
Einstein’s early distain for higher mathematics in physics
Special relativity, light quantum use only calculus of many variables.
Marked reluctance to adopt Minkowski’s four-dimensional methods. He does not
use them until 1912.
Quip: “I can hardly understant Laue’s book” [1911 textbook on special relativity
that used Minkowski’s methods].
Four-dimensional methods disparaged as “superfluous learnedness.”
32
Abraham’s 1912 theory of gravity…
Abraham’s theory is the
simplest mathematically
delivered by fourdimensional methods.
2 2 2 2



 4
x 2 y2 z 2 u2

Fx  
, etc.
x
u=ict
where c=c()
…Einstein’s idea!

…is condemned by Einstein for its purely formal basis.

“…at the first moment (for 14 days) I
too was totally “bluffed” by the beauty
and simplicity of its formulas.” (To Besso)
“[it] has been created out of thin air, i.e.
out of nothing by considerations of
mathematical beauty, and is completely
untenable.” (To Besso)
“totally untenable” (To Ehrenfest)
“incorrect is every respect” (To Lorentz)
“totally unacceptable” (To Wien)
“totally untenable” (To Zangger)
33
General relativity begins to turn the tide
In 1912, Einstein began work on the precursor to general
relativity, the “Entwurf” theory of 1913 with the
mathematical assistance of Marcel Grossmann, who
introduced Einstein to Ricci and Levi-Civita’s “absolute
differential calculus” (now called tensor calculus).
“I am now working exclusively on the gravitation problem and believe that
I can overcome all difficulties with the help of a mathematician friend of
mine here [Marcel Grossmann]. But one thing is certain: never before
in my life have I toiled any where near as much, and I have gained
enormous respect for mathematics, whose more subtle parts I
considered until now, in my ignorance, as pure luxury. Compared
with this problem, the original theory of relativity is child's play.”
Einstein to Sommerfeld, October 1912
Sommerfeld: edited Minkowski’s papers and wrote
introductory papers on four-dimensional methods.
34
Einstein and Grossmann’s “Entwurf…” 1913
Complete framework of general
theory of relativity. Gravity as
curvature of spacetime geometry.
One thing is missing…
The Einstein equations!
Gik = k (Tik –
(1/2) gik T)
Gik = 0 source free case
Ricci tensor Gik is first contraction
of Riemann curvature tensor Riklm
(Yes--the notation is non-standard.)
35
The “Einstein Equations” are approached…
Riemann curvature tensor
“Christoffel’s four-index-symbol”
Its first contraction as the unique tensor
candidate for inclusion is gravitational
field equations.
“But it turns out that this tensor does not
reduce to the [Newtonian] Dj in the special
case of an infinitely weak, static gravitational
field.”
Einstein and Grossman present gravitational
field equations that are not generally
covariant and have no evident geometrical
meaning.
36
Einstein’s “Zurich Notebook”
A notebook of calculation Einstein
kept while he worked on the
“Entwurf” theory with Grossmann.
Einstein expected the physical and
formal/mathematical approaches to
give the same result.
When he erroneously thought they
did not, he chose
the physical
approach over the formal and
selected equations that would
torment him for over two years.
Einstein worked from both ends.
37
Inside the cover…
38
Einstein connects gravity and curvature of spacetime.
Einstein writes the spacetime
metric for the first time as
ds2 = S Glm dxl dxm
Glm soon becomes glm
Importing of special case of
his 1907-1912 theory in
which a variable c is the
gravitational potential.
First attempts at gravitational
field equations based on
physical reasoning of
1907-1912 theory.
p. 39L
39
The physical approach to energy-momentum conservation…
Equations of motion for a
speck of dust (geodesic)
Expressions for energymomentum density and
four-force density for a
cloud of dust.
Combine: energy-momentum
conservation for dust
gm

1
g


m  0
 ( m m  2 
x
x
m
m m
m

p.5R
Rate of
accumulation
energymomentum
Force
density
40
…and the formal approach to energy-momentum conservation.
Is the conservation law
gm

1
g


m  0
 ( m m  2 
x
x
m
m m
m
of the form
differential 

  0?
 operator 


Check: form
differential metric 



 operator tensor 
It should be 0 or a four-vector.

It vanishes!
Stimmt!
p.5R
41
The formal approach to the gravitational field equations
Einstein writes the Riemann
curvature tensor for the first
time… with Grossmann’s
help.
First contraction formed.
To recover Newtonian
limit, three terms “should
have vanished.”
Following pages: Einstein
shows how to select coordinate
systems so that they do vanish.
p. 14L
42
Failure of the formal approach
Einstein finds multiple
problems with the gravitational
field equations based on the
Riemann curvature tensor.
“Special case [of the 1907-1912
theory] apparently incorrect”
p. 21R
43
“Entwurf” gravitational field equations
Derived from a purely physical approach. Energy-momentum conservation.
44
pp. 26L-R
Einstein’s short-lived methodological moral of 1914
The physical approach is superior to the formal approach.
“At the moment I do not especially feel like working, for I
had to struggle horribly to discover what I described
above. The general theory of invariants was only an
impediment. The direct route proved to be the only
feasible one. It is just difficult to understand why I had to
grope around for so long before I found what was so near
at hand.”
Einstein to Besso, March 1914
45
Einstein snatches triumph from near disaster: Fall 1915.
field equations are wrong and returns
to seek generally covariant equations.
David Hilbert in Göttingen applies
formal methods to general field
equations for Einstein’s theory
... and Einstein knows it.
Communications to the Prussian
Academy:
Communications to the Göttingen
Academy:
Einstein realizes his “Entwurf”
Nov. 4 Almost generally covariant
field equations
Nov. 11 Almost generally
covariant field equations
Nov. 18 Explanation of Mercury’s
perihelion motion
Nov. 20 Something very close
to Einstein’s equations
Nov. 26 Einstein equations
46
Einstein’s new methodological moral
Triumph of formal methods over physical considerations.
“Hardly anyone who has truly
understood it can resist the charm of
this theory; it signifies a real
triumph of the method of the general
differential calculus, founded by
Gauss, Riemann, Christoffel, Ricci
and Levi-Civita.”
Communication to Prussian
Academy of Nov. 4, 1915
“I had already taken into consideration the only
possible generally covariant equations, which
now prove to be the right ones, three years ago
with my friend Grossmann. Only with heavy
hearts did we detach ourselves from them, since
the physical discussion had apparently shown
their incompatibility with Newton's law.”
Einstein to Hilbert Nov 18, 1915
“This time the most obvious was
correct; however Grossmann and I
believed that the conservation laws
would not be satisfied and that Newton's
law would not come out in the first
approximation.”
Einstein to Besso, Dec. 10, 1915
47
Einstein’s manifesto of June 10, 1933
Herbert Spenser Lecture, "On the Methods of
Theoretical Physics," University of Oxford
“If, then, it is true that the axiomatic basis of theoretical physics cannot be extracted from
experience but must be freely invented, can we ever hope to find the right way? Nay,
more, has this right way any existence outside our illusions? Can we hope to be guided
safely by experience at all when there exist theories (such as classical mechanics) which to a
large extent do justice to experience, without getting to the root of the matter?
I answer without hesitation that there is, in my opinion, a right way, and that we are capable of
finding it. Our experience hitherto justifies us in believing that nature is the
realization of the simplest conceivable mathematical ideas. I am convinced that
we can discover by means of purely mathematical constructions the concepts and
the laws connecting them with each other, which furnish the key to the
understanding of natural phenomena.
Experience may suggest the appropriate mathematical concepts, but they most certainly
cannot be deduced from it. Experience remains, of course, the sole criterion of the physical
utility of a mathematical construction. But the creative principle resides in
mathematics. In a certain sense, therefore, I hold it true that pure thought can
grasp reality, as the ancients dreamed.”
48
Conclusion
49
“The reciprocal relationship of epistemology and
science is of noteworthy kind. They are
dependent upon each other.
Epistemology without contact with science
becomes an empty scheme.
Science without epistemology is -- insofar as it is
thinkable at all -- primitive and muddled.”
Einstein, “Autobiographical Notes--Remarks Concerning the Essays Brought
together in this Co-operative Volume." p. 683
50
Read all about it…
51
52
www.pitt.edu/~jdnorton
53
54
philsci-archive.pitt.edu
55
56
Finis
57
Einstein Recalls the Decisive Moment
“Why are these two things inconsistent with each other? I felt that I was
facing an extremely difficult problem. I suspected that Lorentz’s ideas had
to be modified somehow, but spent almost a year on fruitless thoughts.
And I felt that was puzzle not to be easily solved.
But a friend of mine living in living in Bern (Switzerland) [Michele Besso] helped me by
chance. One beautiful day, I visited him and said to him: ‘I presently have a problem that
I have been totally unable to solve. Today I have brought this “struggle” with me.’ We
then had extensive discussions, and suddenly I realized the solution. The very next day, I
visited him again and immediately said to him: ‘Thanks to you, I have completely solved
my problem.”
My solution actually concerned the concept of time. Namely, time cannot be absolutely
defined by itself, and there is an unbreakable connection between time and signal
velocity. Using this idea, I could now resolve the great difficulty that I previously felt.
After I had this inspiration, it took only five weeks to complete what is now known as
the special theory of relativity.”
From a lecture given in Kyoto, Dec. 14, 1922. Notes by Jun
Ishiwara; translation Akira Ukawa; revised John Stachel.
58
Physical approach illustrated
Principle of relativity requires that the electromagnetic
field manifests as different mixtures of magnetic field
B and electric field E according to motion of observer.
Based on Einstein’s
1905 magnet-conductor
thought experiment.
59

Formal approach illustrated
Write Maxwell’s equations using four-vector
F *ik
and six-vector (now antisymmetric second rank
0
k
x
tensor) quantities and operators of Minkowski’s
Fik Fli Fkl
1908 spacetime, geometrical approach.


0
l
k
i
x
x
x
Satisfaction of the principle of relativity is

automatic.

 0

B'
F' ik   z
B' y
 0

B' z
B' y
0
B' x
B' x
0
0
0
Pure magnetic
field
0

0
0
0

Lorentz
transformation
Hyperbolic rotation
in spacetime mixes
E’s and B’s
 0

Bz
Fik  
By
iE
 x

Bz
0
Bx
iE y
By
Bx
0
iE z
iE x 

iE y 
iE z 
0 

Mixed magnetic
and electric field
Frame dependence of decomposition of electromagnetic field is a consequence of
spacetime geometry.
Sign and coordinate conventions after Pauli, Theory of Relativity, p. 78.
60
1902-1904 statistical physics
1905 Brownian motion
1905 Light quantum
1905 Special relativity
1906 Specific heats
1909 Wave particle duality
1907-1915 General relativity
1916 A and B coefficients
1917 Relativististic cosmology
1924-25 Bose-Einstein
statistics
1935 EPR
Five dimensional
unified field 1922-41
Distant parallelism 1928
Bivector fields 1932-33
Formal
Physical
Evolution of Einstein’s approaches
Unified field via nonsymmetric connection 1925- 1955
61
Einstein’s search for unified field theory
“I have learned something else from the theory of gravitation:
no collection of empirical facts however comprehensive can ever lead to
the setting up of such complicated equations [as non-linear field equations
of the unified field]. A theory can be tested by experience, but there is no
way from experience to the construction of a theory. Equations of such
complexity as are the equations of the gravitational field can be found
only through the discovery of a logically simple mathematical
condition that determines the equations completely or almost
completely. Once one has obtained those sufficiently strong formal
conditions, one requires only little knowledge of facts for the construction
of the theory; in the case of the equations of gravitation it is the fourdimensionality and the symmetric tensor as expression for the structure of
space that, together with the invariance with respect to the continuous
transformation group, determine the equations all but completely.”
Autobiographical Notes, 1946
62