Einstein’s
Discovery
of the General Theory of
Relativity
John D. Norton
Department of History and Philosophy of Science
Center for Philosophy of Science
University of Pittsburgh
1
What is general relativity?
2
General Theory of Relativity
Kinematics:
study of motion
Geometry of space
Gravity
3
General Theory of Relativity
Kinematics:
study of
motion
Gravity
Geometry of space
Combined
Gravity
is the
curvature
of the
geometry of
spacetime.
4
General Theory of Relativity
Newton
Einstein
5
Eight
Years
6
Eight Years
1907
1908
Einstein begins investigation
of relativistic theories of
gravity.
Principle of equivalence.
1909
1910
1911
1912
Novel theory of static
gravitational fields.
Speed of light c is the
gravitational potential.
!
Einstein
moves from
Prague to
Zurich and
collaborates
with Marcel
Grossmann.
Gravity is
connected with
the curvature of
spacetime.
First sketch of the
general theory of
relativity.
Field equations are
not generally
covariant.
Completed theory.
Mercury explained.
1913
1914
November
1915
7
Getting
Started
8
9
“I am not able to
acquaint myself with
everything published on
the subject, because the
library is closed during
my free time.
…You would therefore
do me a great favor if
you could bring to my
attention other
publications, if you
know about such.”
Einstein to Johannes Stark,
September 25, 1907.
10
11
Galileo’s Law of Fall
All bodies fall alike…
independently of their
masses.
12
Galileo’s Law of Fall
All bodies fall alike…
independently of their
horizontal motion.
13
What Einstein Found in his simplest
special relativistic theories
Horizontal motion
affects the rate of fall.
But only minutely.
(second order small in v/c)
14
Einstein’s Worry
A hot body falls
differently than a
cold body.
15
“The
Happiest
Thought of
My Life”
16
Principle of Equivalence
Uniform acceleration
in distant space
is
equivalent
to
a homogeneous
gravitational field.
17
18
Light is bent by gravity
Uniform acceleration
in distant space
is
equivalent
to
a homogeneous
gravitational field.
19
Clocks are slowed by gravity
Light from the sun is slightly
red shifted.
20
Light is slowed by gravity
Light velocity c remains isotropic.
Speed is proportional to distance in field.
c is the new gravitational potential.
21
All these effects…
… are recoverable as
“coordinate effects” in
Rindler coordinates in a
Minkowski spacetime.
Make physical by generalization.
First instance of the “gauge argument.”
22
1907-1912
Theory of Static
Gravitational Fields
23
1907
1911
1912
Received 26 February
1912
Received 23 March
24
Generalize properties
of homogenous field to all static fields
Clocks, light slowed by
Gravitational
force on a test
mass m.
Gravitational

field equation

gravitation.
d  xÝ
c
m  m
dt  c 
x
and same for y, z
 1
q2
1 2
c
2 c 2 c 2c
c  2  2  2  kc
x y z
This factor, since c can be
determined only up to a
multiplicative constant.
25
An important
Oops…
force
c  kc
Masses self-accelerate into motion.
Violation of equality of action and
reaction.
force
Requiring that a field stress
tensor is definable forces a
quite definite modification.
Field energy becomes an

1  c c 
c  k
c  2kc  x x 

 m m m 

equal source of gravity.
Field equation is non-linear.
The computation becomes the
Energy density of the
gravitational field.
template for generating the field
equations of the spacetime theories.
26
Pictured geometrically (later perspective!)
ds2 = -dx2 - dy2 - dz2 + c2(x, y, z) dt2
spatial slices are
Euclidean
possible deviations
from Minkowski
spacetime reside here
1 0 0

0


0
1
0
0

g  
0 0 1

0


2
0 0 0 c (x, y, z)

Free falls are geodesics
of the spacetime metric.
27
1912-1913
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Rotating Disk 1912
circumference = p x diameter
measured by laying
out rods end to end.
Extend to frames of
reference in rotation.
circumference
> p x diameter
measured by laying out
moving, contracted rods
end to end.
Geometry found on a rotating disk is non-Euclidean.
29
Ready for Assembly
Minkowski 1907.
Kinematics of
special relativity is
the geometry of a
spacetime.
Gravity bends light,
slows time.
Account of static
gravitational fields.
Gravity (rotation)
produces non-Euclidean
geometry.
Gravity is a
geometrical
curvature of
spacetime.
30
1912-1915
From Physical
to Formal
31
Two approaches
Physical
thinking
physical principles
(relativity, conservation)
special cases
(Newton, static)
physical intuition
(thought experiments)
1907
1908
1909
1910
1911
1912
1913
1914
1915
Formal
thinking
mathematical properties
(covariance)
mathematical theorems
(construct invariants)
mathematical naturalness
(simplicity)
…
32
Marcel Grossmann
33
Essentially the
complete general
theory of relativity
in 1913.
…but the
gravitational field
equations are not
generally covariant.
Over two years of
misery awaits while
Einstein fixes the
problem.
34
The Search for Gravitational Field Equations Begins
Poisson equation of
Newtonian theory
j = 4pk
generalize
Entwurf theory
kQ = G
Stress energy
tensor of nongravitational
matter
Field tensor built
from second and first
derivatives of the
metric tensor g.
35
Grossmann’s mathematical part
Riemann tensor identified as the basic
structure from which gravitational field
equations should be built.
36
Ricci tensor formed
by contraction
“However it turns out that this
tensor does not reduce to the
[Newtonian] expression j in the
special case of infinitely weak
static gravitational fields.”
??!??
37
What went wrong?
Infinitely weak static
gravitational fields.
=
The gravitational fields
of the 1912 theory.
…plus more that historians of science are still debating.
38
Entwurf gravitational
field equations
Restricted covariance of unknown
extent.
Constructed by physical arguments
based on the calculation of 1912.
G
39
Zurich
Notebook
40
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42
64=8x8
65=5x13
64=65
43
Teaching himself
introductory
Minkowskian
four-dimensional
electrodynamics.
44
ds 2  G dx dx
Line element of spacetime
written for the first time.
Special case of static
gravitational field.
Gravitational
field equation of
the 1912 theory.
45
Moving body in
Newtonian
mechanics confined
to a surface traces a
geodesic of the
spatial geometry.
46
“Grossmann tensor
fourth rank”
“should vanish”
47
Einstein gets lost in long
calculations.
“Too involved”
48
Einstein uses the
harmonic coordinate
condition to reduce the
Ricci tensor to a
Newtonian form.
“Result [is] certain.
Holds for coordinates
that satisfy the equation
j=0”
49
“Static special case”
1912 stress tensor of
the gravitational field
“special case [is]
apparently wrong”
50
1913-1914
51
52
Focus on Covariance
preservation of form of equations
under coordinate transformations
Extended
Covariance
for Einstein
Arguments that show that
general covariance is
physically uninteresting.
(“Hole argument.”)
Generalization of the
principle of relativity
Development of variational
methods that show that the
Entwurf theory has the
maximum covariance possible.
53
November
1915
54
November 4
“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 LeviCivita.”
55
November 11
56
November 18
57
November 26
58
"In light of knowledge attained, the
happy achievement seems almost a
matter of course, and any intelligent
student can grasp it without too much
trouble. But the years of anxious
searching in the dark, with their intense
longing, their alterations of confidence
and exhaustion and the final emergence
into the light -- only those who have
experienced it can understand it.”
Notes on the Origin of the General
Theory of Relativity
59
More…
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