1
Einstein's 1912-1913 struggles with Gravitation Theory:
Importance of Static Gravitational Fields Theory
Galina Weinstein
January 31, 2012
In December 1911, Max Abraham published a paper on gravitation at the basis of
which was Albert Einstein's 1911 June conclusion about a relationship between the
velocity of light and the gravitational potential. In February 1912, Einstein published
his work on static gravitational fields, which was based on his 1911 June theory. In
March 1912, Einstein corrected his paper, but Abraham claimed that Einstein
borrowed his equations; however, it was actually Abraham who needed Einstein's
ideas and not the other way round. Einstein thought that Abraham converted to his
theory of static fields while Abraham presumed exactly the opposite. Einstein then
moved to Zurich and switched to new mathematical tools. He examined various
candidates for generally covariant field equations, and already considered the field
equations of his general theory of relativity about three years before he published
them in November 1915. However, he discarded these equations only to return to
them more than three years later. Einstein's 1912 theory of static fields finally led him
to reject the generally covariant field equations and to develop limited generally
covariant field equations.
1. Static Fields and Polemic with Max Abraham
In June 1911 Einstein's published in the Annalen der Physik his paper, "Uber den
Einfluβ der Schwerkraft auf die Ausbreitung des Lichtes" ("On the Influence of
Gravitation on the Propagation of Light"). An important conclusion of this paper is
that the velocity of light in a gravitational field is a function of the place: if it is c0 at
the origin of the coordinates, then at a place with a gravitational potential it is given
by the equation:
c = c0(1 + /c2).1
The above equation signifies that there exists a relationship between the velocity of
light and the gravitational potential; the latter influences the first.
Accordingly, beginning in 1912 Einstein claimed that the velocity of light determined
the field and thus he offered a theory of static fields which violated his own light
postulate from the special theory of relativity,2 and as a consequence "this result
excludes the general validity of the Lorentz transformation; it must not deter us from
further pursuing the chosen path". 3
*
This paper was written while I was a visiting scholar in the Center for Einstein Studies, Boston
University.
2
Einstein's 1911 Annalen paper drew the attention of other scientists to develop their
own gravitation theory. In December 1911, a short time after the publication of
Einstein's 1911 Annalen paper, Max Abraham from Göttingen submitted a paper to an
Italian journal and translated it to German for the Physikalische Zeitschrift, "Zur
Theorie der Gravitation" ("On the Theory of Gravitation"). Abraham formulated his
theory in terms of Hermann Minkowski's four-dimensional space-time formalism and
Einstein's above 1911 relation between the variable velocity of light and the
gravitational potential. 4
Since Minkowski took the constancy of the speed of light to be one of his
fundamental principles, Abraham Pais wrote that Abraham tried the impossible: to
incorporate Einstein's idea of a non-constant light velocity into the special theory of
relativity.5 Eat one's cake and have it too…
In February 1912 Abraham published a Berichtigung (correction) to his paper:6
In lines 16 and 17 of my note 'On the Theory of Gravitation" an oversight has to be
corrected which was brought to my attention by a friendly note from Mr. A. Einstein.
Hence one should read there 'we consider dx, dy, dz, and du = idl = icdt as
components of a displacement ds in four dimensional space".
Abraham then got the following idea as a result of Einstein's correction: 7
"Hence:
ds2 = dx2 + dy2 + dz2 – cdt2
is the square of the four-dimensional line element, where the speed of light c is
determined by eq (6)".
Jürgen Renn explained that, "Abraham had effectively introduced […] the general
four-dimensional line element involving a variable metric tensor. However, for the
time being Abraham's expression remained an isolated mathematical formula without
context and physical meaning which, at this point, was indeed neither provided by
Abraham's nor by Einstein's physical understanding of gravitation."8
From Abraham's above Berichtigung we can infer that Einstein read Abraham's paper
of 1912 on the theory of gravitation, "Zur Theorie der Gravitation"; he corrected it,
and then responded to it by a theory of his own. In February 1912, simultaneously to
Abraham's correction, Einstein submitted his paper, "the Speed of Light and the
Statics of the Gravitational Fields".9
Einstein explained that after he had published his 1911 Annalen paper,10
"Abraham has created a theory of gravitation which renders information and draws
conclusions from my first paper as a special case. But we shall see in what follows
that Abraham's system of equations cannot be reconciled with the equivalence
3
hypothesis, and that his conception of time and space already does not hold up from a
purely formal, mathematical point of view".
On February 20, 1912, Einstein wrote Ludwig Hopf "Abraham's theory is completely
wrong" And Einstein anticipated differences between him and Abraham.11 Einstein
attacked Abraham's theory because of what he considered as the incompatibility
between Abraham's simultaneous implementation of both a variable speed of light and
Minkowski's formalism; and because Abraham's theory could not be reconciled with a
theory based on the equivalence principle.
On February 24, 1912, Einstein wrote the editor of the Annalen, Wilhelm Wien, 12
"I am sending you here a paper for the Annalen. Viewed on it many drops of sweat,
but I now have complete confidence in the matter. Abraham's theory of gravitation is
completely unacceptable. How could anybody be so lucky to guess the correct
equations without any effort! Now I am searching for the dynamics of gravitation. It
will proceed but not so quickly!"
On February 26, 1912 Einstein's paper was received. The "many drops of sweat" were
probably left here: Einstein used his technique from his 1911 Annalen paper of
comparing an accelerated system K(x, y, z, t) and a system of constant gravitational
potential (). The axis coincides with the xaxis and the axis is parallel
to the zaxis. Einstein found that the following equations hold good in system K,
x + act2/2, y, z, ct,
where,
c = c0 + ax.
a/c0 = acceleration of the origin of K with respect to .13
According to the above equation, the equation,
c = 0
is satisfied in .
This is a linear equation that corresponds to Poisson's equation in Newtonian theory:
c = c,
where denotes the universal gravitational constant, and the matter density ( is the
Laplacian operator).
Einstein now wanted to obtain the law of motion of a material point in a static
gravitational field.14 Guided by the equivalence principle, the same equation at which
4
he had arrived for K should hold good for . Einstein thus obtained for , or for a
motion of a material point in a static gravitational field the same form of equation,
= c2
where, K = c, a gravitational constant, and = K/c2 is the universal gravitational
constant as before.15
Einstein derived equations that include the energy of the material point in a stationary
gravitational field. According to special relativity this is related to the mass of the
material point. The kinetic energy cannot be distinguished from the gravitational
energy, and it depends on the mass, the velocity of the material point, and on the
velocity of light. Thus it depends on the gravitational potential. The energy of a point
at rest in the gravitational field is mc.16
In section §3 Einstein reformulated his findings from his 1911 Annalen paper
considering the time, "If we measure time in S1 [lower gravitational potential] with a
clock U, we must measure the time in S2 [higher gravitational potential] with a clock
that goes 1 + /c2 slower than the clock U if you compare it with the clock U in the
same place".17 We can summarize this insight as "gravity bends time".18
And Einstein also realized that since mass and energy are equivalent (different forms
of the same thing), "The energy E1 arriving at S1 is greater than the energy E2,
measured by the same means, which was emitted in S2, and that being the potential
energy of the mass E2/c2 in the gravitational field";19 and "The increase in
gravitational mass is equal to E2/c2, and therefore is equal to the increase in inertial
mass resulting from the theory of relativity".20 Related to this finding is another
finding, the above finding of Einstein that the kinetic energy cannot be distinguished
from the gravitational energy, and it depends on the mass.
On February 29, 1912, Einstein wrote his friend Heinrich Zangger, "During my
research on gravitation I discovered that Abraham's theory (Phys. Zeitschr. No. 1) is
completely untenable".21
On March 11, 1912 Einstein wrote Wien again and asked him to return him back and
not publish the manuscript (submitted on February 24 1912). Einstein very likely
already realized that the linear field equation ∆c = kc should be replaced by a nonlinear field equation. And he wrote Wien that he discovered that not everything in the
paper was proper. 22 He finally decided to let publish this paper, and to immediately
submit another correcting paper. He informed Wien that he was now almost finished
with the static field, and was going to discover the laws of the dynamic field. He did
not forget Abraham, and he wrote Wein, "So einfach, wie Abraham meint, ist die
Angelegenheit aber nicht" (the matter is not as easy as Abraham thinks).23
Sometime between February 1912 and March 1912, Einstein arrived at a small
breakthrough in the theory of the static gravitational field: a non-linear field equation
5
and the realization that the theory is a dynamical theory. Einstein realized that energy
would exert gravitational influence just as mass would. Energy contained within the
gravitational field itself would also exert gravitational influence. Any change in the
strength of the gravitational field would produce an extra variation as the change in
the energy within the field fed back into the system as a whole. In other words,
Einstein's February 1912 linear equations were not consistent with the principle of
action and reaction and the principles of energy and momentum.
On March 20, 1912, Einstein submitted a correcting paper to the Annalen,24 and six
days later on March 26, he wrote to his eternal sounding board, his best friend
Michele Besso, about his new finding. He started his letter to Besso by telling him,
"In recent times, I have been working like a mad on the problem of gravitation.
Finally, I have come to finish with the statics. I do not know anything about the
dynamic field that will follow only now. I write to you a few results".25
Einstein defined a "Fesrsetzung" according to which "While the vel. of light c
depends on the location, however it does not depend on the direction". When c =
const., we arrive at the ordinary theory of relativity. He explained to Besso that his
new theory of gravitation perfectly corresponded with special theory (equivalence of
mass-energy).26
Einstein included the basic electromagnetic equations in his theory of the static
gravitational field. In this case the source of the gravitational field is the density of
ponderable matter augmented with locally measured energy density.27 In his letter to
Besso, and in his March 20 paper on static gravitational fields, in section §3, "Great
thermal and Gravitational Field", Einstein first presented the linear equation for the
static gravitational field from his first paper, but with a little change,
c = for space free of matter, and when there is matter,c = c
where, σ denotes the mass density and the energy density. 28
Then he discovered that the principle of equivalence is valid only locally, 29
"this important first step is therefore difficult because I am departing for it from the
foundations of the unconditional principle of equivalence. It seems that the latter
holds only for infinitely small fields. Our derivation of the equations of motion of the
material point and of the electromagnetic equations is therefore not illusory, since [the
above equations] apply only to infinitely small space".
For the more general case Einstein arrived at a non-linear equation for the static
gravitational field:
cc – 1/2(gradc)2 = c2
The second term of the left hand side of the equation is the energy density of the
gravitational field multiplied by c.30
6
Einstein concluded, 31
"If every energy density (σc) generated a (negative) divergence of the lines of force of
gravitation, then this must also hold for the energy density of gravitation itself. We
write [the above equation] in the form,
c = {c +1/2gradc2 c/c}.
Then one recognizes immediately that the second term in the brackets is the energy
density of the Gravitational field".
Einstein did not forget Abraham; he wrote to Besso in the same letter from March 26,
"Abraham's theory was created of the top of his head, i.e., from mere mathematical
beauty considerations, torn off and completely untenable".32 That was almost
Abraham's opinion of Einstein's theory, except for the mathematical beauty.
On May 17, 1912, Einstein wrote Wien, and he also wrote Zangger three days later,
that Abraham wrote him that he no longer adhered to his own equations, but he
converted to Einstein's theory.33 On June 5, 1912, Einstein again wrote Zangger that
he was engaged in an amusing polemic with Abraham but "Abraham had accepted my
main new results concerning gravitation".34
However, Abraham understood that Einstein converted to his theory. According to
Abraham's understanding, Einstein corrected his February theory because he
borrowed some equations from him. In his June 1912 reply to Einstein, Abraham said
that it would be careless to reject Einstein's results, some of which (expression for the
energy density) were precisely similar to those found in Abraham's theory; results that
Einstein independently formulated by the equivalence hypothesis. 35
Abraham started his attack by saying, "Already a year ago, A. Einstein has given up
the essential postulate of the constancy of the speed of light by accepting the effect of
the gravitational potential on the speed of light, in his earlier theory;36 in a recently
published work [February 1912]37 the requirement of the invariance of the equations
of motion under Lorentz's transformations also falls, and this gives the death blow to
the theory of relativity. Those who repeatedly went after the sirens songs of this
theory should be warned that they might be pleased to note that even its author
himself is now convinced by its inconsistency".38
Abraham's final criticism was of Einstein's March 20 paper.39 Abraham did not like
Einstein's way of arriving at his results, even after the March correction. He did not
like Einstein's use of the "Equivalence Hypothesis", and the correspondence between
reference systems. It appeared to Abraham as a fluctuating basis, because Einstein did
not yet adopt the Space-Time formalism of relativity (that is, he did not formulate his
theory on the basis of Minkowski's space-time formalism). 40
7
On July 4, 1912, Einstein replied to Abraham and explained to him that the theory of
relativity is correct to the extent to which its two underlying principles are accepted.41
"As it stands now", asks Einstein in his reply to Abraham, "what is the limit of the
two principles" of the theory of relativity? Einstein thinks that the principle of the
constancy of the velocity of light can be maintained only insofar as one restricts
oneself to spatio-temporal regions of constant gravitational potential. "This is, in my
opinion, not the limit of validity of the principle of relativity, but is that of the
constancy of the velocity of light, and thus of our current theory of relativity".42
Einstein explained to Abraham, "This situation, in my opinion, by no means implies
the failure of the principle of relativity, just as the discovery and correct interpretation
of Brownian motion did not lead to the consideration of Thermodynamics and
Hydrodynamics as heresies". The present theory of relativity would always retain its
significance as the simplest theory for the important limiting case of spatio-temporal
events in a constant gravitational potential.43
Einstein described his equivalence principle of 1912, it could only apply consistently
to infinitely small spaces, and Einstein added that he knew that it did not supply a
satisfactory basis, "But therein I do not see any reason for also rejecting the
equivalence principle because it applies to the infinitely small, no one can deny that
this principle is a natural extrapolation of the most general experimental propositions
of physics".44
Finally, Einstein answered Abraham's plagiarism blames,45
"[…] this result contradicts the fundamental equations of Abraham's theory […]
Abraham further claims that I had used his expressions for the energy density and the
stresses in a gravitational field. This is not true"; and Einstein briefly demonstrated
why his expression actually contradicted Abraham's premises. According to Einstein's
theory one obtains a certain expression for the energy density in a static gravitational
field, while according to Abraham's theory the expression for the energy density is
completely different.46
On July 25, 1912, Abraham replied, "I cannot understand what sense has Hr.
Einstein's reply, if he gives up the 'equivalence hypothesis'".47 Abraham then showed
that his expression for the energy density in a static field, which followed from his
1912 theory of gravitation, exactly coincided with Einstein's expression for the energy
density in the field.48
On August 16, 1912, Einstein wrote to Ludwig Hopf, "Recently, Abraham – as you
may have seen – slaughtered me along with the theory of relativity in two massive
attacks, and wrote down (phys. Zeitschr.) the only correct theory of gravitation (under
the 'nostrification' of my results) – a stately steed, that lacks three legs! He noted that
the knowledge of the mass of energy comes from – Robert Mayer".49
8
Marcel Grossmann, Einstein's loyal friend from school, the Zürich Polytechnic, now
called Eidgenössische Technische Hochschule (ETH), was the dean of the department
of mathematics and physics of the institute. He assisted Einstein and persuaded his
colleagues to offer Einstein a professorship in the ETH. In winter 1911-1912 the
decision was made, and Einstein left Prague, after he stayed there less than two years.
In July 1912 he returned to Zürich, the place he loved so much, to his youth school,
there he stayed a professor until he left to Berlin in the spring of 1914.
In Zürich Einstein decided to be publicly silent. He did not want to continue the
"amusing polemic" with Abraham. Einstein thus sent a very short note under the title
"Comment on Abraham's Preceding Discussion 'Once Again, Relativity and
Gravitation'", to the Annalen. He wrote that since each of them has presented his own
stand point, he did not think that it was relevant to respond to Abraham's note again.
Einstein asked the reader not to interpret his silence as an agreement.50
2 The Zurich Notebook
Recall that Abraham criticized Einstein for not yet adopting Minkowski's space-time
formalism.51 Einstein understood that Minkowski's formalism was crucial for the
further development of his theory of gravitation; however he applied Minkowski's
formalism once he had recognized that the gravitational field should be described by
the metric tensor field, a mathematical object of ten independent components.
Starting in August 1912, Einstein went through a long odyssey in the search after the
correct form of the field equations of his new theory. Einstein began to collaborate
with his school-mate the mathematician Grossmann on the theory of gravitation, and
Grossmann gradually gave Einstein more and more mathematical tools. The first trial
of Einstein's efforts appear to be documented in a blue bound notebook – known as
the "Zurich Notebook" – comprised of 96 pages, all written in Einstein's hand. The
notebook was found within Einstein's papers after his death. 52
The back cover of the Notebook bears the title "Relativität" in Einstein's hand
(probably an indication that he began his notes at the end). Two pieces of paper were
probably taped later to the front of the notebook by Einstein's secretary, Helen Dukas.
The subject matter of the calculations in the notebook includes statistical physics,
thermodynamics, the basic principles of the four-dimensional representation of
electrodynamics, and the major part of the notebook is gravitation.
The calculations that Einstein had done in the final pages of the Notebook indicate
continues path towards the paper of 1913 written with Grossmann, "Entwurf einer
verallgemeinerten Relativitätstheorie und einer Theorie der Gravitation" ("Outline of
a Generalized Theory of Relativity and of a Theory of Gravitation") – the "Entwurf"
paper.53
Most of the calculations in the Zurich Notebook are extremely sketchy, display a lot
of false starts, and of course come with no explanatory text. The gravitation part of
9
the Notebook consists of two parts beginning from the back and from the front,
respectively. Einstein seemed to have started from the back of the notebook and he
also filled the front of the notebook. Hence the scholars had numbered the pages
according to their understanding of Einstein's proceeding in the notebook. The two
parts of the notebook meet on pages 43LA and 43 LB, which are upside down relative
to one another. The letter "L" signifies "left-hand" pages and the letter "R" – "righthand" pages. Part A was written beginning from the back of the notebook and consists
of pages 32L – 43L. Part B comprises the pages beginning from the front of the
notebook, pages 01L – 31L.
2.1 The Back Side of the Notebook
Minkowski's Line Element
The back side of the notebook begins with doodles and then follows a page that deals
with Minkowski. In this page, numbered 32R, Einstein wrote the four space-time
coordinates x, y, z, ict, and x1, x2, x3, x4.54 On page 32R Einstein wrote "Diesen
Skalar…weider Vierervektor". Einstein considered Minkowski's four-vectors. He
made calculations on "Elektrodynamik" and relativity, calculated and erased for
another ten or more pages,55 and then on page 39L wrote the general four-dimensional
line element:
ds2 = Gdxdx56
This is very likely the first time Einstein has written down this expression, because the
coefficients G of the "metric tensor" are written with an upper case G. After page
40R Einstein switched to the lower case g.57
Gravitational equations and Static Gravitation Theory
In the lower half of page 39L Einstein wanted to find the gravitational equations, how
g , the components of the gravitational field or the metric tensor, are generated by
source masses. There Einstein considers a "Spezialfall für G" – a special case
excepting G44 = c2, the case of a coordinate system in which the metric is flat. If c2 is
constant then this metric represents the Minkowski space-time of special relativity. If
c2 is a function of the spatial coordinates, c2 = c2(x1, x2, x3), it represents the metrical
generalization of Einstein's 1912 theory of static gravitational theory.
At the bottom of the page, Einstein tried to apply the gravitational field equation from
his 1912 theory of static gravitational fields. He expected that in the special case of a
static field, and with an appropriate choice of coordinates, the gravitational field
equations would reduce to his March 20, 1912 nonlinear static field equation:
cc – 1/2(gradc)2 = c2.58
11
Einstein thought the number of gravitational potentials would reduce from ten to a
single potential, but he confronted the more difficult problem, to reverse this
reduction. He wanted to put this equation into malleable form which allowed it to be
one component of the ten component metric tensor field equation. Einstein then
rewrote the left hand-side of the above equation in terms of the components of the
metric tensor of G44 = c2 from the "Spezialfall".59
On the facing page, 39R Einstein continued to examine this approximation but he
failed to find covariant formulation in this direction. He then wrote the following
operation from Minkowski's four-dimensional space-time formalism: Div = 0,
where denoted the metric tensor, and he asked: "Ist dies invariant?" and he
continued to calculate and arrived at the conclusion that it was not invariant.60
A Geodesic Curve and a Mirror Page
Einstein continued with more calculations for a few more pages until page 41R.61 On
page 41R he showed that – using Newtonian equation of motion of a particle which is
not subject to external forces but is constrained to move on a curved surface – the
trajectory traced by the particle in the surface is a geodesic, a curve of shortest
distance.62
Page 43LA is almost an exact mirror image of page 43 LB. Until page 43LA Einstein
was calculating from one side of the notebook. And until page 43 LB he was
calculating from the back of the notebook upside down, and then the two pages meet
in the middle of the notebook. The notebook now proceeds to pages full of
calculations of heat theory and thermal physics.63
2.2 The Front Side of the Notebook
Gravitational Field Affects Matter and Matter Generates Gravitational Fields
Einstein came back to gravitation after a few pages on page 05R, and he started to
calculate from the front of the notebook. He wrote the title: "Gravitation" and by now
Grossmann had probably given Einstein more mathematical tools, because the
mathematical style has changed abruptly as compared to the "beginning" of the
notebook. 64 Einstein was seeking to find the stress-energy tensor for the gravitational
field associated with a gravitation tensor he had already introduced. 65
At the top of page 05R Einstein started with the equations of motion of a point particle
in a metric field from an action principle. The action integral is the proper length of the
particle's worldline. The equations of motion of the point particle are written in the
form of the Euler-Lagrange equations. Einstein then applied this to a cloud of
pressureless dust particles in the presence of a gravitational field: he generalized these
results to expressions for the momentum density and the force density in the case of
11
the cloud of dust, and identified the expression for momentum density as part of the
stress-energy tensor for pressureless dust.
Subsequently Einstein inserted this stress-energy tensor and a similar expression for
the density of force acting on the cloud of dust into his initial equations of motion. On
p. 05R he thus arrived at a candidate for the law of energy-momentum conservation in
the presence of a gravitational field; and he wrote an equation that expresses the
vanishing of the covariant four-divergence of the stress-energy tensor T (The
equation expresses the requirement that the stress-energy tensor has zero divergence).
Thus conservation of energy-momentum of the matter field is satisfied. The equation
is generally covariant.66
In the next pages Einstein had become more sophisticated mathematically, but still
was not acquainted with the Riemann tensor. What was very noticeable was that
Einstein used methods due to Eugenio Beltrami. These calculations could either be
part of the field equations or play a role in their construction, but they did not yet lead
to any promising candidates for the left-hand side of the field equations; they led to
several important techniques, results and ideas that Einstein was able to put to good
use once he learned about the Riemann tensor. He was now going to investigate the
covariance properties of various expressions in order to find generally covariant field
equations. Einstein tried to form invariant quantities from the metric tensor, and he
tried to extract field equations in this way. 67
The "Grossmann Tensor": Riemann Tensor
On page 14L Einstein systematically started to explore the Riemann tensor. At the top
of page 14L Einstein wrote on the left: "Grossmann Tensor vierter Mannigfaltigkeit"
("Grosmann's tensor four-manifold"), and next to it on the right he wrote the fully
covariant form of the Riemann tensor.68 Therefore, Grossman very likely conveyed to
Einstein the knowledge about the Riemann tensor.
This signified a new stage in Einstein's search for gravitational field equations. In the
course of this exploration he considered candidate field equations based on the Ricci
tensor which he would come back to only on November 4, 1915. He explored the
Ricci tensor for a few pages, and on page 26L he rejected his results from the few
pages starting from p. 14L. He finally ended with limited generally covariant field
equations, the "Entwurf" field equations.69
Pais says that the transition to Riemannian geometry must have taken place during
the week prior to August 16, 1912 – and Pais dates this according to the letter that
Einstein had written to Ludwig Hopf, and he adds "these conclusions are in harmony
with my own recollections of a discussion with Einstein […]".70 On August 16, 1912,
Einstein wrote Hopf the following: "With gravitation it is going brilliant. If I am not
deceived, I have now found the most general equations".71 "The most general
equations" are the ones that Einstein wrote on page 14L of the Zurich Notebook.
12
Thus the 14L page of the Zurich Notebook – in which the Riemann tensor made its
first appearance – could very likely be dated to the week prior to August 16, 1912
and sometime around this date. The pages preceding the 14L page – in which
Einstein was searching for candidates – could presumably be dated to the beginning
of August 1912.
Newtonian Limit – "Too Complicated"
Einstein contracted the fourth rank Riemann tensor to form the covariant form of the
second rank Ricci tensor. At the bottom of the page Einstein checked whether the
Ricci tensor reduces in the weak field approximation to the Newtonian limit.
Einstein wrote this in the form of an equation in which three of its four second order
derivative terms vanish in the weak field approximation. He then wrote below the
expression: "Sollte verschwinden" ("should vanish").72
The problem – how to cause the Newtonian limit to appear – was going to bring
Einstein straight to pages 26L and 26R and the limited covariant field equations.
Meanwhile Einstein tried to manipulate the Riemann tensor.73 At some stage he was
frustrated with the mathematics that Grossmann had brought to him and he wrote on
page 17L, "zu umstaendlich" (too complicated).74
Newtonian Limit – the Harmonic Coordinate Condition
Generally covariant equations hold in all coordinate systems, whereas the equations of
Newtonian gravitation theory do not. In the process of recovering the Poisson equation
of Newtonian theory for weak static fields from a generally covariant theory, it is thus
necessary to restrict the set of coordinate systems under consideration. This is
achieved through coordinate conditions that must also be satisfied by the final
solution. On page 19L Einstein then tried to recover the Newtonian limit from his
generally covariant field equations with the harmonic coordinate condition (used to
eliminate unwanted second order derivative terms from the Ricci tensor). 75
Calculations under the heading, "Nochmalige Berechnung des Ebenentensors"
("Repeated Calculations of Surface-tensors"): the Ricci tensor from page 14L again,
and the calculation "bleibt stehen" (terminates) at the top of the page. The field
"Gleichungen" that Einstein got on page 19L in first order approximation were written
again on page 19R:
The left hand side is the core-operator term of the reduced Ricci tensor. The right
hand side gives the covariant stress-energy tensor for pressureless dust multiplied by
the gravitational constant .
13
Einstein constructed the field equations out of the Ricci tensor that satisfy the case of
first order, weak field approximation; Einstein thus recovered the equations of the
Newtonian limit. 76
Energy-Momentum conservation – the "Hertz Condition"
However, as Einstein was examining the generally-covariant Ricci tensor expression
to determine whether a physically acceptable field equation could be extracted from
it, he assumed that he needed an additional condition, not just to recover the Poisson
equation for weak static fields, but also to guarantee that the equations be
compatible with the law of energy-momentum conservation.
On page 19R Einstein checked energy and momentum conservation for the resulting
gravitational field equations in the case of weak fields. But he then discovered a
problem. He began by writing:
"For the first approximation our additional condition is.
This is the harmonic coordinate condition.
Einstein then conjectured that the harmonic coordinate condition "can perhaps be
decomposed into" two extra conditions.
The first condition is called by scholars the "Hertz condition":
because it was later mentioned by Einstein in a letter to Paul Hertz from August 22,
1915.77
This condition is related to the requirement of compatibility of the field equations and
energy-momentum conservation,
And the second condition is related with compatibility of the field equations and the
Newtonian limit, a condition on the trace of the weak field metric,
g
= konst.
Entanglement of the Newtonian Limit and Energy-Momentum Conservation
Einstein wanted to satisfy the energy-momentum conservation and at the same time
to satisfy the Newtonian limit. On page 19L he introduced the harmonic coordinate
condition to satisfy the weak field approximation. However, the Hertz condition was a
14
troublesome condition for the above condition on the trace of the weak field metric:
The trace of the metric tensor being constant was incompatible with the metric tensor
diagonal (– 1, – 1, – 1, c2) of a weak static gravitational field. In addition, it was
incompatible with the field equations with the stress-energy tensor of matter for dust.
On page 19R the combination of the Hertz condition with the harmonic coordinate
condition led to the unacceptable equation:
g
= konst. Energy conservation
was found to hold, but the two conditions into which the harmonic coordinate
condition had been split were not compatible with energy-momentum conservation;
this meant an entanglement between the Newtonian limit and energy-momentum
conservation.78
Einstein thus wrote at the bottom of page 19R, "Beide obige Bedingungen sind
aufrecht zu erhalten" (Both the above conditions must be maintained).79
On page 20L Einstein again wrote the two conditions:
g
The Hertz condition, and the trace of the metric tensor:
kk
= 0,
and erased them by crossed lines, because the combination of these two conditions
caused problems. 80
Einstein used the harmonic and Hertz conditions to eliminate various terms from
equations of broad covariance and looked upon the truncated equations of severely
restricted covariance rather than upon the equations of broad covariance he started
from as candidates for the fundamental field equations of his theory. Since coordinate
conditions used in this manner are ubiquitous in the Zurich Notebook the scholars
introduced a special name for them and called them coordinate restrictions.81
Removing the "Hertz Coordinate Condition"
Einstein then imposed the harmonic coordinate condition to reduce the Ricci tensor to
the d'Alembertian acting on the metric in the weak-field case. The Hertz condition was
added to make sure that the divergence of the stress-energy tensor vanishes in the
weak-field case. But the combining of these two conditions implies that the trace of
the metric has to vanish. Therefore, from the weak-field equations that Einstein
obtained on page 19L (and written again on page 19R):
15
it follows that the stress-energy tensor would be traceless. In order to avoid this
problem Einstein modified these field equations and made their right hand side
traceless. With his new weak-field equations Einstein managed to keep the stressenergy tensor and conservation principle. But this solved only part of the problem
caused by the combination of the harmonic coordinate and the Hertz conditions. It
takes care of the problem that a traceless metric would imply a traceless energymomentum tensor, but it does not address a second problem, that a metric field of the
form Einstein used to represent static fields was not traceless. Einstein thus crossed
out his new field equations and modified again the weak-field equations. The
harmonic coordinate condition was imposed and the Hertz condition was removed.82
Einstein was finally able to extract from the Ricci tensor linearized "gravitational
equations" and recover the Newtonian limit successfully:
gkk = U,
Einstein's above modified weak-field equations had removed the need for the Hertz
condition.
These weak field equations have exactly the same form as the weak-field equations
found in Einstein's final general theory of relativity of November 25, 1915. The lefthand side is the linearized version of the Einstein tensor: R – (1/2)gR. There is no
indication in the notebook that Einstein tried to find the exact equations corresponding
to these weak-field equations.83
Getting Rid of the Harmonic Coordinate Condition
Again the contradiction between the coordinate conditions, which led to the
troublesome additional condition for the linearized metric from page 19L; the latter
was not satisfied by a metric of the form diagonal (– 1, – 1, – 1, c2). On page 20L
Einstein tried to avoid this problem by adding a trace term to the weak field equations,
but then he confronted another problem: the metric of the form diagonal (– 1, – 1, – 1,
c2) was not anymore a solution of these modified weak field equations.
On page 21R Einstein returned to his static gravitational field equation from page 39L,
and wrote "Statischer Spezialfall" ("static special case"). He considered the special
case of the metric of static field. He expected to recover from his new metric tensor his
static gravitational field equation: cc – 1/2(gradc)2 = c2 for the metric of the form
(– 1, – 1, – 1, c2). He wrote, "so müssen im statischen Felde g11, g22 etc.
verschwinden" ("g11, g22 etc must vanish in the static field").84 However, the flat
metric of the form (– 1, – 1, – 1, c2) represents the limit of special relativity, and the
weak field equations no longer allow a solution with a metric of this form.85
16
But before giving up his new field equations Einstein wanted to check again whether
his weak field metric is compatible with Galileo's experimental law of free fall and
the equivalence principle. Einstein arrived at the conclusion that the metric of the
form diagonal (– 1, – 1, – 1, c2) is essential to Galileo's law.
Einstein wrote, "if the force is supposed to vary like the energy, then g11, g22 must
vanish for the static field".86 According to Galileo's experimental law of free fall, all
bodies fall with the same acceleration in a given gravitational field. The gravitational
force is then proportional to the inertial mass. According to special relativity, the
inertial mass is proportional to energy. And therefore the field is represented by a
flat metric of the form diagonal (– 1, – 1, – 1, c2).87
Finally Einstein deleted the calculation of the "Static special case" and he wrote at the
bottom of the page "Spezialfall wahrscheinlich unrichtig" ("special case probably
incorrect)."88 With that Einstein gave up the modified field equations from pages
19L-20L. At this early stage until 1915, he no longer considered field equations from
the Ricci tensor with the help of the harmonic coordinate condition; nor did he
examine modified weak-field equations, what was similar to the Einstein equations of
November 1915 in linearized form: gkk = U.89
The November Tensor
Einstein was not ready to give up his attempt to extract the left hand-side of the field
equations from the Riemann tensor. He took another approach to the problem of
constructing a candidate generally covariant tensor from the Riemann curvature tensor.
On page 22R Einstein wrote the heading "Grossmann". Thus perhaps at the suggestion
of Grossmann, Einstein wrote another form of the Ricci tensor. This time, the Ricci
tensor was in terms of the Christoffel symbols and their derivatives, rather than in
terms of the metric tensor and its derivatives – as it appeared until then in the
notebook. This was a fully covariant Ricci tensor in a form resulting from contraction
of the Riemann tensor:
where,
,
Einstein wrote, "G ein Skalar ist", if G is a scalar, then the first term in the expansion
of the Ricci tensor is itself a tensor of the first rank.
He divided the Ricci tensor into two parts – "Tensor 2. Ranges" (tensor of second
rank) and "Vermutlicher Gravitations-Tensor" (presumed gravitational tensor):
17
The second term in the above equation which Einstein called "presumed gravitation
tensor" is called by scholars the "November tensor":
.
Setting the November tensor equal to the energy-momentum tensor, multiplied by the
gravitational constant , one arrives at the field equations of Einstein's first paper of
November 4, 1915.90
Comeback of the "Hertz Condition"
On page 22L Einstein investigated the behavior of the November tensor. Einstein did
not need any more the harmonic coordinate condition and he could impose the Hertz
coordinate condition to eliminate all unwanted second-order derivative terms.91 He
wrote, "Weitere Umformung des Gravitationstensors" (further rewriting of the tensor
of gravitation). The Hertz condition also ensured that the divergence of the linearized
stress-energy tensor vanished. And so energy-momentum conservation law was
satisfied, "Genugt, wenn
verschwindet" (suffices, if it vanishes).92
"Hertz Condition" Not Necessary
At the bottom half of page 22R Einstein had arrived at a candidate for the left-hand
side of the field equations extracted from the November tensor by imposing the Hertz
condition. 93 But the conservation of energy-momentum could not be satisfied; and so
on the next page 23L he went on to another novel method to eliminate terms with
unwanted second-order derivatives of the metric and by which he could extract the
Newtonian limit from the Riemann tensor. Einstein thus abandoned the Hertz
condition:
sei = 0 ist nicht nötig." (not necessary). 94
Einstein already accumulated coordinate conditions – or coordinate restrictions – to
eliminate the terms from his equations. On pages 19L-23L Einstein extracted
expressions of broad covariance from the Ricci tensor. He then truncated them by
imposing additional conditions on the metric to obtain candidates for the left-hand side
of the field equations that reduce to the Newtonian limit in the case of weak static
fields.95 But this model entangled the Newtonian limit and conservation of
18
momentum-energy. One page after the other in the Zurich Notebook, he turned from
one candidate equation to another to find the suitable left hand side of the field
equations that would be compatible with energy-momentum conservation. The
equations satisfied conservation of energy-momentum in the weak field level, but the
source term – stress-energy of matter – of the gravitational field was incompatible
with what he had obtained for the Newtonian limit.96
Field Equations Found Through Conservation of Energy and Momentum
On page 24R Einstein tried to extract yet another candidate for the left-hand side of the
field equations. He did not extract the candidate from the Ricci tensor while
imposing coordinate conditions. He established field equations while starting from the
requirement of the conservation of momentum and energy. The equations could be
covariant with respect to linear transformations, and they satisfied both the Newtonian
limit and conservation of momentum-energy.
Einstein checked this candidate using the rotation metric. He thought that his
expression vanished for the rotation metric, a necessary condition for the rotation
metric to be a solution of the vacuum field equations. This was a mistake, but Einstein
was to discover this much later (October 1915). His expression vanishing for the
rotation metric could signify to Einstein that the new field equations satisfied the
relativity principle and the equivalence principle. Einstein came to believe that his
expression vanishes for the rotation metric because of a sign error in and97
He would do quite the same sign error and come to a similar belief with respect to the
rotation metric a year later in the "Einstein-Besso manuscript".98 He there checked
whether the rotation metric is a solution of the newly "Entwurf" equations he was
developing. 99
Rather than now making the correction of the error above made, Einstein on pages 25L
and 25R was trying to find a way to recover the new field equations from the
November tensor of page 22R which he wrote again on top of page 25L.100
Connecting Between New Field Equations and the November Tensor
Einstein still hoped to connect his new field equations found through energymomentum considerations to the November tensor of page 22R. On pages 25L-25R,
he explicitly tried to recover field equations along this argument from this tensor. At
some point he came to reject his efforts at recovering his new field equations found
through energy-momentum considerations from the November tensor, and he
abandoned general covariance. He failed to connect the November tensor to his new
field equations of page 24R. Einstein then wrote in the lower left corner of page 25R
"Unmöglich" (impossible).101
The "Entwurf" Field Equations
19
This brought Einstein on the very next pages, 26L and 26R, straight to the field
equations, the "Entwurf" equations, which he also established by the same method:
through energy-momentum considerations. The problem remained whether these
equations were covariant enough to enable extending the principle of relativity for
accelerated motion and to satisfy the equivalence principle.
Under the title "System der Gleichungen für Materie" (System of Equations for
Mater), and "Ableitung der Gravitationsgleichungen" (Derivation of the gravitational
equations) Einstein derived gravitational field equations of limited covariance that
were not derived from the Riemann tensor.102 These equations spread over two
facing pages, 26L and 26R and are displayed with a neatness and order rare among the
other pages of the notebook, suggesting that they were transcribed from another place
(probably from Einstein's and Grossmann's joint 1913 "Entwurf" paper) after the result
was known. Einstein ended his gravitation calculations at page 26R with the left-hand
side of the "Entwurf" gravitation tensor.103
As seen from examining the Zurich Notebook, three years before November 1915,
Einstein had written on page 22R the November tensor, when he considered the Ricci
tensor as a possible candidate for the left hand-hand side of his field equations.
Einstein got so close to his November 1915 breakthrough at the end of 1912, that he
even considered on page 20L another candidate – albeit in a linearized form – which
resembles the final version of the November 25, 1915 field equation of general
relativity [R – (1/2)gR]. Einstein therefore first wrote down a mathematical
expression close to the correct field equation and then discarded it, only to return to it
more than three years later.
Why did Einstein reject in 1912-1913 gravitational field equations of much broader
covariance, only to come back to these field equations in November 1915? Einstein
believed that the special principle of relativity for uniform motion could be
generalized to arbitrary motion if the field equations possessed the mathematical
property of general covariance (that is, a form which remained unchanged under all
coordinate transformations). If the principle of relativity is generalized then the
equivalence principle is satisfied. Accordingly, Einstein examined candidates for
generally covariant field equations.
Einstein's earlier work on static fields led him to conclude (on page 21R) that in the
weak field approximation, the spatial metric of a static gravitational field must be
flat.104 This statement appears to have led him to reject the Ricci tensor of page 22R,
and fall into the trap of "Entwurf" limited generally covariant field equations. The
"Entwurf" gravitational equations were thus incompatible with a general principle of
relativity. Einstein said in the introduction to his November 4, 1915 paper, 105
"For these reasons, I completely lost trust in my established field equations, and
looked for a way to limit the possibilities in a natural manner. Thus I arrived back at
the demand of a broader general covariance for the field equations, from which I
21
parted, though with a heavy heart, three years ago when I worked together with my
friend Grossmann. As a matter of fact, we then have already come quite close to the
solution of the problem given in the following".
Since Einstein thought that the Ricci tensor should reduce in the limit to his static
gravitational field theory from 1912 and then to the Newtonian limit, if the static
spatial metric is flat, then this prevented the Ricci tensor from representing the
gravitational potential of any distribution of matter, static or otherwise.106
John Stachel explained that Einstein attempted to formulate in the best way he could
his physical insights about gravitation and relativity already, and incorporate them in
the equivalence principle. Einstein's attempt was hampered by the absence of the
appropriate mathematical concepts. Until 1912 Einstein lacked the Riemanian
geometry and the tensor calculus as developed by the turn of the century, i.e., based
on the concept of the metric tensor; and after 1912 when he was using these, he then
lacked more advanced mathematical tools (the affine connection) 107; these could be
later responsible for inhibiting him for another few years.108
Stachel concludes that in the absence of the affine approach, more-or-less heuristic
detours through the weak field, fast motion (i.e., special-relativistic) limit followed by
a slow motion approximation basically out of step with the fast-motion approach, had
to be used to "obtain" the desired Newtonian results. 109
Judged from the historical point of view of his time, Einstein did not make a mistake,
because he lacked the appropriate mathematical tools to correctly taking the
Newtonian limit of general relativity. Actually with hindsight the story is more
complicated. What was eventually mere coincidence for Einstein would later turn to
be a consequence derived by new mathematical tools, the affine connection, which
was invented after Einstein had arrived at generally covariant field equations. 110
Indeed after 1916 the mathematical tools were elaborated. Stachel summarizes,111
"Had he known about the connection representation of the inertio-gravitational field,
he would have been able to see that the spatial metric can go to a flat Newtonian limit,
while the Newtonian connection remains non-flat without violating the compatibility
conditions between metric and connection. As it was, […], he was amazed to find that
the spatial metric is non-flat".
2 The "Entwurf" Theory – Zurich 1913
In Einstein and Grossmann's "Entwurf" paper Grossmann wrote the mathematical part
and Einstein wrote the physical part. The paper was first published in 1913 by B. G.
Teubner (Leipzig and Berlin). And then it was reprinted with added "Bemarkungen"
(remark) in the Zeitschrift für Mathematik und Physik in 1914. The "Bemarkungen"
was written by Einstein and contained the well-known "Hole Argument".112
21
Einstein and Grossmann developed a new theory of gravitation which was based on
absolute differential calculus. They first established the system of equations for
material processes when the gravitational field was considered as given. These
equations were covariant with respect to arbitrary substitutions of the space-time
coordinates. After establishing these equations, they went on to establish a system of
equations which were regarded as a generalization of the Poisson equation of
Newton's theory of gravitation. These equations determine the gravitational field,
provided that the material processes are given. In contrast to the equations for material
processes, Einstein and Grossmann could not demonstrate general covariance for the
latter gravitational equations. Namely, their derivation was assumed – in addition to
the conservation laws – only upon the covariance with respect to linear substitutions,
and not upon arbitrary transformations.
Einstein felt that this issue was crucial, because of the equivalence principle. His
theory depended upon this principle: all physical processes in a gravitational field
occur just in the same way as they would without the gravitational field, if one related
them to an appropriately accelerated (three-dimensional) coordinate-system. This
principle was founded upon a fact of experience, that of the equality of inertial and
gravitational masses.
Einstein's desire was that acceleration-transformations – nonlinear transformations –
would become permissible transformations in his theory. In this way transformations
to accelerated frames of reference would be allowed and the theory could generalize
the principle of relativity for uniform motions. Einstein thus understood that it was
desirable to look for gravitational equations that are covariant with respect to arbitrary
transformations.
2.1 The Equivalence Principle
The "Entwurf" paper is divided into two parts: a physical part written by Einstein and
a mathematical part written by Grossmann. Einstein begins the physical part with his
new law of nature: the equality of inertial and gravitational masses. From this Einstein
was led to the hypothesis that, from a physical point of view an (infinitesimally
extended, homogeneous) gravitational field can be completely replaced by a state of
acceleration of the reference system. The law of nature is the equality of inertial and
gravitational masses and the equivalence principle is a hypothesis.113
Einstein explained the "equivalence hypothesis" by using the predecessor of the
elevator thought experiment: an observer enclosed in a box can in no way decide
whether the box is at rest in a static gravitational field, or whether it is in accelerated
motion, maintained by forces acting on the box, in space that is free of gravitational
fields.114
22
2.2 The Static Gravitational Field
In section §1 Einstein presented the equations of motion of the material point in the
static gravitational field. He did not abandon his 1912 static fields theory (section §1
of the "Entwurf" paper), but rather developed it, and it became a starting point and a
limiting case of the new theory, and also a kind of inevitable trap of his theory.
Einstein stated that he had shown in previous papers that the equivalence hypothesis
leads to the consequence that in a static gravitation field the velocity of light c
depends on the gravitational potential. This led him to think that the usual (special)
theory of relativity provides only an approximation to reality. It should apply only in
the limiting case where differences in the gravitational potential in the space-time
region under consideration are not too great. He found that c should not be conceived
as a constant. It rather should be a function of the special coordinates that represent a
measure for the gravitational potential. Einstein thus arrived at equations of motion
for the material point and to the conclusion that in the static case c plays the role of
the gravitational potential.115
2.3 Einstein's Static Field and Mach's Idea
Einstein ended section §1 with the conclusion that the momentum and kinetic energy
are inversely proportional to c. Or, the inertial mass is m/c and independent of the
gravitational potential.116 This conforms to Mach's idea that inertia has its origin in an
interaction between the mass point under consideration and all of the other mass
points. Einstein explained that if other masses are accumulated in the vicinity of the
mass point, the gravitational potential c decreases. And then the quantity m/c
increases which is equal to the inertial mass. In the static fields theory Einstein
presented the predecessor to Mach's principle.117
2.4 The Metric Degenerates in Static Fields to g44 = c2 and Space is Flat
In section §2 Einstein was dealing with correspondence to his static gravitational
fields theory. Einstein planned to generalize the principle of relativity in such a way
that the theory of static gravitational fields from section §1 will be contained in his
new theory presented in this paper as a special case.
Here Einstein was trying to do what he has done in the Zurich Notebook on page
21R.118 And so he concluded that in the general case, the gravitational field is
characterized by ten space-time functions (of the metric tensor); g are functions of
the coordinates (x).119 In the case of the usual (special) theory of relativity this
reduces to g44=c2, where c denotes a constant. And Einstein said that the same
degeneration occurs in the static gravitational field of the kind he considered in
section §1, except that in the latter case, this reduces to a single potential g44=c2,
where g44=c2 is a function of spatial coordinates, x1, x2, x3. 120
2.5 Equations of Motion of a Mass Point in an Arbitrary Gravitational Field
23
Einstein then established the equations of motion of a material point in an arbitrary
gravitational field. He followed his calculations from the Zurich Notebook at the top
of page 05R. There he had started with the equations of motion of a point particle in a
metric field from an action principle.121
In section §1 he dealt with the system K, in which the gravitational field was static.
He presented another space-time system K' in which the gravitational field was
arbitrary. Following Max Planck in 1906, with respect to K', Einstein derived the
equations of motion of the freely moving material point from a variation principle in
its Hamiltonian form:
{∫ds} = {∫(– dx2 – dy2 – dz2 + c2dt2)} = 0
where,
ds2 = gdxdx
After further calculations of the momentum of the material point, Einstein concluded
that the quantities g form a covariant tensor of the second rank, which he called the
"covariant fundamental tensor". This tensor determines the gravitational field. He
further arrived at the results that the momentum and energy of the material point form
together a covariant tensor of the first rank, i.e. a covariant vector. Subsequently he
referred the reader to Grossman's mathematical part for further explanations of this
issue.122
2.6 The "Natural Interval"
Already in the Zurich Notebook on page 05R the equations of motion of the point
particle were written in the form of Euler-Lagrange. The action integral was the
proper length of the particle's worldline. What is a worldline from a physical point of
view? Einstein was looking for physical meaning to the mathematical quantities and
so in Section §3 he dealt with the significance of the fundamental tensor of the g for
the measurement of space and time. He said that from what he laid down in section §2
it is obvious that one can use measuring rods and clocks in much the same way as
one can do in the usual (special) theory of relativity. Einstein thus sought for the
physical meaning – the measurability – of the space-time quantities x1, x2, x3, x4.
Sometime around November 18, 1915, Einstein found out that in general relativity
one could not use measuring rods and clocks in the same way as one would do in
special relativity, because "time and space are deprived of the last trace of objective
reality".123 Thus space and time coordinates have no meaning in general relativity.
In his 1913 "Entwurf" theory Einstein considered two infinitely close space-time
points. ds possesses a physical meaning that is independent of the chosen reference
system. Einstein assumed that ds is the "naturally measured" interval between the
24
two space-time points, or the square of the four-dimensional interval between two
infinitely close space-time points. It is measured by means of a rigid body that is not
accelerated in a system which is introduced by means of linear transformations with
respect to the immediate vicinity system of the point (dx1, dx2, dx3, dx4), and by
means of unit measuring rods and clocks at rest relative to it.
For given dx1, dx2, dx3, dx4, the natural interval that corresponds to these differentials
can be determined only if one knows the quantities g that determine the
gravitational field. Or the gravitational field influences the measuring bodies and
clocks in a determinate manner.
Einstein then concluded that from the fundamental equation (the line element):
ds2 = gdxdx
one sees that in order to fix the physical dimensions of the quantities g and x,
another stipulation is required. ds has the dimensions of length, and so does x and x4
("time"), and he did not ascribe any physical dimension to the quantities g.124
2.7 The Conservation Law of Energy Momentum for Matter
In section §4 Einstein started with continues incoherent masses moving in arbitrary
gravitational fields; the cloud of pressureless dust particles in the presence of a
gravitational field from page 05R of his Zurich Notebook. For this case Einstein
applied the conservation of momentum law. He arrived at a form for the conservation
law of energy momentum for matter, which was already found in on page 05R125:
The first three of these equations = 1, 2, 3) express the momentum law, and the last
one ( = 4) the energy law.126
These equations are covariant with respect to arbitrary substitutions. And Einstein
again referred to Grossmann's mathematical part.127
2.8 The Contravariant Stress-Energy Tensor
Einstein called the tensor the "(contravariant) stress-energy tensor of the material
flow". He then said that he ascribed to the above equation a validity that goes far
beyond the special case of the flow of incoherent masses. The above equation
represents in general the energy balance between the gravitational field and an
arbitrary material process; one had only to substitute for the stress-energy tensor
corresponding to the material system under consideration. Einstein further explained
that the first term in the equation contains the space derivatives of the stresses or of
the density of the energy flow, and the time derivatives of the momentum density or
25
the energy density. The second term is an expression for the effects exerted by the
gravitational field on matter.128
2.9 The Field Equations
Einstein now advanced in section §5 to the differential equations of the gravitational
field itself. Einstein started from his new tensor, , the stress-energy tensor, for the
material processes. He then asked: What differential equations permit us to determine
the quantities gik, that is, the gravitational field? He wanted to find equations from
which he would be able to calculate the quantities gik when the quantities of the
material processes are known.
Einstein sought the generalization of Poisson's equation:
k.
The generalization Einstein was seeking would likely have the form:
,
where is a constant, analogous to the Newtonian gravitation constant G, is
analogous to the source mass-density of the Poisson equation , and is a secondrank contravariant tensor derived from the fundamental metric tensor g by
differential operations.129
In line with the Newton-Poisson equation Einstein now presumed the above equations
would be second order. But then he confronted the problem of being unable to find a
differential expression that is a generalization of and that proves to be a tensor
with respect to arbitrary transformations. And again Einstein referred to Grossmann's
mathematical part for this defect.130
As quoted in extension further below, Grossman explained that the covariant
differential tensor of second rank – the Ricci tensor – would have been the natural
candidate for . But it turns out that in the special case of the infinitely weak, static
gravitational field this tensor does not reduce to the expression .
Einstein said that it cannot be negated that the final, exact equations of gravitation
could be of higher than second order. And then there will be a possibility that the
perfectly exact differential equations of gravitation would be after all covariant with
respect to arbitrary transformations. But given the present state of his knowledge of
the physical properties of the gravitational field, the attempt to discuss such
possibilities would be premature. "Therefore, given the limitation of the second order,
we must forgo establishing gravitational equations that are covariant with respect to
arbitrary transformations. It should be emphasized, incidentally, that we have no
evidence for general covariance of the gravitational equations".131
26
Einstein decided to follow the spirit of Poisson's equation, and to give up searching
for generally covariant field equations, equations of gravitation that are covariant with
respect to arbitrary transformations.
Following the correspondence principle, the Newtonian limit, the field equations are
covariant only with respect to a particular group of transformations, which group was
as yet unknown to Einstein at this stage. But given the usual (special) theory of
relativity, Einstein reasoned that it was naturally to assume that the transformation
group he was seeking also includes the linear transformations. Hence he required that
be a tensor with respect to any or arbitrary linear transformations. Einstein could
now obtain an expression for a covariant tensor of the second rank with respect to the
linear transformations.132
2.10 The Contravariant Form of the Field Equations
Einstein wrote the conservation law for the gravitational field: 133
() is given by:
And,
is the contravariant stress-energy tensor of the gravitational field, which
enters the conservation law for the gravitational field in exactly the same way as the
tensor of the material process enters the conservation law for this process. It is
given by:
134
See page 28L of the Zurich Notebook. 135
Einstein arrived at the (contravariant) form of the gravitational equations:
,
() =
).
The field equations show that the stress-energy tensor
of the gravitational field
acts as a field generator in the same way as the tensor of the material process. 136
Einstein next showed that the conservation laws hold for the matter and the
gravitational field together by including them in the equation: 137
27
2.11 The Covariant Form of the Field Equations
In their covariant form the gravitational equations are: 138
D(g) = tT),
where, tis the covariant stress-energy tensor of the gravitational field:139
and Tis the covariant stress-energy tensor of matter:
D(g) is given by:
Einstein ended section §5 by writing the conservation law of energy momentum for
matter in covariant form:140
2.12 Grossmann's Mathematical Part
Grossmann started his mathematical part with the invariance of the line element:
ds2 = gdxdx
Grossmann said that the mathematical tools concerning this line element are found in
Christoffel's paper from 1869, "On the Transformation of the Homogeneous
differential Forms of the Second Degree".141
In sections §1 to §3 Grossmann gave a preliminary introduction to tensor analysis.142
In section §4 he started with the mathematical supplements to Einstein's physical part
section §4. He first supplied a proof of the covariance of the momentum-energy
equations, which Einstein presented in the physical part. He arrived at a covariant
equation and concluded, "The divergence of the (contravariant) stress-energy tensor
of the material flow, or of the physical process vanishes".143
28
Grossmann next complemented mathematically Einstein's section §5. The problem of
constructing the differential equations of the gravitational field is connected with the
differential tensors that are given with a gravitational field. Grossmann said that the
complete system of these differential tensors (with respect to arbitrary
transformations) goes back to the covariant differential tensor of fourth rank found in
Riemann's and Christoffel's works. He wrote Riemann's differential tensor.144
Subsequently, Grossmann wrote the complementary mathematical argument to
Einstein's physical argument in section §5, 145
"The extraordinary importance of these conceptions for the power of differential
geometry of a line element that is given by its manifold makes it a priori probable that
these general differential tensors may also be of importance for the problem of the
differential equations of a gravitational field. It is possible, in fact, at first to specify a
covariant differential tensor of second rank and second order for Gim to specify which
one could enter into these equations [the Ricci tensor…]
But it turns out that in the special case of the infinitely weak, static gravitational field
this tensor does not reduce to the expression . We must therefore leave open the
question to what extent the general theory of the differential tensors associated with a
gravitational field is connected with the problem of the gravitational equations. Such a
connection would have to exist, provided that the gravitational equations are to permit
arbitrary substations; but in that case, it seems that it would be impossible to find
second-order differential equations. On the other hand, if it were determined that the
gravitational equations permit only a certain group of transformations, then it would
be understandable if one could not manage to provide the differential tensors by the
general theory. As has been explained in the physical part, we are not able to take a
stand on these questions".
Grossmann appeared to have been influenced by Einstein's conception that, in the
weak field approximation, the spatial metric of a static gravitational field must be flat.
This prevented the Ricci tensor from representing the gravitational potential.146
3 The Hole Argument – 1913-1914
3.1 Dissatisfaction with Limited Covariant Field Equations
On August 14, 1913, Einstein was dissatisfied with his "Entwurf" field equations, for
he wrote to Lorentz, 147
"But, unfortunately, this matter hooks me so much, that my confidence in the
admissibility of the theory is still shaky. The Entwurf is satisfactory so far, insofar as
it concerns the effect of the gravitational field on other physical processes. For the
absolute differential calculus permits the setting up of equations here that are
covariant with respect to arbitrary substitutions. The gravitational field (g) seems to
be the skeleton, so to speak, on which everything hangs. But unfortunately, the
29
gravitation equations themselves do not possess the property of general covariance.
Only their covariance with respect to linear transformations is certain. Now, however,
all of the confidence in the theory rests on the conviction that an acceleration of the
reference system is equivalent to a gravitational field. So if not all of the system of
equations of the theory, and thus also the gravitational equations permit other than
linear transformations, then the theory refutes its own starting point; then it stands in
the air".
Two days later Einstein wrote Lorentz and found that assuming the law of momentum
and energy conservation, his gravitational equations are never absolutely covariant, 148
"I also found out yesterday to my greatest satisfaction the opposite to the doubts
regarding the gravitation theory, which appear in my last letter, as well as those
expressed in the paper. The matter seems to me solved as follows. The expression for
the energy principle for matter & gravitation field taken together […] starting out
from this assumption, I set up equations [… the gravitational equations]. Now,
however, a consideration of the general differential operators of the absolute
differential calculus shows that such an equation is never constructed absolutely
covariant. As we thus postulated the existence of such an equation, we tacitly
specialized the choice of the reference system. We restricted ourselves to the use of
such reference systems with respect to which the law of momentum and energy
conservation holds in this form. It appears that if one privileges such reference
systems, then only more general linear transformations remain as the only right
choice.
Hence, in a word: By postulating the conservation law, one arrives at highly
particular choice of the reference system and the admissible substitutions.
Only now, does the theory give me pleasure after this ugly dark spot seems to have
been eliminated".
On September 9, 1913, in Einstein's lecture before the annual meeting of the
Naturforschende Gesellschaft in Frauenfeld, "Physikalische Grundlagen einer
Gravitationstheorie" ("Physical Foundations of a Theory of Gravitation"), Einstein
spoke of "a general consideration",149
"It has been possible to demonstrate by a general consideration that equations that can
be covariant determine the gravitational field with respect to non arbitrary
substitutions. This fundamental knowledge is therefore especially remarkable because
all other physical equations, […] are not covariant with respect to arbitrary, but only
with respect to linear transformations. We will therefore also have to request for the
desired field equations only the covariance with respect to linear transformations.
Added to these considerations, it has been found that we can completely perform
certain equations that must emerge from the equations in the special case as an
approximation of the Poisson equation".150
31
The general consideration could be the explanation presented to Lorentz on August 16
or a new consideration. Einstein ended his paper by saying, 151
"By the theory outlined an epistemological defect is removed, which is inherent not
only to the original theory of relativity, but also to the Galilean mechanics, and had
been emphasized especially by E. Mach. […] It has been shown that in fact the
[gravitational] equations indicate the behavior of the inertial resistance, which we can
denote as the inertia of relativity. This fact is one of the main pillars of the outlined
theory".
Although Einstein felt that he managed for the time being to remove the
epistemological defect, he could not deal with the mathematical defect of general
covariance.
Immediately after Einstein's talk, Grossman presented his talk in the same
Naturforschende Gesellschaft in Frauenfeld, "Mathematische Begriffsbildungen zur
Gravitationstheorie" ("Mathematical Concepts of Gravitational Theory"). Grossman
opened his talk by mentioning the works of Christoffel from 1869, Levi-Civita and
Ricci from 1901, and the developments since then to the theory of invariants and the
new treatments by Minkowski, Sommerfeld and Laue.152 However, Grossman did not
solve the mathematical defect in Einstein's theory: the general covariance problem of
the gravitational field equations.
Two weeks later, on September 23, 1913, Einstein attended the 85th congress of the
German natural Scientists and Physicists in Vienna. There he presented another talk,
"Zum gegenwärtigen Stande des Gravitationsproblems" ("On the Present State of the
Problem of Gravitation") pertaining to his "Entwurf" theory. He also engaged in a
dispute after this talk with scientists who opposed to his theory. A text for this lecture
with the discussion was published in the December volume of the Physikalische
Zeitschrift. Einstein rewrote in section §7 of his Vienna paper the gravitational
equations he had obtained in the "Entwurf" paper. 153 And thus there was little new
under the sun in the Vienna talk. Before presenting these equations he wrote, 154
"The whole problem of gravitation would therefore be solved satisfactorily, if one
were also able to find such equations covariant with respect to any arbitrary
transformations that are satisfied by the quantities g that determine the gravitational
field itself. We have not succeeded in solving that problem in this manner".
But after the words "in this manner" Einstein added the following footnote, which
indicated that he did arrive at some new idea, "2) In the last few days, I have found a
proof that such a generally covariant solution cannot exist at all".155
This footnote appeared in the printed version of the Vienna lecture – published on
December 15, 1913. Hence says Stachel the footnote could be added only later, after
September 1913. If it was added just before December 1913, then a short time before
December 1913, in November 1913, Einstein arrived at an ingenious idea.156
31
Einstein told Ludwig Hopf on November 2, 1913:
"I am now very happy with the gravitation theory. The fact that the gravitational
equations are not generally covariant, which bothered me some time ago, has proved
to be unavoidable; it can easily be proved that a theory with generally covariant
equations cannot exist if it is required that the field be mathematically completely
determined by matter".157
And the proof was the Hole Argument.
3.2 The Hole Argument
"Hole argument" is the English translation of the German phrase "Lochbetrachtung".
Einstein developed this argument against the possibility of generally-covariant
equations for the gravitational field. The argument was first published in the
"Bemarkungen" (remarks"), which forms the addendum to the 1913 "Entwurf"
paper.158
The Hole Argument reappeared again in Einstein's paper, "Prinzipielles zur
verallgemeinerten Relativitätstheorie und Gravitationstheorie" ("On the Foundations
of the Generalized Theory of Relativity and the Theory of Gravitation"), published in
January 1914 in Physikalische Zeitschrift. Einstein was already defensive at this stage,
because his failure to offer generally covariant field equations was a great worry and
embarrassment for him.159 Einstein wrote, " 'Very true', thinks the reader, 'but the fact
the Messrs. Einstein and Grossmann are not able to give the equations for the
gravitational field in generally covariant form is not a sufficient reason for me to
agree to a specialization of the reference system'. But there are two weighty
arguments that justify this step, one of them of logical, the other one of empirical
provenance". The logical argument is the Hole Argument.160
These two versions are the first versions of the Hole Argument.
4 The Einstein-Grossman second "Entwurf" paper
Around March 1914 – Einstein was about to leave Zurich and start his Berlin period.
His collaboration with Grossmann was going to end. Before Einstein left he wrote his
last joint paper with Grossmann that included just another excuse for not presenting
generally covariant field equations, "Kovarianzeigenschaften der Feldgleichungen der
auf die verallgemeinerte Relativitätstheorie gegründeten Gravitationstheorie"
("Covariance Properties of the Field Equations of the Theory of Gravitation Based on
the General Theory of Relativity").161 The paper was published in May 1914 when
Einstein was already in Berlin.
4.1 A New Condition that saves the 'Entwurf" theory
In section §2 of the paper Einstein brought a new "simple consideration" according to
which g that characterize the gravitational field could not be completely determined
32
by generally-covariant equations. Einstein's new argument not only justified the
restricted covariance of the field equations; it even supplied reasoning for why
generally covariant field equations would be unacceptable. 162
Einstein first explained the argument to his close friend Michele Besso on March 10,
1914. Einstein wrote Besso that his novelty in the gravitation theory was the
following. From the gravitation equations:
(1)
And from the conservation law it follows that:
(2)
Einstein wrote these in short: B = 0.
These are 4 third order equations for the g (or ) which can be conceived as the
conditions for the special choice of the reference system. Einstein told Besso that by
means of a simple calculation he can prove that the gravitation equations hold for
every reference system that is adapted to this condition.
Einstein thus concluded, 163
"This shows that there exists acceleration transformations of varied kinds, which
transform the equations to themselves (e.g. also rotation), so that the equivalence
hypothesis is preserved in its original form, even to an unexpectedly large extent".
In section §1 "An Stelle der Gravitationsgleichung (21) bzw. (18) des 'Entwurfes'" of
Einstein and Grossmann's paper, Einstein wrote the above gravitational equations
(1).164 In section §2 he also obtained the second equation (2) appearing in Einstein's
letter to Besso.165
Einstein was so happy that he wrote Besso, "Now I am perfectly satisfied and no
longer doubt the correctness of the whole system, regardless of whether the
observation of the solar eclipse will be successful or not. The logic of the thing is too
evident".
Einstein said in his 1914 joint paper with Grossmann, "the gravitational equations
established by us are generally covariant just to the degree, which is possible under
the condition that the fundamental tensor g is completely determined. It follows in
particular that the gravitational equations are covariant with respect to acceleration
transformations (i.e., nonlinear transformations) of about various kinds".166
33
4.2 Falling Deeper into the Hole
In addition to the above new "simple consideration" brought in section §2 of the
paper, Einstein wrote,167
"We want to show now at first that, completely independent of the gravitational
equations that we established, a complete determination of the fundamental tensor
of the gravitational field at a given by a generally-covariant system of
equations is impossible.
Namely, we can prove that if a solution for the at a given is already known,
then the existence of further solutions can be deduced from the general covariance of
the equations". And then immediately after this sentence Einstein reproduced the Hole
Argument.
After the Hole Argument Einstein wrote, "Having thus recognized that the useable
theory of gravitation requires a necessary specialization of the coordinate system, we
also see that the gravitational equations given by us are based upon special coordinate
system".168
In section §5 Einstein concluded, "the gravitational equations [(1)] are covariant
with respect to all admissible transformations of the coordinate systems, i.e., with
respect to all transformations between coordinate systems which satisfy the conditions
[… (2)]". 169 And the proof for this claim was that, since the conditions B = 0, by
which one restricted the coordinate systems, are direct consequence of the
gravitational equations, therefore the covariance of the equations is far-reaching.170
4.3 Restricting Covariance is the Obvious
In March 1914, after presenting to him equations (1) and (2), Einstein wrote Besso,
"So I am going to live in Dahlem and have a room in Haber's institute […]. At the
moment I do not especially feel like working, for I had to struggle horribly to discover
the above matter. The general theory of invariants appeared only as an impediment.
The direct route proved to be the only feasible one. It is difficult to understand why I
had to grope around so long before I found what was so obvious".171
I wish to thank Prof. John Stachel from the Center for Einstein Studies in Boston University
for sitting with me for many hours discussing special and general relativity and their history.
Almost every day, John came with notes on my draft manuscripts, directed me to books in his
Einstein collection, and gave me copies of his papers on Einstein, which I read with great
interest. I also wish to thank Prof. Alisa Bokulich, Director of the Boston University Center
for History and Philosophy of Science, for her kind assistance while I was a guest of the
Center. Finally I would like to thank her Assistant, Dimitri Constant, without whose advice
and help I would not have been able to get along so well at BU and in Boston in general!
34
Endnotes
1
Einstein, Albert, "Uber den Einfluβ der Schwerkraft auf die Ausbreitung des Lichtes", Annalen der
Physik 35, 1911, pp. 898-908; The Collected Papers of Albert Einstein. Vol. 3: The Swiss Years:
Writings, 1909–1911 (CPAE 3), Klein, Martin J., Kox, A.J., Renn, Jürgen, and Schulmann, Robert
(eds.), Princeton: Princeton University Press, 1993, Doc 23), p. 906.
2
Einstein, Albert (1912b), "Lichtgeschwindigkeit und Statik des Gravitationsfeldes", Annalen der
Physik 38, 1912, pp. 355-369 (The Collected Papers of Albert Einstein Vol. 4: The Swiss Years:
Writings, 1912–1914 (CPAE 4), Klein, Martin J., Kox, A.J., Renn, Jürgen, and Schulmann, Robert
(eds.), Princeton: Princeton University Press, 1995, Doc. 3), p. 360.
3
Einstein, 1912b, p. 355.
4
Abraham, Max (1912a) "Zur Theorie der Gravitation", Physikalische Zeitschrift 13, 1912, pp. 1-4,
"Berichtigung", p. 176; English translation, "On the New theory of Gravitation" and "Correction", in
Renn (ed), 2007, pp. 331-339.
5
Pais, Abraham, Subtle is the Lord. The Science and Life of Albert Einstein, 1982, Oxford: Oxford
University Press, p. 230.
6
Abraham, 1912a, p. 176; Abraham, 1912a, in Renn (ed), 2007, p. 339.
7
Abraham, 1912a, p. 176; Abraham, 1912a, in Renn (ed), 2007, p. 339.
8
Renn, Jürgen, "The Summit Almost Scaled: Max Abraham as a Pioneer of a Relativistic Theory of
Gravitation", in Renn, Jürgen, ed., The Genesis of General Relativity, Vol. 3 Gravitation in the
Twighlight of Classical Physics: Between Mechanics, Field Theory, and Astronomy, 2007, New York,
Berlin: Springer, pp. 305-330.pp. 311-312.
9
Einstein, 1912b.
10
Einstein, 1912b, p. 355.
11
Einstein to Hopf, February 20, 1912, The Collected Papers of Albert Einstein Vol. 5: The Swiss
Years: Correspondence, 1902–1914 (CPAE 5), Klein, Martin J., Kox, A.J., and Schulmann, Robert
(eds.), Princeton: Princeton University Press, 1993, Doc 364.
12
Einstein to Wien, February 24, 1912, CPAE, Vol. 5, Doc 365.
13
Einstein, 1912b, p. 359.
14
Einstein, 1912b, p. 361.
15
Einstein, 1912b, p. 362.
16
Einstein, 1912b, p. 363.
17
Einstein, 1911, p. 906.
18
Einstein, 1912b, pp. 365-366.
19
Einstein, 1911, p. 902.
20
Einstein, 1911, p. 903.
21
Einstein to Zangger, before February 29, 1912, CPAE, Vol. 5, Doc 366.
22
Einstein to Wien, March 11, 1912, CPAE, Vol. 5, Doc 371.
35
23
Einstein to Wien, March 11, 1912, CPAE, Vol. 5, Doc 371.
24
Einstein, Albert (1912c), "Zur Theorie des statischen Gravitationsfeldes", Annalen der Physik 38,
1912, pp. 443-458 (CPAE, Vol 4, Doc. 4).
25
Einstein to Besso, March 26, 1912, CPAE, Vol, 5, letter 377.
26
Einstein to Besso, March 26, 1912, CPAE, Vol, 5, letter 377.
27
Pais, 1982, p. 205.
28
Einstein, 1912c, p. 450-453.
29
Einstein, 1912c, p. 455-456.
30
. Einstein, 1912c, p. 456
31
Einstein, 1912c, p. 457.
32
Einstein to Besso, March 26, 1912, CPAE, Vol. 5, letter 377.
33
Einstein to Wien, May 17, 1912, CPAE, Vol. 5, Doc 395; Einstein to Zangger, May 20, 1912, CPAE,
Vol. 5, Doc 398.
34
Einstein to Zangger, June 5, 1912, CPAE, Vol. 5, Doc 406.
35
Abraham, Max (1912b) "Relativität und Gravitation. Erwiderung auf eine Bemerkung des Herrn. A.
Einstein", Annalen der Physik 38, 1912, pp. 1056-1058; p. 1058.
36
Einstein, 1911.
37
Einstein, 1912b.
38
Abraham, 1912b, p. 1056.
39
Einstein, 1912c, p. 456.
40
Abraham, 1912b, p. 1058.
41
Einstein, Albert (1912e), "Relativität und Gravitation. Erwiderung auf eine Bemerkung von M.
Abraham", Annalen der Physik 38, 1912, pp. 1059-1064 (CPAE, Vol. 4, Doc 8), pp. 1061-1062.
42
Einstein, 1912e, p. 1062.
43
Einstein, 1912e, p. 1063.
44
Einstein, 1912e, p. 1063.
45
Einstein, 1912c.
46
Einstein, 1912e, p. 1064.
47
Abraham, Max, (1912c) "Nochmals Relativität und Gravitation Bemerkungen zu A. Einstein
Erwiderung", Annalen der Physik 39, 1912, pp. 444-448; p. 446.
48
Abraham, 1912c, pp. 447-448.
49
Einstein to Hopf, August 16, 1912, CPAE, Vol. 5, Doc 416.
50
Einstein, Albert, "Bemerkung zu Abraham's vorangehender Auseinandersetzung, 'Nochmals
Relativität und Gravitation'", Annalen der Physik 39, 1912, p. 704.
36
51
Abraham, 1912b, p. 1058.
Vol.1 and 2: The Zurich Notebook Transcription and Facsimile. And: Einstein’s Zurich Notebook,
Commentary and Essays, Janssen, Michel, Renn, Jürgen, Sauer, Tilman, Norton, John, and Stachel,
John, in Renn, ed., The Genesis of General Relativity: Sources and Interpretation: Boston Studies in
the Philosophy of Science, 2007, Springer.
52
The Zurich Notebook was thoroughly and comprehensively analyzed by the major Einstein scholars
and was fully reconstructed. For an analysis of the Zurich Notebook see also
http://www.pitt.edu/~jdnorton/Goodies/Zurich_Notebook/index.html
53
Einstein, Albert, and Grossmann, Marcel, Entwurf einer verallgemeinerten Relativitätstheorie und
einer Theorie der Gravitation I. Physikalischer Teil von Albert Einstein. II. Mathematischer Teil von
Marcel Grossman, 1913, Leipzig and Berlin: B. G. Teubner. Reprinted with added "Bemerkungen",
Zeitschrift für Mathematik und Physik 62, 1914, pp. 225-261.
54
Zurich Notebook transcription and facsimile (2007), Vol. 1, pp. 318-321 (will be written in short
"Zurich Notebook").
55
Zurich Notebook (2007), Vol. 1, pp. 321-345.
56
Zurich Notebook (2007), Vol. 1, pp. 346; 348.
57
Janssen, Renn, Sauer, Norton and Stachel, 2007, in Renn ed (2007), p. 503.
58
Zurich Notebook (2007), Vol. 1, pp. 346; 348.
59
Janssen, Renn, Sauer, Norton and Stachel, 2007, in Renn ed (2007), pp. 504-505.
60
Janssen, Renn, Sauer, Norton and Stachel, 2007, in Renn ed (2007), p. 506; Zurich Notebook (2007),
Vol. 1, pp. 347; 349.
61
Zurich Notebook (2007), Vol. 1, pp. 350-354; 356.
62
CPAE, Doc. 10, editorial note, p. 209; Zurich Notebook (2007), Vol. 1, pp. 355; 357.
63
Zurich Notebook (2007), Vol. 1, pp. 362-365.
64
Zurich Notebook (2007), Vol. 1, pp. 383; 385.
65
Zurich Notebook (2007), Vol. 1, pp. 383; 385.
66
Janssen, Renn, Sauer, Norton and Stachel, 2007, in Renn ed (2007), p. 516; Zurich Notebook (2007),
Vol. 1, pp. 383; 385.
67
Janssen, Renn, Sauer, Norton and Stachel, 2007, in Renn ed (2007), p. 523; Zurich Notebook (2007),
Vol. 1, pp. 386-417.
68
Zurich Notebook (2007), Vol. 1, pp. 418; 420.
69
Janssen, Renn, Sauer, Norton and Stachel, 2007, in Renn ed (2007), p. 603.
70
Pais, 1982, p. 212.
71
Einstein to Hopf, August 16, 1912, CPAE, Vol. 5, Doc. 416.
72
Zurich Notebook (2007), Vol. 1, pp. 418; 420.
73
Zurich Notebook (2007), Vol. 1, pp. 421-429.
37
74
Zurich Notebook (2007), Vol. 1, pp. 430; 432.
75
Zurich Notebook (2007), Vol. 1, pp. 433-437.
76
Janssen, Renn, Sauer, Norton and Stachel, 2007, in Renn ed (2007), p. 627; Zurich Notebook (2007),
Vol. 1, pp. 438-441.
77
Renn, Jürgen and Sauer, Tilman, ''Pathways out of Classical Physics. Einstein's Double Strategy in
his Search for the Gravitational Field Equation'', 2007, in Renn (2007), Vol. 1, pp. 113-312; p. 184;
Einstein to Paul Hertz, August 22, 1915, The Collected Papers of Albert Einstein. Vol. 8: The Berlin
Years: Correspondence, 1914–1918 (CPAE 8), Schulmann, Robert, Kox, A.J., Janssen, Michel, Illy,
Jószef (eds.), Princeton: Princeton University Press, 2002, Doc. 111.
78
Janssen, Renn, Sauer, Norton and Stachel, 2007, in Renn ed (2007), p. 627; Zurich Notebook (2007),
Vol. 1, pp. 439; 441; Renn and Sauer, 2007, in Renn ed (2007), Vol. 1, p. 214.
79
Zurich Notebook (2007), Vol. 1, pp. 439; 441.
80
Zurich Notebook (2007), Vol. 1, pp. 442; 444.
81
Janssen Michel and Renn, Jürgen, "Untying the Knot: How Einstein Found His Way Back to Field
Equations Discarded in the Zurich Notebook", in Renn (2007), Vol. 1, pp. 839-925; Norton, John, "How
Einstein Found His Field Equations: 1912-1915," Historical Studies in the Physical Sciences 14, 1984,
pp. 253-315. Reprinted in D. Howard and J. Stachel (eds.), Einstein and the History of General
Relativity: Einstein Studies Vol. I, Boston: Birkhauser, pp 101-159, p. 254.
82
Janssen, Renn, Sauer, Norton and Stachel, 2007, in Renn ed (2007), p. 633.
83
Janssen, Renn, Sauer, Norton and Stachel, 2007, in Renn ed (2007), p. 634.
84
Zurich Notebook (2007), Vol. 1, pp. 447; 449.
85
Zurich Notebook (2007), Vol. 1, pp. 346; 348.
86
Zurich Notebook (2007), Vol. 1, pp. 447; 449.
87
Janssen, Renn, Sauer, Norton and Stachel, 2007, in Renn ed (2007), p. 641.
88
Zurich Notebook (2007), Vol. 1, pp. 447; 449.
89
Janssen, Renn, Sauer, Norton and Stachel, 2007, in Renn ed (2007), p. 644.
90
Janssen and Renn, 2007 in Renn ed (2007), Vol. 1, p. 853; Norton, 1984, pp. 269-270; Zurich
Notebook (2007), Vol. 1, pp. 451; 453.
91
Renn and Sauer, 2007, in Renn ed (2007), Vol. 1, pp. 219-220.
92
Zurich Notebook (2007), Vol. 1, pp. 450; 452.
93
Janssen, Renn, Sauer, Norton and Stachel, 2007, in Renn ed (2007), p. 652; Zurich Notebook (2007),
Vol. 1, pp. 451; 453.
94
Janssen, Renn, Sauer, Norton and Stachel, 2007, in Renn ed (2007), p. 554; Zurich Notebook (2007),
Vol. 1, pp. 454; 456.
95
96
Janssen, Renn, Sauer, Norton and Stachel, 2007, in Renn ed (2007), p. 497.
Renn and Sauer, 2007, in Renn (2007), Vol. 1, pp. 183-184; pp. 214-215; Zurich Notebook (2007),
Vol. 1, pp. 438-456; 458.
38
97
Janssen, Renn, Sauer, Norton and Stachel, 2007, in Renn ed (2007), pp. 687; Zurich Notebook
(2007), Vol. 1, pp. 459; 461.
98
In June 1913 Besso visited Einstein in Zurich and they both tried to solve the "Entwurf" field
equations to find the perihelion advance of Mercury in the field of a static sun in what is known by the
name, the "Einstein-Besso manuscript". CPAE, Vol. 4, Doc. 14.
99
CPAE, Vol. 4, Doc. 14, p. 41; CPAE, Vol. 4, Doc. 14, note 186, p. 445.
100
Janssen, Renn, Sauer, Norton and Stachel, 2007, in Renn ed (2007), p. 682; CPAE, Vol. 4, Doc. 14,
p. 41.
101
Janssen, Renn, Sauer, Norton and Stachel, 2007, in Renn ed (2007), pp. 681; 683; 704; Zurich
Notebook (2007), Vol. 1, pp. 462-465.
102
Zurich Notebook (2007), Vol. 1, pp. 466; 468.
103
Norton, 1984, p. 270; CPAE, vol. 4, Doc. 10, editorial note, p. 263; Zurich Notebook (2007), Vol. 1,
pp. 466-469.
104
Zurich Notebook (2007), Vol. 1, pp. 447; 449.
105
Einstein, Albert (1915a), "Zur allgemeinen Relativitätstheorie", Königlich Preuβische, Akademie
der Wissenschaften (Berlin). Sitzungsberichte, 1915, pp. 778-786 (The Collected Papers of Albert
Einstein. Vol. 6: The Berlin Years:Writings, 1914–1917 (CPAE 6), Klein, Martin J., Kox, A.J., and
Schulmann, Robert (eds.), Princeton: Princeton University Press, 1996, Doc. 21), p. 778.
106
Stachel, John, "Einstein's Search for General Covariance 1912-1915", in Howard, Don and Stachel
John (eds), Einstein and the History of General Relativity, Einstein Studies, Vol. 1, 1989, Birkhäuser,
pp. 63-100; reprinted in Einstein from ‘B’ to ‘Z’, 2002, Washington D.C.: Birkhauser, pp. 301-338;
pp.304-306.
107
Stachel, John, "the Meaning of General Covariance. The Hole Story", in philosophical problems of
the Internal and external Worlds, Essays of the philosophy of Adolf Grünbaum, Earman, John, Janis,
Allen, I., Massey, Gerald, J., and Rescher, Nicholas (eds), 1993, Pittsburgh: University of
Pittsburgh/Universitätsverlag Konstanz, pp. 134, 136.
A dynamical physical theory purporting to describe events that can be characterized by their positions
in space at certain moments of time employs certain mathematical structures. Some of these structures,
used to characterize the behavior of (ideal) clocks and measuring rods, are called chronometrical and
geometrical structures, respectively, or collectively, the chronogeometrical structures of the theory.
Other structures, used to characterize the behavior of freely falling (ideal, force-free, sphericallysymmetrical, nonrotating) test particles are referred to as affine structures. The ten component metric
tensor encodes all information about the chronogeometrical structure. In general relativity, the affine
structure of space-time is identical with the inertio-gravitational field.
108
Stachel, John, "The First-two Acts", 2002/2007, reprinted in Stachel (2002), pp. 261-292, p. 265.
109
Stachel, John, "The Story of Newstein: Or is Gravity Just Another Pretty Force?", in Renn, Jügen
and Schimmel, Matthias (eds.), The Genesis of General Relativity vol. 4, Gravitation in the Twilight of
Classical Physics: The Promise of Mathematics , 2007, Berlin, Springer, pp. 1041-1078; p. 1058.
110
Stachel, 2007, in Renn (2007), p. 1058.
111
Stachel, 2007, in Renn (2007), p. 1059.
112
Einstein and Grossman, 1913.
113
Einstein and Grossmann, 1913, pp. 3-4.
39
114
Einstein and Grossmann, 1913, p. 3.
115
Einstein and Grossmann, 1913, p. 5.
116
Einstein and Grossmann, 1913, pp. 4-6.
117
Einstein, Albert (1912d), "Gibt es eine Gravitationswirkung, die der elektrodynamischen
Induktionswirkung analog ist?", Vierteljahrsschrift für gerichtliche Medizin und öffentliches
Sanitätswesen 44, pp. 37-40 (CPAE 4, Doc. 7), p. 39.
118
Zurich Notebook (2007), Vol. 1, pp. 447; 449.
119
Einstein and Grossmann, 1913, p. 6.
120
Einstein and Grossmann, 1913, p. 7.
121
Zurich Notebook (2007), Vol. 1, pp. 386-417.
122
Einstein and Grossmann, 1913, pp. 7-8.
123
Einstein, Albert (1915b), "Erklärung der Perihelbewegung des Merkur aus der allgemeinen
Relativitätstheorie", Königlich Preuβische Akademie der Wissenschaften (Berlin). Sitzungsberichte,
1915, pp. 831-839; p. 831 (CPAE 6, Doc 23); Stachel, 1989, in Stachel (2002), p. 232.
124
Einstein and Grossmann, 1913, pp. 8-9.
125
Zurich Notebook (2007), Vol. 1, pp. 383; 385.
126
Einstein and Grossmann, 1913, p. 10.
127
Einstein and Grossmann, 1913, p. 10.
128
Einstein and Grossmann, 1913, p. 11.
129
Einstein and Grossmann, 1913, p. 11.
130
Einstein and Grossmann, 1913, p. 12.
131
Einstein and Grossmann, 1913, p. 12.
132
Einstein and Grossmann, 1913, p. 12.
133
Einstein and Grossmann, 1913, p. 16.
134
Einstein and Grossmann, 1913, p. 15.
135
Zurich Notebook (2007), Vol. 1, pp. 475; 477.
136
Einstein and Grossmann, 1913, pp. 15;17.
137
Einstein and Grossmann, 1913, p. 17.
138
Einstein and Grossmann, 1913, pp. 16-17.
139
Einstein and Grossmann, 1913, pp. 16-17.
140
Einstein and Grossmann, 1913, p. 17.
141
Einstein and Grossmann, 1913, p. 23.
142
Einstein and Grossmann, 1913, pp. 24-34.
143
Einstein and Grossmann, 1913, pp. 34-35.
41
144
Einstein and Grossmann, 1913, p. 35.
145
Einstein and Grossmann, 1913, p. 36.
146
Stachel, 1989, in Stachel (2002), pp.304-306.
147
Einstein to Lorentz, August 14, 1913, CPAE, Vol. 5, Doc. 467.
148
Einstein to Lorentz, August 16, 1913, CPAE, Vol. 5, Doc. 470.
149
Einstein, Albert (1913b), "Physikalische Grundlagen einer Gravitationstheorie", Vierteljahrsschrift
der Naturforschenden Gesellschaft in Zürich, 1914, pp. 284-290 (CPAE, Vol. 4, Doc. 16), p. 289.
150
Einstein, 1913b, p. 288.
151
Einstein, 1913b, p. 290.
152
Grossman, "Mathematische Begriffsbildungen zur Gravitationstheorie", Vierteljahrsschrift der
Naturforschenden Gesellschaft in Zürich, 1914, pp. 291-297; p. 291.
153
Einstein, Albert (1913a), "Zum gegenwärtigen Stande des Gravitationsproblems", Physikalische
Zeitschrift 14, 1913, pp. 1249-1262 (CPAE, Vol. 4, Doc. 17), pp. 1258-1259.
154
Einstein, 1913a, p. 1257.
155
Einstein, 1913a, p. 1257, footnote 2.
156
Stachel, 1989, in Stachel (2002), pp. 308-309.
157
Einstein to Hopf, November 2, 1913, CPAE, Vol. 5, 1913, Doc. 480.
158
Einstein and Grossman, 1913, p. 260; Stachel, 1989, in Stachel (2002), pp. 309-310.
"Let there be in our four dimensional manifold a portion L, in which a "material process" is not
occurring, in which therefore the [the components of the stress-energy tensor] vanish. The
given outside L, as requested by our assumptions, therefore also determines completely everywhere the
[components of the metric tensor] inside L. We now imagine that, instead of the original
coordinates, x, new coordinates x' are introduced of the following type. Outside of L x = x'
everywhere; inside L, however, for at least a part of L and for at least one index , x x'. It is clear
that by means of such a substitution it can be achieved that, at least for a part of L, ' . On the
other hand, ' = everywhere, namely outside of L, because in this region x = x', inside L, but
because for this region, = 0 = '. It follows that in the case considered, if all substitutions are
allowed as justified, namely, to the system of belongs more than one system . So if – as has been
the case in the paper – one maintains the requirement that the should be completely determined by
the , then one is forced to restrict the choice of reference system".
159
Howard, Don and Norton, John, "Out of the Labyrinth? Einstein, Hertz, and the Göttingen Answer to
the Hole Argument" in Earman, John, Janssen, Michel, Norton, John (ed), The Attraction of
Gravitation, New Studies in the History of General Relativity, Einstein Studies Vol 5, 1993, Boston:
Birkhäuser, pp. 30-61; p. 32.
160
Einstein, Albert, "Prinzipielles zur verallgemeinerten Relativitätstheorie und Gravitationstheorie",
Physikalische Zeitschrift 15, 1914, pp. 176-180 (CPAE, Vol. 4, Doc.25), p. 178.
"If the reference system is chosen totally arbitrarily, then the g can by no means be completely
determined by the [stress-energy tensor] T. For imagine that the T and gare given everywhere,
and that all the T vanish in a region of of the four-dimensional space. I can now introduce a new
41
reference system, which agrees completely with the original outside of but is different from it
inside (without a violation of continuity)If one now refers everything to this new reference system,
in which matter is represented by T' and the gravitational field by g', then, even though we do have,
T' = T
everywhere, the equations
g' = g
are certainly not all satisfied inside [footnote 1] This proves the assertion.
If one wants to make it possible for the g (gravitational field) to be completely determined by the T
(matter), then this could only be achieved by restricting the choice of the reference system.
Footnote 1: The equations are thus to be understood, that each of the independent variables x' , on the
left-hand side are to be given the same numerical values as the variables x on the right-hand side".
161
Einstein, Albert and Grossmann, Marcel, "Kovarianzeigenschaften der Feldgleichungen der auf die
verallgemeinerte Relativitätstheorie gegründeten Gravitationstheorie", Zeitschrift für Mathematik und
Physik 63, 1914 pp. 215-225.
162
Einstein and Grossmann, 1914, p. 216.
163
Einstein to Besso, March 10, 1914, CPAE, Vol. 5, Doc. 514.
164
Einstein and Grossmann, 1914, p. 217.
165
Einstein and Grossmann, 1914, p. 219.
166
Einstein and Grossmann, 1914, p. 216.
167
Einstein and Grossmann, 1914, pp. 217-218.
168
Einstein and Grossmann, 1914, p. 218.
169
Einstein and Grossmann, 1914, p. 224.
170
Einstein and Grossmann, 1914, p. 224.
171
Einstein to Besso, March 10, 1914, CPAE, Vol. 5, Doc. 514.
Galina Weinstein