# Einstein and the problem of confirmation by previously known evidence: A
comment on Michel Janssen and Jürgen Renn’s _Einstein and the Perihelion
Motion of Mercury_

Galina Weinstein  Reichman University, Herzliya, Israel. The Department of
Philosophy, University of Haifa, Haifa.

###### Abstract

In this paper, I comment on a recent paper by Michel Janssen and Jürgen Renn
([24]). In his published paper of November 18, 1915, Einstein presented a
solution to the problem of the perihelion motion of Mercury and obtained the
correct result of 43 seconds of arc per century. Before 1915, Einstein had
established non-covariant field equations (the _Entwurf_ field equations). But
in 1915, he changed his mind and dropped these equations and was led to
general covariance. In a manuscript written in 1913, Einstein and his best
friend Michele Besso tried to solve the _Entwurf_ field equations to attain
the perihelion advance of Mercury. The end result arrived at by Einstein was
$1821^{{}^{\prime\prime}}$. To make a long story short, the field of a static
Sun produced an advance of the perihelion of Mercury of
$18^{{}^{\prime\prime}}$. Einstein kept quiet about this result and continued
to work on the _Entwurf_ theory. Janssen and Renn ask: If we follow the
prescriptions of Karl Popper, why did Einstein not accept that his _Entwurf_
theory had been falsified? They further ask: Given the undeniable importance
of Besso’s earlier calculations, why did Einstein not invite Besso as a co-
author of his November 1915 paper on the perihelion motion of Mercury? Besso
did not even get an acknowledgment in Einstein’s paper ([24], p. 7, p. 9).
After a brief review of Einstein’s and Besso’s early attempts at calculating
the advance of the perihelion of Mercury, I discuss falsification and
confirmation by previously known evidence and then propose an answer to the
above two questions.

Keywords— Einstein, general relativity, perihelion motion of Mercury, Besso,
Einstein-Besso manuscript

##  1 Introduction

In this paper, I comment on a recent paper by Michel Janssen and Jürgen Renn.
Their paper discusses the _Einstein-Besso manuscript_ and Einstein’s 1915
paper on the perihelion motion of Mercury ([24]). It is not the first time
that Janssen has written on the _Einstein-Besso manuscript_ (for instance,
[21], [22]).

I will first provide a short introduction to the most important steps in
Einstein’s route to his field equations. Starting in 1912, Einstein began
collaborating with his schoolmate, Marcel Grossmann, who was a professor of
mathematics in Zurich. Einstein started to work with tensors and jotted
gravitational equations in a manuscript called the _Zurich Notebook_ , while
Grossmann gradually updated him with new mathematical tools. In 1912, Einstein
wrote a form of the Ricci tensor in terms of the Christoffel symbols and their
derivatives. This was a fully covariant Ricci tensor in a form resulting from
contraction of the Riemann tensor. He considered candidate field equations
with a gravitational tensor constructed from the Ricci tensor. The
gravitational tensor - called by scholars the ”November tensor” - transforms
as a tensor under unimodular transformations. It is called the November tensor
because setting this tensor equal to the stress-energy tensor, multiplied by
the gravitational constant, one arrives at the field equations of Einstein’s
first paper of November 4, 1915 ([11]). In 1912, Einstein hoped he could
extract the Newtonian limit from the November tensor. But he found it
difficult to recognize that the November tensor reduces to the Newtonian
limit. Thus, he finally chose non-covariant field equations.

In March 1913, Einstein and Grossmann wrote a joint paper in which they
established non-covariant field equations through energy-momentum
considerations instead of the November tensor; these equations could also
reduce to the Newtonian limit ([10]). Like Einstein’s general theory of
relativity of November 1915, in the Einstein-Grossmann theory, the
gravitational field was represented by a metric tensor and the action of
gravity on other physical processes was represented in a form which remained
unchanged under all coordinate transformations, i.e. by generally covariant
equations. The only fly in the ointment was that the theory contained field
equations that were not generally covariant. This theory is called by scholars
the Outline theory, and in German, the _Entwurf_ theory. Actually, Einstein
also used this name (Einstein to Lorentz, August 14, 1913, [3], Doc. 467).

Since the Mercury anomaly could not be easily explained in the framework of
Newtonian gravitation theory, it presented a good opportunity to theoretically
check the _Entwurf_ theory of gravitation. So, in June 1913, Einstein’s
closest friend and conﬁdant, the engineer Michele Besso, came to visit him in
Zurich, and they both tried to solve the new _Entwurf_ field equations of
Einstein to attain the perihelion advance of Mercury in the field of the
static Sun. Besso was much more than just an engineer. In 1926 Einstein wrote
that Besso had extraordinary knowledge in pure science (Einstein to Zangger,
December 21, 1926 [5], Doc. 436). Besso had a deep knowledge of physics and
mathematics. He visited Einstein again in August 1913. During this time,
Einstein and Besso continued to work on the project. Their joint work consists
the _Einstein-Besso manuscript_ , which contains calculations with little
explanations.

During 1913-1914, Einstein elaborated the _Entwurf_ theory to the point that
its field equations were valid for adapted coordinate systems and he proposed
a few versions of the hole argument against general covariance.

Around October-November 1915, Einstein changed his mind and dropped the
_Entwurf_ field equations. He postulated that his gravitational tensor should
be invariant under unimodular transformations. This finally led him to general
covariance. The long and devious route Einstein took from 1912 until the above
step finally led him back to his starting point, namely to the November tensor
of 1912. Einstein gradually expanded the range of the covariance of his
gravitational field equations and on November 18, 1915 he presented a solution
of the problem of the perihelion motion of Mercury, i.e. he obtained the
correct result of 43 seconds of arc per century. All in all Einstein published
a series of four papers in November 1915 (see [29], chapter D).

For now, let us concentrate on the _Einstein-Besso manuscript_ and Einstein’s
November 18, 1915 paper on the perihelion motion of Mercury. Janssen writes
that in 1988, the Einstein Papers Project obtained a copy of the _Einstein-
Besso manuscript_ in possession of descendants of Besso. In the 1970s, Pierre
Speziali edited the Einstein-Besso correspondence ([27]) and became friendly
with Besso’s son Vero who gave him the manuscript. Speziali approached the
director of the Einstein Papers Project and made a copy of the manuscript
available to the project. A portion of the manuscript (which consists of 53
pages, half of them in Einstein’s hand and the other half in Besso’s, but
there might well be more pages) was published in the _Collected Papers of
Albert Einstein_ ([2]). Historians of science then began to analyze the
manuscript ([22]). After discovering the _Einstein-Besso manuscript_ ,
scholars realized that ([2]):

1) Einstein had already calculated the perihelion motion of Mercury in 1913 in
collaboration with Besso.

2) The method used in the _Einstein-Besso manuscript_ is virtually identical
to the method Einstein used in his November 18, 1915 paper on the perihelion
motion of Mercury.

In their new paper, Janssen and Renn argue that if Einstein had followed the
prescriptions of Karl Popper, he should have accepted that his _Entwurf_
theory had been falsified. But during 1913-1914, Einstein did nothing of the
sort. According to the _Entwurf_ theory, the field of a static Sun produced an
advance of the perihelion of Mercury of $18^{{}^{\prime\prime}}$ (18 seconds
of arc per century). Einstein kept quiet about this result and continued to
work on the _Entwurf_ theory. Of course, the story is not so simple and the
end result which was given in the _Einstein-Besso manuscript_ by Einstein was:
$1821^{{}^{\prime\prime}}$ and it was corrected. This is discussed in section
2.

Janssen and Renn further claim that given the undeniable importance of Besso’s
earlier calculations, ”one can legitimately ask why Einstein did not invite
Besso as a co-author” of his November 18, 1915 paper on the perihelion motion
of mercury. ”As it happened”, say Janssen and Renn, ”Besso did not even get an
acknowledgment” in Einstein’s November 18, 1915 paper ([24], p. 7).

After a brief review of Einstein’s and Besso’s early attempts at calculating
the advance of the perihelion of Mercury, I discuss falsification (section 2)
and confirmation (section 3) of a theory by evidence. A problem in
confirmation theory which was mainly considered by Clark Glymour, and which
has been the subject of many papers, is confirmation by old evidence. Einstein
derived out of general relativity the perihelion motion of Mercury discovered
more than half a century earlier. I end section 3 with a discussion of
confirmation of a theory by previously known evidence.

In sections 3 and 4 I propose an answer to two questions:

1) Why did Einstein not reject his _Entwurf_ field equations in the face of
the wrong result $18^{{}^{\prime\prime}}$?

2) Why was Besso not invited as a co-author of the 1915 paper on the
perihelion motion of Mercury, even though he contributed to the calculations
in the _Einstein-Besso manuscript_?

##  2 Falsification

In the summer of 1913, in the _Einstein-Besso manuscript_ , Besso and Einstein
tried to solve the _Entwurf_ vacuum field equations and calculate the
perihelion advance of Mercury in the field of the static Sun. According to the
_Entwurf_ theory, the field of a static Sun produces an advance of the
perihelion of Mercury of ([2], p. 351):

$1.25\pi\frac{A}{a\left(1-e^{2}\right)}\vspace{1mm}$ (1)

per revolution, where $A$ is proportional to the gravitational constant and
the mass of the Sun, $a$ represents the semi-major axis and $e$ \- the
eccentricity of the elliptical orbit.

According to the _Entwurf_ theory, the field of a static Sun produces an
advance of the perihelion of Mercury of $18^{{}^{\prime\prime}}$ but the end
result which is given by Einstein is: $1821^{{}^{\prime\prime}}$. Next to this
result Einstein wrote in the _Einstein-Besso manuscript_ : “independently
checked” ([2], p. 28). Einstein’s result was a factor $100$ too large. The
Sun, which in Newton’s theory produces no perihelion motion at all, in the
_Entwurf_ theory, produces a perihelion motion of more than three times the
size of the total perihelion motion that is observed. Fortunately, Einstein
and Besso found a mistake in the numerical calculation – an error factor of 10
for the mass of the Sun. Since the perihelion motion is proportional to the
square of this quantity, the final result was an error factor of 100. Several
pages ahead there is a correction in Besso’s hand of the erroneous value for
the mass of the Sun that Einstein had used. On another page, Einstein himself
corrected the old value for the mass of the Sun.111More specifically, on page
30 of the _Einstein-Besso manuscript_ another attempt is made by Einstein to
find the perihelion advance of Mercury in the field of a static sun, because
Einstein found a mistake in the previous calculation regarding the evaluation
of the mass of the Sun. He wrote the new value as: $M=1.96\times 10^{33}$. But
the new calculation contains several other errors. The end result of this
calculation, in fractions of $\pi$ per half a revolution, is a perihelion
advance of: $1.65\times 10^{-8}$. Below this number Einstein wrote another
number: $3.4\times 10^{-8}$. This is probably the number that Einstein
expected to obtain with the new value of the mass of the Sun. After correcting
the errors in the calculation this is indeed the number one obtains and this
gives an advance of the perihelion of Mercury of $18^{{}^{\prime\prime}}$ per
century ([2]).

Janssen and Renn explain that Einstein found the precession per half a
revolution in seconds of arc and the precession in 100 years in seconds. ”The
correct result of $18^{{}^{\prime\prime}}$ is not stated anywhere in the
manuscript, although, […] there are clear indications that Einstein and Besso
discovered the source of the erroneous factor 100 in the result stated on” p.
28. It seems that Besso was the first one who found the source of the error,
say Janssen and Renn ([24], p. 7, p. 46).

Be that as it may, $1821^{{}^{\prime\prime}}$ is erroneous because of
Einstein’s computation error and $18^{{}^{\prime\prime}}$ is wrong because of
Einstein’s _Entwurf_ theory. Janssen and Renn conclude: ”The theory would have
been decisively refuted had it truly predicted $1800^{{}^{\prime\prime}}$;
that it could only account for less than half the missing seconds was
undoubtedly disappointing but not particularly troublesome” ([24], p. 6). The
point is that the _Entwurf_ hypothesis of non-covariance is refuted by the
evidence, i.e. $18^{{}^{\prime\prime}}$. Yet Einstein never mentioned the
above disappointing result until he recalculated it on the basis of the
correct November 1915 theory.

So, let us try to answer the question:

Why did Einstein not reject his _Entwurf_ field equations in the face of the
wrong result $18^{{}^{\prime\prime}}$?

I shall now try to answer this question starting with rotation and ending with
falsification.

On pages 41-42 of the _Einstein-Besso manuscript_ , Einstein checked whether
the Minkowski metric in rotating coordinates was a solution of his _Entwurf_
vacuum field equations. To find this, he used the same approximation procedure
he had used to calculate the field of the Sun for the perihelion advance of
Mercury. He first substituted the components of the rotation metric into the
_Entwurf_ vacuum field equations in a first-order approximation. He thus
checked whether the metric field of the rotating system was a solution of the
field equations of the _Entwurf_ theory. However, in substituting the first-
order components of the rotation metric into the _Entwurf_ vacuum field
equation, Einstein made mistakes. As a result of his mistakes he obtained the
answer he wanted. He asked whether the solution for the (44-component of the
metric tensor) $g_{44}$ found from the _Entwurf_ field equations was the same
as the $g_{44}$ one would obtain through transformation of the Minkowski
metric from an inertial system to the rotating system. Einstein’s answer in
the _Einstein-Besso manuscript_ was yes, and next to the solution he wrote
that it was correct ([2]).

Janssen and Renn note that Einstein made several mistakes in the above
calculation. He incorrectly read off the components of the metric from the
line element and wrote a plus sign which should have been a minus sign, and
thus made a sign error. When these mistakes are corrected, the metric field of
the rotating system is not a solution of the field equations of the _Entwurf_
theory ([24], p. 47).

The point is that, Einstein related between the problem of rotation and the
perihelion of Mercury. For small angular velocities, the metric field of a
rotating system has the same general form as the metric field of the Sun –
considered by Einstein and Besso in their 1913 manuscript for calculating the
perihelion of Mercury. Thus, Einstein could use the approximation procedure
used to find the metric field of the Sun to second-order to find the metric in
a rotating coordinate system. He could also do the opposite; refer from his
calculation of the rotating coordinate system about the case of the metric
field of the Sun and Mercury’s perihelion.

The _Einstein-Besso manuscript_ remained with Einstein in Zurich and in early
1914, he sent it to Besso, urging his friend to finish their project. Besso
added more calculations. It seems that in 1913 Einstein ”did not follow the
prescriptions of Karl Popper” [26] or Carl Hempel (Hempel’s version of
falsification) [18] because he was under the impression that the Minkowski
metric in rotating coordinates was a solution of his _Entwurf_ vacuum field
equations. Since this problem is intimately related to the perihelion advance
of Mercury, it is reasonable to assume that Einstein hoped the
$18^{{}^{\prime\prime}}$ value would eventually be corrected by adding other
terms.

In 1913, Gunnar Nordström constructed an alternative theory to Einstein’s
_Entwurf_ theory. Nordström’s theory was simpler than the one suggested by
Einstein and more related to special relativity.222Nordström left the speed of
light constant and adapted his theory of gravitation to the special principle
of relativity in such a way that gravitational and inertial masses were equal.
He thus accepted a version of Einstein’s equality of inertial and
gravitational masses and included it in his theory. Unlike Einstein’s tensor
theory, Nordström proposed a scalar theory. Nordström’s and Einstein’s
theories had almost the same empirical consequences. Both Nordström’s and
Einstein’s theories predicted a red shift of spectral lines. But according to
Nordström’s theory, there was no bending of light rays in a gravitational
field. In fact, the _Entwurf_ theory predicted a value for the deflection of
light in a gravitational field of the Sun, 0.83 arc seconds, which much later
turned out to be wrong. On the other hand, Einstein’s 1915 generally covariant
theory yields twice that value, 1.7 arc seconds. But Einstein had no way to
know that the value 0.83 was wrong. The fact of the matter is that neither
theories could yield the correct value of the perihelion motion of Mercury. On
page 53 of the _Einstein-Besso manuscript_ (the calculations added by Besso in
1914), Besso calculated the perihelion motion of Mercury using Nordström’s
theory, and the strategy followed in these calculations was the same as that
followed by Einstein and Besso when they had both solved the _Entwurf_ field
equations in 1913 to find the perihelion advance for Mercury of 18 seconds of
arc per century ([2]).

Thus, in 1913 there were two incompatible theories, each could not account for
the motion of the perihelion of Mercury. Before 1915, the _Entwurf_ theory
remained without empirical support and a decision in favor of one or the other
theory – the _Entwurf_ or Nordström’s – was impossible on empirical grounds.
But from Einstein’s point of view, Nordström’s theory had a big flaw: it did
not satisfy Ernst Mach’s ideas, i.e. it did not account for rotation. As we
have already seen, neither did Einstein’s _Entwurf_ theory explain rotation.

Upon revisiting his calculations in September-October 1915, Einstein was
willing to accept his mistakes.

First, already in an exchange of letters and postcards that began in March
1915 and ended in May 1915, Tulio Levi-Civita presented his objections to
Einstein’s adapted coordinate systems. Einstein did not agree with Levi-Civita
and stubbornly clung to what had remained from his _Entwurf_ theory. There was
even a moment in which Einstein regained full confidence in his _Entwurf_
theory and he admitted to Levi-Civita, through the deeper considerations to
which the latter interesting letters have led him, that he became even more
firmly convinced that his theory was correct (see [29], Chapter D).

In the autumn of 1915, Einstein eventually accepted that he had chosen a
problematic Lagrangian function. Einstein’s 1914 ten field equations determine
the ten functions $g^{\mu\nu}$ but the $g^{\mu\nu}$ must also satisfy four
conditions for the special choice of adapted coordinate systems. In 1914, he
thought that the selection of the above Lagrangian could be supported by this
restriction and he provided a proof that, the above Lagrangian is uniquely
selected by the requirement that it is invariant for adapted coordinate
systems. Einstein believed that the variation of the Lagrangian uniquely led
to the _Entwurf_ field equations. Therefore, he thought he had demonstrated
that his _Entwurf_ field equations were the only equations that were invariant
for adapted coordinate systems. Around October-November 1915, he realized that
his Lagrangian function did not uniquely lead to the _Entwurf_ field equations
that were invariant for adapted coordinate systems. Consequently, covariance
with respect to adapted coordinate systems was a wrong path and Einstein
understood that he had to require general covariance.

Second, Janssen and Renn argue that in 1913 Besso discussed with Einstein
rotation and the possibility that the Minkowski metric in rotating coordinates
was not a solution of his _Entwurf_ field equations. Einstein though would
nonetheless continue to cling to his _Entwurf_ theory (I will further
elaborate on this suggestion later in this section).

In the autumn of 1915, Einstein finally recognized that the metric field
describing a rotating system was not a solution of the _Entwurf_ vacuum field
equations. Einstein corrected his initial mistakes and found that his field
equations were not satisfied and the metric field describing a rotating system
was not a solution of these equations. He obtained a solution for the $g_{44}$
from the _Entwurf_ field equations and demonstrated that it was not the same
as the one obtained through direct transformation of the Minkowski metric to
the rotating coordinate system.

On September 30, 1915 Einstein sent a letter to Erwin Freundlich saying he had
found a blatant contradiction in his theory of gravitation. He demonstrated to
Freundlich that the rotation metric was not a solution of his _Entwurf_ field
equations. He realized that he must have done a calculation error somewhere in
his work. The above realization seemed to have stimulated what Einstein had
already discovered earlier that, Mercury’s perihelion motion was too small,
namely $18^{{}^{\prime\prime}}$. Indeed, Einstein told Freundlich that he had
no doubt that his solution for the perihelion motion of Mercury (from 1913)
was also suffering from the same problem. He therefore immediately related
between the problem of rotation and the perihelion motion of Mercury (Einstein
to Freundlich, September 30, 1915, [4], Doc. 123). But recall that in the
_Einstein-Besso manuscript_ Einstein had already connected the two problems.

The day afterwards, Einstein sent a letter to Otto Naumann, in which he redid
the calculation of pages 41-42 of the _Einstein-Besso manuscript_ and found
that the metric field describing a rotating system was not a solution of the
_Entwurf_ field equations. He also demonstrated that the solution obtained
from the _Entwurf_ field equations, was not the same as the one obtained
through direct transformation of the Minkowski metric to the rotating
coordinate system (Einstein to Naumann, October 1, 1915, [4], Doc 124, note
5). This indicated to Einstein that his solution for the perihelion motion of
Mercury suffered from the same fault as the solution for a rotating coordinate
system.

Janssen writes: ”The story suggests that it was the problem with rotation
rather than the other two problems […] that dealt the decisive blow to the
_Entwurf_ theory” ([20], p. 127). It seems to me that for Einstein, the two
problems - that of the perihelion advance of Mercury and that of rotation -
were interconnected. But what led Einstein to give up his _Entwurf_ theory in
1915 was first and foremost the entanglement between rotation and Mercury. In
1915 Einstein redid the calculation of pages 41-42 of the _Einstein-Besso
manuscript_ and discovered his arithmetic error and thus, found out by himself
that the metric field describing a rotating system was not a solution of the
_Entwurf_ field equations. He then related this problem to Mercury’s
perihelion.

Now let us return back to 1913. It appears that the reason Einstein did not
accept that his _Entwurf_ theory was falsified, even though it predicted the
wrong result for the perihelion motion of Mercury, was that he related between
the problem of rotation and that of the perihelion motion of Mercury.
Einstein’s calculations in the _Einstein-Besso manuscript_ concerning rotation
involve the same approximation procedure Einstein had used to calculate the
field of the Sun for the perihelion advance of Mercury.

The point is that in his 1935 book, _Logik der Forschung_ [26], Popper
proposes an absolute notion of falsification which demands a refutation of a
theory by a single individual counter instance. Although the result Einstein
and Besso had obtained $18^{{}^{\prime\prime}}$ was a single counter example
to Einstein’s _Entwurf_ theory, in the _Einstein-Besso manuscript_ Einstein
and Besso nonetheless tried to resort to additional effects that might
contribute to the perihelion advance of Mercury. But these effects did not
produce the desired value of $43^{{}^{\prime\prime}}$ ([2]). Nevertheless,
Einstein was not bothered of the small perihelion advance of Mercury
$18^{{}^{\prime\prime}}$, as long as he believed that the rotating metric was
a solution of his _Entwurf_ field equations. The reason is that from his point
of view, these problems were entangled.

Janssen and Renn argue that, despite warnings recorded by Besso in his memo -
notes that Besso wrote of discussions with Einstein during their attempt to
account for the perihelion anomaly of Mercury on the basis of the _Entwurf_
theory - Einstein was nonetheless firmly convinced that the rotation metric
was a solution of the _Entwurf_ field equations ([24], p. 47).

Besso wrote on the first page of his notes of discussions with Einstein,
called _Besso’s memo_ : ”One can therefore not think of rotational forces as
produced by the rotation of the fixed stars according to the Einsteinian
gravitational equations”. In the top-right corner of the first page of the
memo, Besso wrote: “28 VIII 13.” ([23], p. 806, p. 787). Hence, Besso dated
the first page of the memo to be from August, 28, 1913. Accordingly, Janssen
and Renn interpret this as indicating that Besso told Einstein that the
rotation metric was not a solution of the _Entwurf_ field equations. Einstein
though would often stubbornly stick to his errors and mistakes and he was
unwilling to listen to warnings that people tried to bring to his attention.

But we know that in August, 1913 – on the second time Besso visited Einstein
in Zurich – Einstein was dissatisfied with his _Entwurf_ field equations. On
August 14, 1913, he wrote to Hendrik Antoon Lorentz that the problem hooked
him so much, that his confidence in the admissibility of the _Entwurf_ theory
was still shaky. Einstein was disturbed by the thought of the non-general
covariance of his gravitational field equations. If his field equations
permitted only linear transformations and were not generally covariant, then
Einstein felt the theory refuted its foundations and it thus stood unsupported
with no basis. Two days later, however, Einstein wrote to Lorentz that he had
found a way out of this muddle. Using the law of momentum and energy
conservation, Einstein limited the choice of reference systems to certain
allowed ones. With respect to these reference systems, the law of momentum and
energy conservation holds and the general linear transformations remain as the
only possible choice. Einstein was most satisfied and he told Lorentz that now
the theory gave him pleasure after he managed to eliminate this ugly dark spot
(Einstein to Lorentz, August 14, 16, 1913, [3], Doc. 467, 470).

This could hint that for a very short while, Einstein could have thought that
his _Entwurf_ theory was falsified. Besso had probably pinpointed the problem
of rotation. He was the one that put his hand on the problem. It is then
reasonable to assume that in August 1913, rotation was at least one of the
reasons why _for a brief moment_ Einstein was dissatisfied with his _Entwurf_
theory. Another reason could be Mercury’s perihelion. But very soon
afterwards, Einstein fell straight into the trap of the hole argument against
general covariance. Two years later, in January 1915, Einstein told Lorentz
that an argument he presented him has already been stressed by Ernst Mach.
”But it is best illustrated by a comparison that my friend Besso had thought
up last year” (Einstein to Lorentz, January 23, 1915, [4], Doc. 47). It seems
that at that time, Einstein tried to find a solution to the problem that his
rotation metric was not really a solution of his _Entwurf_ field equations.
Doing so enabled him to cling even more tightly to his hole argument and his
_Entwurf_ theory.

There is still the possibility that before 1915, Einstein refused to accept
that the rotating metric was not a solution of his _Entwurf_ field equations.
And only in 1915, when he redid the calculation of pages 41-42 of the
_Einstein-Besso manuscript_ , did he discover his arithmetic error and found
out by himself that the metric field describing a rotating system was not a
solution of the _Entwurf_ field equations. This scenario seems to me the most
reasonable one.

After 1913, Einstein’s belief in the _Entwurf_ theory was strengthened by
deriving the Newtonian law of gravitation from the linear approximation of the
_Entwurf_ field equations for the static case (for which the spatial metric
was flat) and by developing the hole argument. After all, a year earlier,
Einstein failed to extract the Newtonian limit from the November tensor and
his calculations entangled the Newtonian limit and conservation of momentum-
energy. On the other hand, the _Entwurf_ field equations satisfied both the
Newtonian limit and conservation of momentum-energy and so, it was difficult
to free Einstein from the clutches of the hole argument and later from his
adapted coordinate systems (see [29], Chapter D).

##  3 Confirmation

As said in the previous section, sometime in October 1915 Einstein dropped the
_Entwurf_ theory. Starting on November 4, 1915, he gradually expanded the
range of the covariance of his field equations [11]. On November 11, 1915,
Einstein wrote the field equations of gravitation in a generally covariant
form. In the addendum of November 11, 1915, he wrote these equations with the
source term [12] and on November 18, 1915, the equations were vacuum field
equations [13]. In his November 18 paper [13], Einstein solved the problem of
the perihelion of Mercury. He transferred the basic framework of the
calculation from the _Einstein-Besso manuscript_ and corrected it according to
his November 18, 1915 generally covariant vacuum field equations and geodesic
equation written in terms of the Christoffel symbols. Einstein used these
vacuum field equations for calculation through successive approximations of
the gravitational field of the Sun. He calculated the metric field of the Sun
using these vacuum field equations in a first-order approximation, substituted
the result of this calculation back into the vacuum field equations, obtained
a second-order approximation and solved the vacuum field equations to obtain
more accurate approximations for the metric field of the Sun.

Einstein’s solution was an ellipse that had a major axis that was not constant
and rotated. The field of a static Sun produced a precession of the
perihelion, an advance of the perihelion of $43$ seconds of arc per century
([13], p. 838):

$3\pi\frac{A}{a\left(1-e^{2}\right)}$ (2)

per revolution.

Compare equation (2) with equation (1). The $1.25\pi$ equation leads to an
advance of the perihelion of $18^{{}^{\prime\prime}}$. Multiply this by 2.4
and you get the advance of the perihelion of $43^{{}^{\prime\prime}}$ and the
$3\pi$ equation.

In the _Entwurf_ theory, the components of the gravitational field were
expressed as the gradient of the metric tensor. On the other hand, in November
1915, Einstein expressed the components of the gravitational field in terms of
the Christoffel symbols of the second kind. So, in 1915, the components of the
gravitational field of the static Sun were the Christoffel symbols.

Einstein then wrote equations of motion for a point mass moving in the
gravitational field of the Sun. A planet in free fall in the gravitational
field of the Sun moves on a geodesic line according to the geodesic equation.
The point mass moves on a geodesic line under the influence of the
gravitational field of the Sun. Einstein calculated the equations of the
geodesic lines in this space and compared them with the Newtonian equations of
the orbits of the planets in the solar system.

Einstein discovered two errors in the 1913 calculations:

1) In the _Einstein-Besso manuscript_ , Einstein calculated the metric field
of the Sun using the _Entwurf_ field equations in a first-order approximation,
and found that the static spatial metric was _flat_. Thus to first-order,
Mercury moves along a geodesic curve that depends on _flat_ components of the
metric. In November 1915, when Einstein redid this calculation, he calculated
the metric field of the Sun using the November 18, vacuum field equations of
the general theory of relativity in a first-order approximation. Einstein
wrote to Besso that he was surprised by the occurrence of _non-flat_
components of the first-order (static spatial) metric and he discovered that
Mercury moves along a geodesic curve that depends on _non-flat_ components of
the metric (Einstein to Besso, December 10, 1915, [4], Doc. 162; [28], pp.
304-306). Unlike the _Entwurf_ field equations, the 1915 ones are generally
covariant and a material point in a gravitational field moves on a geodesic
line where both equations are written in terms of the Christoffel symbols.

2) Besso managed to write his ”area law” in the form of the Newtonian angular
momentum conservation, but Kepler’s second law (the radius vector from the sun
to the planet sweeps out equal areas in equal times) did not hold for the
_Entwurf_ theory. On the other hand, in his November 18, 1915 paper, Einstein
emphasized that Kepler’s laws hold exactly for his generally covariant general
theory of relativity. Hence, Einstein corrected and adjusted Besso’s
derivation of 1913 according to his generally covariant theory. He then
stressed in his November 18, 1915 paper that the equations that determine the
motion of the planet for the case of circular motion give no deviation from
Kepler’s laws ([2];[13], p. 837).

Einstein attempted to obtain a solution, without considering whether or not
there was only one possible unique solution, exactly as he had done with Besso
in 1913. Einstein’s theory was consistent with the evidence. Moreover, unlike
the _Einstein-Besso manuscript_ , the result $43^{{}^{\prime\prime}}$
confirmed Einstein’s general theory of relativity of November 18, 1915 and it
was accepted as an account for the advance of the perihelion of Mercury.

On the very day Einstein submitted his paper on the perihelion motion of
Mercury to the _Proceedings of the Royal Prussian Academy of Sciences, Berlin_
, he wrote to David Hilbert. He told him that he was going to submit ”today” a
paper in which he derived quantitatively out of general relativity, without
any guiding hypotheses, the perihelion motion of Mercury, ”discovered by Le
Verrier” _more than half a century earlier_. Einstein added that this was not
achieved until then by any gravitational theory (Einstein to Hilbert, November
18, 1915, [4], Doc. 148).

From the above letter we learn two things:

_First_ , Einstein explained to Hilbert that his general relativity (the new
November 18, 1915 general relativity) entails (”without any guiding
hypotheses”) the advance of the perihelion of Mercury and therefore, it has to
be considered as confirming this evidence (see definition in [18], p. 102).

_Second_ , the precession of the perihelion of Mercury was, an unexplained
phenomenon that bothered scientists even before Einstein had advanced his
general theory of relativity. If we take into consideration Hempel’s logical
confirmation theory, then unlike Newtonian classical gravitational theory, the
result is evidence in favour of the general theory of relativity and so, as
said above, the correct value of the anomaly confirms Einstein’s 1915 general
theory of relativity. But from the point of view of the Bayesian confirmation
theory, there seems to be a problem here.

In the chapter, “Why I am not a Bayesian”, Glymour asks: “When does a piece of
evidence confirm a hypothesis according to the Bayesian scheme of things? The
natural answer is that it does so when the posterior probability of the
hypothesis is greater than its prior probability […]”. A Bayesian learns new
facts and each time he learns a new fact he revises his degrees of belief by
conditionalizing on the new fact. The discovery that the fact is the case has
confirmed those hypotheses whose probability after the discovery is higher
than their probability before ([17], p. 38, [16], p. 83). Glymour, however,
thinks this account is unsatisfactory for a few reasons the most problematic
of which is the following: “Scientists commonly argue for their theories from
evidence known long before the theories were introduced […] Newton argued for
universal gravitation using Kepler’s second and third laws, established before
the Principia was published” ([16], pp. 85-86).

According to Glymour, Einstein argued for his theory of relativity from
evidence known long before his theory was introduced. Glymour gives the
example of the perihelion of Mercury: in 1915 Einstein derived by general
relativity the anomalous advance of Mercury’s perihelion discovered more than
half a century earlier. Glymour, however, argues that if one attempts to
explain old evidence, one encounters a problem. He shows that old evidence can
in fact confirm new theory, but according to Bayesian conditionalization it
cannot.

By the same token, one might argue that in his 1905 paper on the light quanta,
Einstein had also accounted for old evidence. He explained the photoelectric
effect observed by Philipp Lenard in 1900. In Section $\S 8$ of his 1905 paper
on the light quanta, Einstein treats the photoelectric effect. He is guided by
the following principle: ”the energy of light is distributed discontinuously
in space” and writes: ”The usual conception, that the energy of light is
continuously distributed over the space through which it travels, meets with
especially great difficulties when one attempts to explain the photoelectric
phenomena, which are described in the pioneering work by Mr. Lenard” ([7], p.
133, p. 145). Einstein had read Lenard’s paper on the photoelectric effect
four years earlier in 1901 ([25]; Einstein to Marić, probably May 28, 1901,
[1], Doc. 111). But in his 1905 paper on the light quanta, he insists that his
account of the photoelectric effect is deduced by his light quanta principle
and not induced from Lenard’s experimental result. He concludes his exposition
saying: ”As far as I can see, our conception does not contradict with the
properties of the photoelectric effect observed by Mr. Lenard” (Einstein,
1905c, p. 145, p. 147). Hempel explains that by means of a given hypothesis,
we can deduce special predictions, which have the form of observation laws
([18], p. 101). On the Nobel Prize website it is written: ”The Committee says
that the most important application of Einstein’s photoelectric law and also
its most convincing confirmation has come from the use Bohr made of it in his
theory of atoms, which explains a vast amount of spectroscopic data” ([9]). It
therefore seems that in the case of the photoelectric effect: $P_{t}(e)\neq
1$.

On the other hand, as discussed above, Einstein’s general relativity confirms
the perihelion of Mercury and therefore, to the Bayesian the prior is:
$P_{t}(e)=1$. More specifically, Einstein was already interested in the
problem of Mercury’s perihelion from early on in his search for a new
relativistic theory of gravitation. He wrote to his close friend Conrad
Habicht in 1907 that he hoped his gravitation theory would explain the
anomalous advance of Mercury’s perihelion (Einstein to Habicht, December 24,
1907, [3], Doc. 69). Accordingly, Einstein had long known that the anomalous
advance of Mercury’s perihelion could not be explained by Newtonian theory and
by 1913, he also knew the value for the anomalous secular advance of Mercury’s
perihelion. Hence, evidence $e$ was known to Einstein in 1907, before the
_Entwurf_ theory was introduced by him in 1913. In $t=1915$, Einstein
presented his general theory of relativity $T$. Because $e$ was already known
at $t=1915$, for Einstein $P_{t}(e)=1$. Further, because $P_{t}(e)=1$, the
likelihood of $e$ given $T$, $P_{t}(e,T)$, is also $1$. We then have:

$P_{t}(T,e)=\frac{P_{t}(T)P_{t}(e,T)}{P_{t}(e)}=P_{t}(T).\vspace{1mm}$ (3)

The posterior probability $P_{t}(T,e)$ is therefore the same as the prior
probability $P_{t}(T)$ and $e$ cannot constitute evidence for Einstein’s $T$
and “we have the absurdity that old evidence cannot confirm new theory”, says
Glymour ([16], pp. 85-86).

Many philosophers attempted to solve and dissolve the old evidence problem
(see [6], p. 120 and [15], to name only two).

Bas van Fraassen has responded to Glymour’s claim saying that although the
measurements of the anomalous advance of Mercury’s perihelion had long been
made, when Einstein showed in 1915 that the above measurements do, indeed,
confirm his new theory, _we_ updated our credence ([14], p. 154):

> Indeed, when the scientific community got to the point when it could say
> that the advance in the perihelion of Mercury confirmed Einstein’s theory,
> they had indeed just newly learned _something_ – though not _that_ fact
> about Mercury, but another fact that has to do with it – had conditionalized
> on that something, and the conditionalization had increased their credence
> in Einstein’s theory.

This solution is more reasonable. But it seems to me that the problem in
Bayesian confirmation theory persists.

##  4 Why did Einstein not invite Besso as a co-author in 1915?

Besso collaborated with Einstein on a project (the _Einstein-Besso
manuscript_) that contains a few methods and equations that are similar to the
ones one finds in Einstein’s November 18, 1915 paper on the perihelion motion
of Mercury. Some time in November 1915, Einstein realized that unlike the 1913
calculations, he could now demonstrate that the $43^{{}^{\prime\prime}}$ does,
indeed, confirm his new general theory of relativity. Unlike the _Einstein-
Besso manuscript_ , with Einstein’s new theory and the Christoffel symbols, it
seemed like everything simply fell into place.

With his new 1915 general theory of relativity, Einstein quickly achieved the
correct advance of Mercury’s perihelion in the field of a static Sun by
applying the methods he had already worked out two years earlier with Besso.
Recall that Einstein wrote to Hilbert on the very day he submitted (to the
_Proceedings of the Royal Prussian Academy of Sciences, Berlin_) the November
18, 1915 paper on the perihelion motion of Mercury (see section 3). Hilbert
wrote back the next day saying: ”If I could calculate as fast as you can”
(Hilbert to Einstein, November 19, 1915, [4], Doc. 149). According to Janssen
and Renn, Einstein did not bother to tell Hilbert that he had done very
similar calculations before in the _Einstein-Besso manuscript_. ”He probably
enjoyed giving Hilbert a taste of his own medicine”. It is only because the
_Einstein-Besso manuscript_ ”ended up in Besso’s hands rather than Einstein’s,
who almost certainly would have discarded it, that we can belatedly call
Einstein’s bluff trying to put one over on Hilbert”, say Janssen and Renn
([24], pp. 5-6).

Hence, Einstein did not acknowledge his earlier work with Besso. Neither did
he mention Besso’s name in his 1915 paper explaining the anomalous precession
of Mercury. ”Besso had always been thought of as an important sounding board
for Einstein, never as a serious scientific collaborator”, say Janssen and
Renn ([24], p. 6). When the _Einstein-Besso manuscript_ was discovered,
scholars realized that Einstein had already calculated the perihelion motion
of Mercury in 1913 in collaboration with Besso. Janssen and Renn then write
about the November 18, 1915 paper ([24], p. 7):

> Still, given the undeniable importance of his earlier calculations, one can
> legitimately ask why Einstein did not invite Besso as a co-author. As it
> happened, Besso did not even get an acknowledgment. We suspect that this is
> largely because of the race Einstein perceived himself to be in with
> Hilbert. He was in a hurry and it probably never even occurred to him to ask
> Besso to write a joint paper on the topic.

Janssen and Renn add that ”An unusually high number of typos in the paper
suggests that it was written in haste”.

There seem to me to be two main reasons why Einstein did not invite Besso as a
co-author in 1915:

1) _First_ , it appears that Einstein still considered Besso as a sounding
board, even though Besso undertook calculations with Einstein in the
_Einstein-Besso manuscript_. When Einstein wrote a series of letters to Besso
between 1913 and 1916 ([27]), and described to him, step by step, his
discoveries of general relativity, Besso once again functioned as the
committed sounding board. On the other hand, from 1912, apparently Einstein
considered Grossmann, and not Besso, as his partner and collaborator while
Besso remained Einstein’s closest friend and sounding board.

Let us pause on this point. Janssen and Renn write ([24], pp. 7-8, p. 27):
”Besso nonetheless took responsibility for key parts of their joint project
and corrected some errors in Einstein’s calculations”. Besso corrected
Einstein’s overly crude approximations of Mercury’s precession of Mercury’s
perihelion produced by the field of the Sun. Recall that on page 28 of the
_Einstein-Besso manuscript_ , Einstein recorded the value for the “precession
in 100 years” which was too large by a factor of 100, i.e. $1821^{"}$. On page
35 of the _Einstein-Besso manuscript_ , Besso located the source of the error
in the value Einstein reported on page 28. On page 30 Einstein made the same
correction (see discussion in section 2). Janssen and Renn claim that ”The
exclamation point on [p. 35] suggests that Besso found the mistake first”.

It was not the first time that Einstein made errors in calculations. Recall
that a few pages ahead, on pages 40-41 of the _Einstein-Besso manuscript_ ,
one can see that Einstein checked whether the rotation metric was a vacuum
solution of the field equations he established. He did a sign error and a few
mistakes and came to the belief that the rotation metric was a solution of the
new _Entwurf_ field equations ([2]). Recall from section 2 that according to
Janssen and Renn, in August 1913, Besso told Einstein that the rotation metric
was not a solution of the _Entwurf_ field equations. _But at the end of the
day, before November 1915, Einstein stubbornly clung to his _Entwurf_ field
equations like a butterfly to a flower_.

Think of the following example. Much later, in 1916, Einstein published a
paper on gravitational waves in which he presented a solution that did not
satisfy the unimodular coordinates. In obtaining the above solution, he made a
mathematical error. This error caused him to obtain three different types of
waves compatible with his approximate field equations: longitudinal waves,
transverse waves and a new type of wave. He added an Addendum in which he
suggested a solution to this problem: A comeback of the restrictive condition.
He suggested that for systems in unimodular coordinates waves of the third
type transport energy. Einstein’s colleagues (Willem de Sitter, Tulio Levi-
Civita, Erwin Schrödinger, and Gunnar Nordström) demonstrated to him that for
systems in unimodular coordinates the gravitational wave of a mass point carry
no energy, but Einstein tried to persuade them that he had actually not made a
mistake in his gravitational waves paper.333After 1916 Nordström gave up his
scalar theory of gravitation and started to work on Einstein’s new general
theory of relativity. The stubborn Einstein, however, eventually accepted his
colleagues’ results and dropped his restrictive condition. This finally led
him to discover plane transverse gravitational waves. Einstein confided in
Gustav Mie and told him that his 1916 gravitational wave paper contained
calculation errors, and that he intended to present a cleaned-up version of
the paper to the Prussian Academy (Einstein to Gustav Mie, December 29, 1917,
[4], Doc. 421). Einstein was the sole author of the new paper published in
1918. Neither de Sitter, nor Levi-Civita, Schrödinger, and Nordström were co-
authors of the new 1918 paper on gravitational waves, even though they
corrected Einstein’s errors and calculations from the 1916 paper.

It seems that the reason why Einstein made errors in calculations is his
ability to think in images, thought experiments, and entertain non-verbal
images. Einstein first used images (anchored in the unconscious) to solve
problems in science; he first imagined thought experiments. Words, as they are
written or spoken, came later, when he expressed his discoveries through the
language of logical connections and the discovery of universal formal
principles (see [29], chapter E). Einstein’s sister Maja reported that in the
Gymnasium he was supposed to begin the study of algebra and geometry at the
age of thirteen. By that time however he already had a predilection for
solving complicated problems in applied arithmetic, although the computational
errors he made kept him from appearing particularly talented in the eyes of
his teachers ([1], pp. 1xi, pp. xx). Thus, even as a child Einstein would make
computational errors.

2) _Secondly_ , and most importantly, in the November 1915 perihelion motion
of Mercury paper, Einstein wrote the so-called ”area law” (angular momentum
conservation) and the law of conservation of energy for the orbit of Mercury
in polar coordinates. In the _Einstein-Besso manuscript_ , Besso had already
written the ”area law” - and defined the area constant and the area speed -
and wrote the equation for the law of energy conservation for the orbit of
Mercury ([2], p. 8). However, in the _Einstein-Besso manuscript_ , Besso
copied equations directly from Einstein’s 1913 _Entwurf_ paper. In the
_Entwurf_ paper, Einstein obtained the energy and momentum of the mass point,
respectively. Besso copied the equations of motion for the mass point, the
energy and the momentum of the mass point, and the Lagrangian directly from
Einstein’s _Entwurf_ paper and derived the angular momentum conservation,
which he called the ”area law” ([10], p. 7). But according to Besso’s angular
momentum conservation expression (the ”area law”), Kepler’s second law did not
hold for the _Entwurf_ theory. There was, therefore, no warrant for calling it
an ”area law”.

In 1915, Einstein proceeded by using the ”area law” and the law of
conservation of energy for the orbit of Mercury, the components of the
gravitational field to the first-order approximation and the components of the
gravitational field to the second-order approximation. Subsequently, using the
equations of the geodesic line in terms of the Christoffel symbols, Einstein
obtained the equation of motion of Mercury to the second-order approximation.
Eventually, from Einstein’s derivation it followed that Kepler’s laws hold
exactly.

Accordingly, Einstein need not have invited Besso as a co-author of his 1915
paper. Einstein ends his 1905 relativity paper by noting that when he worked
on the problem discussed in the relativity paper, Besso faithfully stood by
him. Einstein is indebted to him for several ”valuable suggestions” ([8], p.
921). He does not end the November, 18, 1915 perihelion motion of Mercury
paper with a similar acknowledgment to Besso. Einstein though probably felt
guilty about not _thanking_ Besso in his November 18, 1915 paper, because much
later, he told his assistant Leopold Infeld that he should look up the
literature to quote previous scientists: ”Do it by all means. Already I have
sinned too often in this respect” ([19], p. 277).

This work is supported by ERC advanced grant number 834735.

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