7 Nova Geminorum 1912 and the Origin of the
Idea of Gravitational Lensing
Tilman Sauer
Einstein Papers Project
California Institute of Technology 20-7
Pasadena, CA 91125, USA
tilman@einstein.caltech.edu
Abstract
Einstein’s early calculations of gravitational lensing, contained in
a scratch notebook and dated to the spring of 1912, are reexamined.
A hitherto unknown letter by Einstein suggests that he entertained
the idea of explaining the phenomenon of new stars by gravitational
lensing in the fall of 1915 much more seriously than was previously
assumed. A reexamination of the relevant calculations by Einstein
shows that, indeed, at least some of them most likely date from early
October 1915. But in support of earlier historical interpretation of
Einstein’s notes, it is argued that the appearance of Nova Geminorum
1912 (DN Gem) in March 1912 may, in fact, provide a relevant context
and motivation for Einstein’s lensing calculations on the occasion of
his first meeting with Erwin Freundlich during a visit in Berlin in April
1912. We also comment on the significance of Einstein’s consideration
of gravitational lensing in the fall of 1915 for the reconstruction of
Einstein’s final steps in his path towards general relativity.
Introduction
Several years ago, it was discovered that Einstein had investigated the idea
of geometric stellar lensing more than twenty years before the publication
http://arxiv.org/abs/0704.0963v1
of his seminal note on the subject.1 The analysis of a scratch notebook2
showed that he had derived equations in notes dated to the year 1912 that
are equivalent to those that he would only publish in 1936.3 In the notes
and in the paper, Einstein derived the basic lensing equation for a point-like
light source and a point-like gravitating mass. From the lensing equation it
follows readily that a terrestial observer will see a double image of a lensed
star or, in the case of perfect alignment, a so-called “Einstein ring.” Einstein
also derived an expression for the apparent magnification of the light source
as seen by a terrestial observer. The dating for the notes was based on other
entries in the notebook. Some of these entries are related to a visit by Einstein
in Berlin April 15-22, 1912, and it was conjectured that the occasion for the
lensing entries was his meeting with the Berlin astronomer Erwin Freundlich
during this week.
The lensing idea lay dormant with Einstein until in 1936 he was prodded
by the amateur scientist Rudi W. Mandl into publishing his short note in
Science. In the meantime, the idea surfaced occasionally in publications by
other authors, such as Oliver Lodge (1919), Arthur Eddington (1920), and
Orest Chwolson (1924).4 We only have one other piece of evidence that
Einstein thought about the problem between 1912 and 1936. In a letter to
his friend Heinrich Zangger, dated 8 or 15 October 1915, Einstein remarked
that he has now convinced himself that the “new stars” have nothing to do
with the lensing effect, and that with respect to the stellar populations in
the sky the phenomenon would be far too rare to be observable.5
The Albert Einstein Archives in Jerusalem recently acquired a hitherto
unknown letter by Einstein that both corroborates some of the historical
conjectures of the early history of the lensing idea and also adds significant
new insight into the context of Einstein’s early considerations. From this
letter it appears that the phenomenon of “new stars,” i.e. the observation
of this type of cataclysmic variables, played a much more prominent role in
the origin of the idea than was suggested by the side remark in Einstein’s
letter to Zangger. It also adds important new information about Einstein’s
thinking in the crucial period between losing faith in the precursor theory to
1[Renn, Sauer, and Stachel 1997] and [Renn and Sauer 2003].
2Albert Einstein Archives (AEA), call number 3-013, published as [CPAE3, Appendix
A]. A facsimile is available on Einstein Archives Online at http://www.alberteinstein.info.
3[Einstein 1936].
4[Lodge 1919], [Eddington 1920, pp. 133–135], [Chwolson 1924].
5Einstein to Heinrich Zangger, 8 or 15 October 1915 [CPAE8, Doc. 130].
http://www.alberteinstein.info
the general theory of relativity entertained in the years 1913–1915, and the
breakthrough to a general relativistic theory of gravitation in the fall of 1915.6
In fact, the new letter justifies a reexamination of our reconstruction of what
we know about Einstein’s intellectual preoccupations both in April 1912 and
in October 1915, and more generally about the genesis of the concept of
gravitational lensing.
1 Einstein’s letter to Emil Budde
The new letter is a response to Emil Arnold Budde (1842–1921), dated 22
May 1916.7 Budde had been director of the Charlottenburg works of the
company of Siemens & Halske from 1893 until 1911.8 He was the author
of a number of scientific publications, among them a monograph on ten-
sors in three-dimensional space [Budde 1914a]9 and of a critical comment on
relativity published in 1914 in the Verhandlungen of the German Physical
Society.10
In an unknown letter to Einstein, Budde apparently had written about the
possibility of observing what are now called Einstein rings, i.e. ring shaped
images of a distant star that is in perfect alignment with a lensing star and
a terrestial observer. The subject matter of Budde’s initial letter can be in-
ferred from Einstein’s response in which he pointed out that one would expect
the phenomenon to be extraordinarily rare, and that it could not be detected
on photographic plates “as little circles” since irradiation would diffuse the
images that would hence only appear as bright little discs, indistinguishable
from the image of a regular star.
6For historical discussion, see [Norton 1984], [Janssen et al. 2007], and further refer-
ences cited therein.
7AEA 123-079. The letter will be published in the forthcoming volume of the Collected
Papers of Albert Einstein.
8Budde had studied catholic theology and science, and had worked as a secondary
school teacher and as a correspondent for the German daily Kölnische Zeitung in Paris,
Rome, and Constantinople. In 1887, he became a Privatgelehrter in Berlin, edited the
journal Fortschritte der Physik, and entered the company Siemens & Halske as a physicist
in 1892. In 1911, he retired and moved to Feldafing, near Lake Starnberg, since he had been
advised by his physicians to live at an altitude of at least 600m [Laue 1921, Werner 1921].
9In [Norton 1992, pp. 309–310] this textbook is cited as evidence for the argument that
Grossmann’s generalization of the term ‘tensor’ in [Einstein and Grossmann 1913] was an
original development.
10[Budde 1914b], [Budde 1914c].
The interesting part of Einstein’s response follows after this negative com-
ment. Einstein continued to relate that he himself had put his hopes on a
different aspect, namely that “due to the lensing effect” the distant star
would appear with an “immensely increased intensity,” and that he initially
had thought that this would provide an explanation of the “new stars.” He
went on to list three reasons why he had given up this hope after more
careful consideration. First, the temporal development of the intensity of a
nova is asymmetric. The luminosity increases much faster than it declines
again. Second, the color of the novae usually changes towards the red and,
in general, its spectral character changes in a distinct and characteristic way.
Third, the phenomenon would be very unlikely for the same reasons that the
observation of an Einstein ring would be unlikely.
In the beginning of his letter, Einstein pointed out that Budde’s idea con-
cerned the same thing that “about half a year ago” (“vor etwa einem halben
Jahre”) had put him into “joyous excitement” (“freudige Aufregung”). At
the end of the letter, he again wrote that the joy had been “just as short as
it had been great.” Counting back six months from the date of Einstein’s
letter, 22 May 1916, takes us to the 22nd of November 1915, which is just
the time of the final formulation of general relativity. It is also just another
six weeks or so away from the date of his letter to Zangger of early October,
in which he wrote about the very same subject of the possible explanation
of novae as a phenomenon of gravitational lensing.
2 The lensing calculations in the scratch note-
In light of this new letter, let us briefly reexamine the calculations in the
Scratch Notebook that had been dated to April 1912.11 Stellar gravitational
lensing is an implicit consequence of a law of the deflection of light rays in
a gravitational field. Such a law had been obtained by Einstein in 1911 as
a direct consequence of the equivalence hypothesis. The angle of deflection
11The following brief recapitulation refers to [CPAE3, 585–586], or
http://www.alberteinstein.info/db/ViewImage.do?DocumentID=34432&Page=23 and
· · ·&Page=26. For a complete and detailed paraphrase of Einstein’s notes, see the
Appendix below.
http://www.alberteinstein.info/db/ViewImage.do?DocumentID=34432
Figure 1: The geometric constellation for stellar gravitational lensing as
sketched in Einstein’s Scratch Notebook. From [CPAE3, p. 585].
α̃12 was found to be
where k is the gravitational constant, M the mass of the lensing star, c
the speed of light, and ∆ the distance of closest approach of the light ray
measured from the center of the massive star.13 On [p. 43] of the Scratch
Notebook we find the sketch shown in Fig. (1) and underneath it the lensing
equation
r = ρ
R +R′
where R denotes the distance between the light emitting distant star and
the massive star that is acting as a lens, R′ the distance between the lensing
star and the position of a terrestial observer who is located a distance r away
from the line connecting light source and lensing star. ρ is the distance of
closest approach of a light ray emitted by the star and seen by the observer.
α = 2kM/c2 is a typical length (later known as the Schwarzschild radius)
that depends on the mass of the light deflecting star and that determines
12I am using the notation α̃ instead of α (as in [Einstein 1911]) in order to distinguish
this angle from the quantity α (effectively the Schwarzschild radius) in Einstein’s scratch
notebook.
13[Einstein 1911, p. 908]. Qualitatively, Einstein had already derived the consequence of
light bending in a gravitational field when he first formulated his equivalence hypothesis
[Einstein 1907, p. 461]. In the final theory of general relativity, the same relation is
obtained with an additional factor of 2, as observed explicitly in [Einstein 1915c, p. 834].
Incidentally, the relevant formula was printed incorrectly by a factor of 2 in (the first
printing of) Einstein’s 1916 review paper of general relativity [Einstein 1916, p. 822], see
[CPAE6, Doc. 30, n. 36] and also Einstein’s response to Carl Runge, 8 November 1920
[CPAE10, Doc. 195].
the angle of deflection to be α
. The lensing equation can be written in
dimensionless variables as
r0 = ρ0 −
, (1)
after defining r0 and ρ0 as
r0 = r
R′(R +R′)α
ρ0 = ρ
R +R′
. (2)
The fact that equation (1) is a quadratic equation for ρ0 entails that there are
two solutions which correspond to two light rays that can reach an observer,
along either side of the lensing star,14 and hence that a terrestial observer
will see a double image of the distant star. For perfect alignment, the double
image will turn into a ring shaped image, an “Einstein-ring” whose diameter
0 = ρ
= 1 also follows immediately from the lensing equation.
In light of Einstein’s letters to Zangger and Budde, it is interesting that
Einstein went on to compute also the apparent magnification, obtaining the
following expression:
Htot = H
. (3)
Here Htot is the total intensity received by the observer, and H the intensity
of the star light at distance R. ρ1,2 denote the two roots of the quadratic
equation (1). The term in brackets gives the relative brightness, reducing
to 1 if no lensing takes place. Finally, some order of magnitude calculations
on these pages showed that the probability of observing this effect would be
given by the probability of having two stars within a solid angle that would
cover 10−15 of the sky, which is highly improbable given that the number of
known stars at the time was of the order of 106.15
Equations that are entirely equivalent to these were published much later,
in 1936, in Einstein’s note to Science.16
14Since only three points are given, the problem is intrinsically a planar one, as long as
the three points are not in perfect alignment.
15See the discussion in the appendix.
16[Renn, Sauer, and Stachel 1997].
The dating of the lensing notes in the scratch notebook to Einstein’s visit
in Berlin in April 1912 was based on other evidence in the notebook. Most
importantly, p. [36] lists Einstein’s appointments during his Berlin visit. In
addition, pp. [38] and [39] recapitulate very specifically the equations of Ein-
stein’s two papers on the theory of the static gravitational field of February
and March 1912, respectively.17 The calculations that deal with the lensing
problem then appear on pp. [43]-[48], and on pp. [51] and [52] of the note-
book. The sheet containing pp. [44] and [45] is a loose sheet inserted between
p. [43] and p. [45]. After p. [53], three pages have been torn out, and then
follow 37 blank pages, with some pages torn out in between. The remainder
of the notebook contains entries that begin at the other end of the notebook
which was turned upside down. Except for some apparently unrelated and
undated entires on pp. [49], [50],18 and [54], the lensing calculations hence
are at the end of a more or less continuous flow of entries. These physical
characteristics of the notebook lead to an important consequence. All infor-
mation that was pointing to a date of the lensing calculations in the year
1912 preceded the actual lensing calculations. Reexaming pp. [51] and [52]
of the notebook in light of the letters to Zangger and to Budde in fact reveals
that at least these entries were not written in 1912, but rather most likely
at the time of the letter to Zangger, in early October 1915. There are two
reasons for this. First, at the top of p. [51], Einstein wrote down the title
of a book published only in 1914.19 Therefore, the following calculations are
almost certainly to be dated later than the publication of this book. Second,
at the bottom of p. [52], Einstein explicitly refers to the “apparent diameter
of a Nova st[ar].” The calculations on pp. [51] and [52] in fact are a calcula-
tion of the apparent brightness and diameter of a star. We conclude that, in
all probability, the calculations on pp. [51] and [52] were written at the time
of Einstein’s letter to Zangger, early October 1915.
Does the dating of pp. [51] and [52] to October 1915 also compel us to
17[Einstein 1912a, Einstein 1912b].
18On the bottom half of p. [49] there is a sketch of Pascal’s and Brianchon’s Theorems,
which deal with hexagons inscribed in or circumscribed on a conical section. I wish to
thank Jesper Lützen for this identification. Other entries on pp. [49] and [50] also appear
to deal with problems from projective geometry. There is also a sketch of a vessel filled
with a liquid and the words “eau glyceriné” and what appears to be sketch of a magnetic
moment in a sinusoidal magnetic field.
19[Fernau 1914]. Could it be that the book was mentioned to Einstein when he met
with Romain Rolland in Geneva in September 1915, see [CPAE8, Doc. 118]?
revise our dating of the other lensing calculations in the notebook? To answer
this question, we need to consider the broader historical context of the notes.
But before doing so, we first observe that pp. [49] and [50] contain entries
that appear unrelated to the lensing problem. As shown by the detailed
paraphrase given in the appendix, the calculations on pp. [43] to [48] on
the other hand represent a coherent train of thought, as do the calculations
of pp. [51] and [52]. We also note that Einstein used a slightly different
notation on pp. [43]ff. and on pp. [51]-[52]. In the first set, he denoted the
distances between light source and lens and between lens and observer as
R and R′, respectively. On pp. [51]-[52] he used the notation R1 and R2,
respectively. He also reversed the roles of r and ρ. We conclude that there
is a discontinuity between the first set of lensing calculations on pp. [43] to
[48] and the second set on pp. [51] and p. [52].
3 The context of Einstein’s early lensing cal-
culations
From Einstein’s letter to Budde we learn that he had investigated the idea
that stellar lensing might explain the phenomenon of the “new stars,” and
that he had given up this idea after looking more closely into the character-
istic features of novae, especially their light curves and the changes in their
spectral characteristics. Let us therefore briefly look into the astronomical
knowledge about novae at the time.
The observation of a new star is an event that, in the early twentieth
century, occurred only every few years. Between 1900 and 1915, eight novae
were observed:20 Nova Persei 1901 (GK Per), Nova Geminorum (1) 1903 (DM
Gem), Nova Aquilae 1905 (V604 Aql), Nova Vela 1905 (CN Vel), Nova Arae
1910 (OY Ara), Nova Lacertae 1910 (DI Lac), Nova Sagittarii 1910 (V999
Sgr), and Nova Geminorum (2) 1912 (DN Gem) with maximum brightness
of 0.2, 4.8, 8.2, 10.2, 6.0, 4.6, 8.0, 3.5 magnitudes, respectively. At the time,
“the two most interesting Novae of the present century,” [Campbell 1914,
p. 493], were Nova Persei of 1901 and Nova Geminorum of 1912. The next
spectacular nova to occur was the very bright Nova Aquilae 1918 (V603 Aql)
with a maximum brightness of −1.1 mag.
Nova Geminorum (2) was discovered on March 12, 1912, by the as-
20For the following, see [Duerbeck 1987].
Figure 2: The light curve of Nova Geminorum 1912 for the first three
months after its appearance, as put together by Fischer-Petersen on the
basis of 253 individual observations. The points are the magnitudes re-
ported by the individual observers, the solid line is to guide the eye. From
[Fischer-Petersen 1912, p.429].
tronomer Sigurd Enebo at Dombaas, Norway [Pickering 1912]. On a pho-
tographic plate taken at Harvard College Observatory on March 10, showing
stars of magnitude 10.5, it was not visible, but it was visible as a magni-
tude 5 star in the constellation Gemini on a Harvard plate of March 11. On
March 13, a cablegram was received at Harvard and distributed throughout
the United States. In the following days all major observatories as well as
many amateur astronomers pointed their instruments towards the new star.
The maximum brightness of mag 3.5 was reached on March 14 (Einstein’s
33rd birthday!) [Fischer-Petersen 1912]. By March 16, the brightness was
down to a magnitude of 5.5 and in the following weeks it decreased further,
with distinct oscillations. By mid-April 1912, most observers registered a
brightness of mag 6 ≈ 7, see Fig. (2). We now know that the DN Gem is a
fast nova with a t3-time of 37d. Its light curve is type Bb in the classification
of [Duerbeck 1987], i.e. it declines with major fluctuations.
Like all classical novae, Nova Geminorum is, in fact, a binary system of
a white dwarf and main sequence star, where hydrogen-rich matter is being
accreted onto the white dwarf. Recent observations have even determined the
binary period [Retter et al. 1999]. The eruption of a classical nova occurs
when a hydrogen-rich envelope of the white dwarf suffers a thermonuclear
runaway.21 This explanation of classical novae also entails that they display
the same sequence of spectral behaviour as the luminosity decreases, see also
Fig. (3) below. However, our current understanding of classical novae was
suggested only in the fifties.22
The temporal proximity of the appearance of Nova Geminorum 1912 with
Einstein’s Berlin visit during the week of April 15–22, suggests that this
astronomical event was discussed also when Einstein met with Freundlich
for the first time.23 We know that the observatory in Potsdam took a
number of photographs of the new star between March 15 and April 12
[Furuhjelm 1912, Ludendorff 1912], and that Freundlich, among others, was
charged with photometric observations of the nova [Fischer-Petersen 1912,
p. 429]. Einstein and Freundlich had earlier corresponded about the possib-
lity of observing gravitational light deflection through the gravitational field
of the sun.24 The purpose of their meeting was to discuss possible astro-
nomical tests of Einstein’s emerging relativistic theory of gravitation. The
recent observation of the brightest nova since 1901 must have been on Fre-
undlich’s mind, and it seems more than likely that the idea of explaining
the phenomenon in terms of gravitational lensing therefore came up in the
course of their conversation. We conclude that our earlier dating of the first
set of calculations of the lensing problem in the Scratch Notebook to the
time of Einstein’s encounter with Freundlich in April 1912 is the most likely
possibility.
In fact, the context of the observation of Nova Geminorum 1912 provides
an answer to the question as to why Einstein would have done the calculations
at all and, in particular, why he would not have been content at the time
with a calculation of the lensing equation, the separation of the double star
image and, perhaps, the radius of the Einstein ring. Without this context
it might seem a rather ingenious move on Einstein’s part to go ahead and
immediately compute the apparent magnification of the lensed star as well.
But this answer to the question of motivation for the specific details of the
21For a review, see [Shara 1989].
22For a historical overview of previous theories, see [Duerbeck 2007].
23For evidence that Einstein met with Freundlich, see his letter to Michele Besso, 26
March 1912, in which he mentions planned discussions (“Besprechungen”) with Nernst,
Planck, Rubens, Warburg, Haber, and “an astronomer”—presumably Freundlich [CPAE5,
Doc. 377].
24Einstein to Freundlich, 1 September 1911, 21 September 1911, and 8 January 1912
[CPAE5, Docs. 281, 287, 336].
Figure 3: Changes in the spectrum of Nova Geminorum 1912, March 22 to
August 19, 1912. From [Adams and Kohlschütter 1912].
calculations in the Scratch Notebook, immediately raises another question.
Assuming that the first set of lensing calculations were done in spring
1912, why do we have no evidence that this idea was followed up by either
Einstein or by Freundlich until the fall of 1915? To answer this question,
it should first be observed that no summarizing results and analyses of the
observations of Nova Geminorum 1912 were published before the end of the
summer.
Let us briefly recall Einstein’s intellectual preoccupations after his visit
to Berlin in April 1912.25 Shortly before his trip to Berlin he had submitted
his two papers on a theory of the static gravitational field.26 After his return
to Prague in April 1912, Einstein was preparing for his move to Zurich. The
two papers were published in the 23 May issue of the Annalen der Physik.
Einstein wrote an addendum at proof stage to the second one, in which
he showed that the equations of motion could be written in a variational
form, adding that this would give us “an idea about how the equations of
motion of the material point in a dynamic gravitational field are constructed”
[Einstein 1912b, p. 458]. He also entered into a published dispute with Max
Abraham on their respective theories of gravitation.27 At the end of July,
he departed Prague for Zurich. The next thing we know about his work on
gravitation comes from a letter to Ludwig Hopf, dated 16 August 1912, in
which he wrote:
The work on gravitation is going splendidly. Unless I am com-
pletely wrong, I have now found the most general equations.28
These most general equations are, in all probability, equations of motion
in a gravitational field, represented by a metric tensor. After his arrival
in Zurich, Einstein began a collaboration with his former classmate Marcel
Grossmann, now his colleague at the ETH. Their research on a generalized
25We will focus here on his work of gravitation yet for the sake of completeness it should
be noted that Einstein at the same time was also thinking about quantum theory, most
notably about the law of photochemical equivalence and about the problem of zero point
energy, see [CPAE4, Docs. 5, 6, 11, 12].
26[Einstein 1912a], [Einstein 1912b], were received by the Annalen der Physik on 26
February and 23 March, respectively.
27[Einstein 1912c] which was received by the Annalen on 4 July 1912 is a response to a
critique by Abraham.
28Einstein to Hopf, 16 August 1912 [CPAE5, Doc. 416].
theory of relativity is documented in Einstein’s so-called “Zurich Notebook”29
and culminates in the publication of the “Outline [Entwurf] of a generalized
theory of relativity and a theory of gravitation,” in early summer of 1913 co-
authored with Marcel Grossmann.30 This so-called Entwurf-theory contains
all the elements of the final theory of general relativity, except for generally
relativistic field equations. Einstein would hold onto this theory until his
final breakthrough to general relativity in the fall of 1915.
In conclusion, we observe that Einstein’s path toward the general theory
of relativity in 1912 took him deep into the unknown land of the mathematics
associated with the metric tensor, before there was a chance to reconsider
the lensing idea in light of the data for Nova Geminorum 1912. In any case,
he would have to rely on Freundlich or other professional astronomers for a
secure assessment of the possibilities of an observation of the lensing effect
at the time.
Freundlich, on the other hand, continued to think about ways to test
Einstein’s new theory of gravitation.31 But his focus was on observations
of light deflection during a solar eclipse.32 In August 1914, he led a first
(unsuccessful) expedition to the Crimea to observe the eclipse of 21 August
1914. Even these efforts were hampered by the lack of funding and, more
generally, by the difficulties of securing increased research time that would
have allowed Freundlich to freely pursue his collaboration with Einstein.
Given these circumstances, and the fact that order-of-magnitude calcu-
lations may have convinced Einstein already in 1912 that the phenomenon
would be rare, it seems plausible that the lensing idea was not pursued further
for some time after Einstein’s visit in Berlin in April 1912.
Let us finally reexamine the events of fall 1915. Einstein, in the meantime
had left Zurich in the spring of 1914, accepting an appointment as member
of the Prussian Academy in Berlin. In September 1915, Einstein spent a few
weeks in Switzerland where he met, among others, with Heinrich Zangger,
Michele Besso, and Romain Rolland. On 22 September 1915, he left Zurich33
but travelled via Eisenach where he was on the 24th of September.34 By the
29AEA 3-006, see [CPAE4, Doc. 10]. For a comprehensive discussion of this document,
including a facsimile, transcription, and detailed paraphrase, see [Janssen et al. 2007].
30[Einstein and Grossmann 1913].
31See [Hentschel 1994] and [Hentschel 1997].
32See his correspondence with Einstein in [CPAE5].
33[CPAE8, p. 998].
34[CPAE10, Doc. Vol. 8, 122a].
30th of September, at the latest, he was back in Berlin, and wrote a letter
to Freundlich:
I am writing you now about a scientific matter that electrifies me
enormously.35
It is clear from the letter, however, that the excitement indicated to Fre-
undlich is not about the idea of gravitational lensing. Rather, Einstein
had found an internal contradiction in his Entwurf theory that amounted
to the realization that Minkowski space-time in rotating Cartesian coordi-
nates would not be a solution of the Entwurf field equations.36 This insight
undermined his confidence in the validity of the Entwurf theory, and is later
mentioned as one of three arguments that induced Einstein to lose faith in
the Entwurf equations.37 The first of these arguments was the fact that a cal-
culation of the planetary perihelion advance in the framework of the Entwurf
theory did not produce the well-known anomaly that had been established
for Mercury. This problem had been known to Einstein for some time.38
The third argument was realized sometime in early October, a few days after
stumbling upon the problem with rotation, and concerned the mathematical
derivation of the Entwurf field equations in Einstein’s comprehensive review
of October 1914.39 In any case, we know that Einstein asked Freundlich to
look into the problem of the rotating metric, and that they met some time in
early October. This follows from a letter Einstein wrote to Otto Naumann,
35Einstein to Freundlich, 30 September 1915 [CPAE8, Doc. 123]. For a detailed discus-
sion of this letter and its significance for the reconstruction of Einstein’s final breakthrough
to general relativity, see [Janssen 1999].
36Interestingly, the Scratch Notebook contains an entry that is pertinent to this problem.
On p. [66], i.e. on the last page of the backward end of the notebook, Einstein considers the
case of rotation in a calculation that exactly matches corresponding calculations dating
from October 1915, see [Janssen 1999]. Janssen cautiously remarks that he believes this
calculation to date from 1913 [Janssen 1999, p. 139]. It seems possible, however, that these
entries as well as the immediately preceding ones on the perihelion advance (see note 38)
may well date from late 1915 as well.
37See Einstein to Arnold Sommerfeld, 28 November 1915, and to Hendrik A. Lorentz, 1
January 1916 [CPAE8, Docs. 153, 177].
38See [Earman and Janssen 1993] and [CPAE4, pp. 344–359]. The Scratch Notebook
contains some calculations related to the perihelion advance on pp. [61–66], i.e. in the
backward end of the notebook. On p. [61], Einstein there explicitly noted that the advance
of Mercury’s perihelion would be 17′′ which is the value that is obtained on the basis of
the Entwurf-theory. These calculations are undated, see note 36.
39[Einstein 1914].
dated after 1 October 1915, in which Einstein asked about possibilities to al-
low Freundlich more freedom to pursue independent research. In this letter,
Einstein mentioned that Freundlich had visited him “recently.”40
By 12 October, Einstein had realized the third problem with the Entwurf
theory, the unproven uniqueness of the Lagrangian for the Entwurf field equa-
tions, as he reported in a letter to Lorentz. In this letter, he neither men-
tioned the problem with the rotating metric nor the issue of gravitational
lensing.41
For our reconstruction of this episode, the precise date of Einstein’s letter
to Zangger in which he remarked that he had given up the hope of explaining
the “new stars” as a lensing phenomenon is relevant. It could have been
written either on the 8th or the 15th of October.42
The letter to Zangger suggests that they had talked about the idea earlier
since Einstein seems to presuppose that Zangger knew what he was talking
about and did not explain what he meant by “lens effect” (“Linsenwirkung”).
As mentioned before, Einstein had just recently met with Zangger, as well as
with Besso before returning to Berlin. The following scenario seems therefore
plausible:
Upon returning to Berlin some time after the 24th of September 1915,
Einstein realized the problem of the rotating metric solution and wrote to
Freundlich on the 30th, asking him to look into this issue. Shortly afterwards,
the two met in person. Most likely they discussed not only the rotation
problem, but also the lensing idea. Having found troubling indications of an
inner inconsistency in the very foundations of this theory, it would have been
a natural move for Einstein to go back and reconsider early arguments such as
one based safely on the equivalence hypothesis.43 After this meeting, Einstein
40“Letzter Tage war Herr Dr. Freundlich von der Sternwarte N bei mir.” [CPAE8,
Doc. 124].
41In a letter to Hilbert, dated 7 November 1915, Einstein wrote that he realized the flaw
in his proof “about four weeks ago” [CPAE8, Doc. 136].
42The editors of [CPAE8] dated this letter explicitly to the 15th of October. It seems,
however, that the 8th is also a possibility. The letter was written on a Friday between
September 30, when a fire and explosion took place in the comb factory Walter near Lake
Biel took place, mentioned in the letter, and October 22 when Einstein participated in
the first Academy session after the summer break. I see no reason why Einstein could not
have heard of the accidents from Zangger before October 8.
43It seems unlikely that Einstein at that time was already contemplating a quantitatively
different law of light deflection. Einstein first observed in [Einstein 1915c, p. 834] that an
additional factor of 2 would arise from the different first-order approximation for the
wrote to Naumann exploring possibilities to give Freundlich more research
freedom. By October 8, Einstein had convinced himself that gravitational
lensing cannot explain the “new stars.” On 12 October, he realized the third
problem of his mathematical derivation of the Entwurf field equation.
According to this reconstruction of the sequence of events, it is remarkable
that the “joyous excitement” about the lensing idea falls within days after
his being “electrified” about the realization of the rotation problem on 30
September, and his realization of the third problem of the mathematical
derivation of the Entwurf equation, on or before 12 October 1915.44
Some five weeks later, his excitement was even greater and his heart,
allegedly, skipped a beat when he found that he could derive the anomalous
advance of Mercury’s perihelion on the basis of his new field equations. And
after having submitted the last of his four November communications to the
Prussian Academy on 25 November which presented the final gravitational
field equations, the “Einstein equations,” he wrote to Sommerfeld:
You must not be cross with me that I am answering your kind
and interesting letter only today. But in the last month I had
one of the most exciting, exhausting times of my life, indeed also
one of the most successful. I could not think of writing.45
It is interesting to learn from Einstein’s letter to Budde that in addition to
the realization of the problems with the Entwurf theory and the eventual suc-
metric if the Newtonian limit is derived on the basis of generally covariant field equations
in which the Ricci tensor is directly set proportional to the energy-momentum tensor.
These latter equations were published in his second November memoir, presented on 11
November, under the assumption that the trace of the energy-momentum tensor vanish.
In his comment on the factor of 2, Einstein refers to this result as being in contrast to
“earlier calculations” where the hypothesis of vanishing energy-momentum had not yet
been made.
44For completeness, one should point one other intellectual activity of Einstein’s during
those days. In Einstein’s letter to Zangger of 8 or 15 October, he also mentioned that
he wrote “a supplementary paper to my last year’s analysis on general relativity.” The
last year’s analysis is, in all likelyhood [Einstein 1914]; the supplementary paper is, in all
likelihood, an early version of [Einstein 1916b], or, perhaps, an early version of Einstein’s
first November memoir [Einstein 1915a], see [CPAE8, Doc. 130, note 5] and [Janssen 1999,
note 51].
45“Sie dürfen mir nicht böse sein, dass ich erst heute auf Ihren freundlichen und in-
teressanten Brief antworte. Aber ich hatte im letzten Monat eine der aufregendsten,
anstrengendsten Zeiten meines Lebens, allerdings auch der erfolgreichsten.” Einstein to
Sommerfeld, 28 November 1915 [CPAE8, Doc. 153].
cess of his breakthrough to general relativity, an astronomical problem, the
idea of explaining novae in terms of gravitational lensing added to Einstein’s
excitement in the midst of what must indeed have been the most intense
period of intellectual turmoil in his life.
4 Concluding remarks
Einstein’s recollections of his thought concerning the explanation of the “new
stars” as a phenomenon of gravitational lensing in his letter to Budde add
two significant insights to our reconstruction of the genesis of general rel-
ativity. If our dating and context hypothesis of the lensing calculations in
the scratch notebook are correct, we learn that it was an astronomical ob-
servation that triggered the elaboration of a significant consequence of the
equivalence hypothesis and its consequence of gravitational light deflection.
It is also interesting that on his intellectual path from the Entwurf theory to
the final theory of general relativity, Einstein also took a detour in which he
explored further consequences of one of the solid pillars of general relativity,
the equivalence hypothesis.
Appendix: Einstein’s lensing calculations in
the Scratch Notebook AEA 3-013
The following is a self-contained line-by-line paraphrase of Einstein’s lensing
calculations in his scratch notebook, [CPAE3, pp. 585–589]. The pagination
in square brackets refers to the sequence of pages in the notebook.
The calculations start out on p. [43] with Fig. (1) and continue on the
facing page p. [46]. From the more explicit sketch in Fig. (4), we read off the
lensing equation:
r = ρ
R +R′
. (4)
Here R is the distance between the light emitting star S and the lensing star
L; R′ the distance between the massive star L and the projected position of
the observer O on the line connecting light source and lens; ρ is the distance
of closest approach of a light ray emitted from the distant star and seen by
an observer; r is the orthogonal distance of the terrestial observer to the
line connecting light source and lens. The first term in the lensing equation
Figure 4: The geometry of stellar lensing.
(4) is obtained from the similarity of triangles with baseline R and R + R′,
respectively, and the second term is the angle of deflection as given by the law
of gravitational light bending, where α is the Schwarzschild radius 2GM/c2.
If we want to write this equation in dimensionless variables, we need to
multiply it by a factor of
R′(R +R′)α
so that, when we define r0 and ρ0 as
r0 = r
R′(R +R′)α
ρ0 = ρ
R +R′
the lensing equation (4) turns into
r0 = ρ0 −
. (8)
This is a quadratic equation for r0, the two solutions of which correspond to
the two light rays passing above and below L. The observer O therefore sees
two images of S at positions S ′ and S ′′, respectively. To read off the radius
of an “Einstein ring,” obtained for perfect alignment of S, L, and O, one
only needs to set r0 ≡ 1.
In order to get an expression for the apparent magnification, Einstein
proceeded as follows. He first took the square of eq. (8) as
2 + r2 = ρ2 +
. (9)
If we multiply this equation by π and denote the areas of the circles corre-
sponding to the radii r and ρ as f = πr2 and ϕ = πρ2, respectively, we can
write this equation as
2π + f = ϕ+
. (10)
We are not interested in the full circle corresponding to these radii but in the
differential area element associated with these radii. More precisely, we are
interested in the change of the differential area element df associated with f
when we change the differential area element dϕ associated with ϕ. Hence,
Einstein wrote
dϕ. (11)
The intensity H of the brightness received at r is related to the intensity H
of the brightness at ρ by
Hdf = ±Hdϕ, (12)
where the plus and minus signs refer to the two solutions of the quadratic
equation. Since we have from (11)
, (13)
we get
H = ± H
. (14)
or, inserting the explicit solutions, we can write the total brightness at r as
Htot = H
. (15)
As Einstein remarked, the term in brackets gives the relative brightness, if
we take the value for r → ∞ to be 1.46 This result is equation number (3)
46“Klammer gibt relative Helligkeit”
in Einstein’s notes, and most of the following material on pp. [47] and [48],
as well as on the loose sheet containing pp. [44] and [45], will be a discussion
of this expression for the relative brightness.
On p. [47], Einstein first rewrote the reduced lensing equation as
− x, (16)
and then the terms in brackets as
1− x41
x42 − 1
. (17)
The next step is to bring the two terms to a common denominator47
x41 − x42
(1− x41)(1− x42)
. (18)
If one squares the lensing equation (16) twice, one obtains
− 2 + (2 + r2)2 = 1
+ x4. (19)
If we now introduce new variables A and u via
2A = −2 + (2 + r2)2, (20)
A = −1 +
(2 + r2)2 = 1 + 2r2 +
r4, (21)
u = x4, (22)
we can write the quadrupled equation (19) as
2A = u+
. (23)
Multiplication by u and adding A2 on each side gives
u2 − 2Au+ A2 = −1 + A2, (24)
47In the notes, Einstein refers to this step as “Rationalisierung”.
from which one can immediately read off the two solutions of eq. (23) as
u = −A±
A2 − 1. (25)
Given (18), the difference between the two roots,
u1 − u2 = 2
A2 − 1, (26)
provides an expression for the nominator of Hr in (18). With the two roots,
we can also rewrite the quadratic equation in the form
u2 − 2Au+ 1 = (u− u1)(u− u2), (27)
and if we now set u = 1, we obtain
2(1−A) = (1− u1)(1− u2), (28)
which gives us an expression for the denominator of Hr in (18). Combining
the two expressions, as Einstein did on p. [48], we obtain
1 + 1
) , (30)
where we have inserted (21) to obtain the second line.
We now have an explicit expression for the relative brightness as a func-
tion of the dimensionless variable r. We now evidently see that Hr → 1/r
for r → 0, and that Hr approaches 1 asymptotically from above for large r,
see Fig. (5).
Let us now reconstruct Einstein’s order-of-magnitude estimate for the
expected frequency of the phenomenon on p. [45]. The explicit expression
for the relative brightness gives us a measure of the maximal distance r for
which significant magnification is obtained. We can look at specific values of
Hr(r). For instance, for r0 = 12 we find
1 + 1
5 ≈ 2. (31)
0 1 2 3
H (r)
Figure 5: A plot of the expression (30) for the relative brightness Hr as a
function of r. The inset is from [CPAE3, p. 587].
Hence, Einstein concluded that up to a distance of r0 =
one would obtain
an increase of the intensity by a factor of 2. In other words, if we write the
intensity Hr0 asymptotically for small r0 and R′ ≫ R as
R′(R +R′)α
, (33)
we see that for a lensing star at a distance of R, the relative increase in
intensity is given by
= tg ᾱ. (34)
Here ᾱ is the angle that determines how well the distant star has to be aligned
with the lensing star and the observer to produce appreciable magnification.
In order to get an order-of-magnitude estimate for this angle, one needs an
order-of-magnitude estimate for
. In order to obtain such an estimate,
Einstein notes that the ratio of the solar Schwarschild radius α to the solar
equatorial radius Rs is given approximately by
= 3 · 10−6. (35)
The radius of the sun is 2 light seconds, and the distance of the nearest stars
is of the order of 10 light years, or
105 · 365 · 10 ≈ 4 · 108 lightseconds. (36)
It follows that α
for a star of 1 solar mass 10 lightyears away is
≈ 10−14 or
≈ 10−7. (37)
To see the distant star with double intensity, we therefore have
tg ᾱ
, (38)
so that the angle ᾱ is of order 10−7. A linear angle corresponds to a solid
angle roughly by taking its square. Thus, the angular size of the region
where the distant star needs to be found behind a massive star in order to
be magnified in the lens is of order 10−14. In angular units, the total sky has
an area of 4π ≈ 10, so that the angular size of the region in question covers a
fraction of 10−15 of the total sky. This has to be contrasted with the average
density of stellar population in the sky. The Bonner Durchmusterung listed
of the order of 3 · 105 stars to ninth magnitude for the northern hemisphere,
so a reasonable average density of the number of stars would be 1 star per
10−5 of the sky.48
On the back of the loose sheet [p. 44] we find a few more calculations
related to order-of-magnitude estimates that start from (32). Einstein here
again goes back to the definition of r0 and ρ0 in terms of R, R
′, and α.49
Again, he observes that r0 =
would give twice the usual intensity, and
rewrites (6) for this case:
R′(R +R′)α
. (39)
The latter equation for R′ ≫ R turns into
, (40)
and for R ≪ R′ into
αR′. (41)
Einstein concluded that the smaller of the two distances R and R′ determines
the angle r
. In the top right corner of the page, Einstein jotted down another
order-of-magnitude calculation, which I do not fully understand. Apparently,
he computed the distance of 100 lightyears in terms of centimeters
3 · 1010 · 3 · 107 · 102 [cm] ≈ 1020 cm (42)
48On the relevant page under discussion here, we also find a little sketch by Einstein of
a circle and the angle of its radius for a point some distance away. The precise meaning
of this sketch is unclear but the numbers written next to it suggest that Einstein was
considering the order of magnitude for the angular size of the moon. The radius of the
moon is seen under an angle of 15′ from the earth, and the mean distance between the
earth and the moon in units of the lunar radius is about 200, which translates to an angle
of 50o.
49One can see here that Einstein corrected an error in his earlier calculations on [p. 43],
where he had erroneously written the second term of the lensing equation (4) with R
instead of R′, which resulted in a confusion of the factors of R and R′ in expressions (5)
and (6).
Figure 6: A sketch in Einstein’s scratch notebook to obtain eq. (43). From
[CPAE3, p.585].
He also computed the angle x under which the star at distance R′ and
the star at distance R+R′ would be seen by an observer at distance r away
from the connecting line between the two stars if no lensing took place:
x = r
R +R′
R′(R +R′)
. (43)
The first equation can be read off from a little sketch of the geometry of light
source, lensing star, and observer, at the bottom of the page, see Fig. (6).
Let us finally comment on the calculations on pp. [51] and [52]. As men-
tioned in the main text of this article, Einstein here introduced a change
of notation. On p. [51], he sketched again the geometry for stellar lensing.
Here, the geometry has been turned by 90 degrees, and the notation changed
so that R and R′ become R1 and R2, and ρ and r are interchanged to be-
come r and ρ, respectively. This change of notation is reflected in the lensing
equation, written down on p. [52] as
ρ = r +R1
, (44)
where tanw = r/R2. Einstein then immediately proceeded to compute the
magnification by taking the square of the lensing equation and then comput-
ing the derivative as
d(ρ2)
d(r2)
(R1α)
= A ·
. (45)
Instead of pursueing this calculation further, Einstein instead wrote “appar-
ent diameter of a Nova star,” and wrote down the solution of eq. (44) for
ρ = 0, as to obtain the diameter of an Einstein ring:
R1R2α
R1 +R2
. (46)
He computed the angle w0 as
R2(R1 +R2)
. (47)
The calculation ends with an attempt at a numerical order-of-magnitude
estimation which seems to proceed along the same lines as in eqs. (35,36).
The calculation, however, was broken off, and the whole page was struck
through.
Acknowledgments
I wish to thank Diana Buchwald for a critical reading of an earlier version of
this paper, and Hilmar Duerbeck for some helpful comments. Unpublished
correspondence in the Albert Einstein Archives is quoted by kind permission.
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	Einstein's letter to Emil Budde
	The lensing calculations in the scratch notebook
	The context of Einstein's early lensing calculations
	Concluding remarks

  Einstein's early calculations of gravitational lensing, contained in a
scratch notebook and dated to the spring of 1912, are reexamined. A hitherto
unknown letter by Einstein suggests that he entertained the idea of explaining
the phenomenon of new stars by gravitational lensing in the fall of 1915 much
more seriously than was previously assumed. A reexamination of the relevant
calculations by Einstein shows that, indeed, at least some of them most likely
date from early October 1915. But in support of earlier historical
interpretation of Einstein's notes, it is argued that the appearance of Nova
Geminorum 1912 (DN Gem) in March 1912 may, in fact, provide a relevant context
and motivation for Einstein's lensing calculations on the occasion of his first
meeting with Erwin Freundlich during a visit in Berlin in April 1912. We also
comment on the significance of Einstein's consideration of gravitational
lensing in the fall of 1915 for the reconstruction of Einstein's final steps in
his path towards general relativity.

Introduction
Several years ago, it was discovered that Einstein had investigated the idea
of geometric stellar lensing more than twenty years before the publication
http://arxiv.org/abs/0704.0963v1
of his seminal note on the subject.1 The analysis of a scratch notebook2
showed that he had derived equations in notes dated to the year 1912 that
are equivalent to those that he would only publish in 1936.3 In the notes
and in the paper, Einstein derived the basic lensing equation for a point-like
light source and a point-like gravitating mass. From the lensing equation it
follows readily that a terrestial observer will see a double image of a lensed
star or, in the case of perfect alignment, a so-called “Einstein ring.” Einstein
also derived an expression for the apparent magnification of the light source
as seen by a terrestial observer. The dating for the notes was based on other
entries in the notebook. Some of these entries are related to a visit by Einstein
in Berlin April 15-22, 1912, and it was conjectured that the occasion for the
lensing entries was his meeting with the Berlin astronomer Erwin Freundlich
during this week.
The lensing idea lay dormant with Einstein until in 1936 he was prodded
by the amateur scientist Rudi W. Mandl into publishing his short note in
Science. In the meantime, the idea surfaced occasionally in publications by
other authors, such as Oliver Lodge (1919), Arthur Eddington (1920), and
Orest Chwolson (1924).4 We only have one other piece of evidence that
Einstein thought about the problem between 1912 and 1936. In a letter to
his friend Heinrich Zangger, dated 8 or 15 October 1915, Einstein remarked
that he has now convinced himself that the “new stars” have nothing to do
with the lensing effect, and that with respect to the stellar populations in
the sky the phenomenon would be far too rare to be observable.5
The Albert Einstein Archives in Jerusalem recently acquired a hitherto
unknown letter by Einstein that both corroborates some of the historical
conjectures of the early history of the lensing idea and also adds significant
new insight into the context of Einstein’s early considerations. From this
letter it appears that the phenomenon of “new stars,” i.e. the observation
of this type of cataclysmic variables, played a much more prominent role in
the origin of the idea than was suggested by the side remark in Einstein’s
letter to Zangger. It also adds important new information about Einstein’s
thinking in the crucial period between losing faith in the precursor theory to
1[Renn, Sauer, and Stachel 1997] and [Renn and Sauer 2003].
2Albert Einstein Archives (AEA), call number 3-013, published as [CPAE3, Appendix
A]. A facsimile is available on Einstein Archives Online at http://www.alberteinstein.info.
3[Einstein 1936].
4[Lodge 1919], [Eddington 1920, pp. 133–135], [Chwolson 1924].
5Einstein to Heinrich Zangger, 8 or 15 October 1915 [CPAE8, Doc. 130].
http://www.alberteinstein.info
the general theory of relativity entertained in the years 1913–1915, and the
breakthrough to a general relativistic theory of gravitation in the fall of 1915.6
In fact, the new letter justifies a reexamination of our reconstruction of what
we know about Einstein’s intellectual preoccupations both in April 1912 and
in October 1915, and more generally about the genesis of the concept of
gravitational lensing.
1 Einstein’s letter to Emil Budde
The new letter is a response to Emil Arnold Budde (1842–1921), dated 22
May 1916.7 Budde had been director of the Charlottenburg works of the
company of Siemens & Halske from 1893 until 1911.8 He was the author
of a number of scientific publications, among them a monograph on ten-
sors in three-dimensional space [Budde 1914a]9 and of a critical comment on
relativity published in 1914 in the Verhandlungen of the German Physical
Society.10
In an unknown letter to Einstein, Budde apparently had written about the
possibility of observing what are now called Einstein rings, i.e. ring shaped
images of a distant star that is in perfect alignment with a lensing star and
a terrestial observer. The subject matter of Budde’s initial letter can be in-
ferred from Einstein’s response in which he pointed out that one would expect
the phenomenon to be extraordinarily rare, and that it could not be detected
on photographic plates “as little circles” since irradiation would diffuse the
images that would hence only appear as bright little discs, indistinguishable
from the image of a regular star.
6For historical discussion, see [Norton 1984], [Janssen et al. 2007], and further refer-
ences cited therein.
7AEA 123-079. The letter will be published in the forthcoming volume of the Collected
Papers of Albert Einstein.
8Budde had studied catholic theology and science, and had worked as a secondary
school teacher and as a correspondent for the German daily Kölnische Zeitung in Paris,
Rome, and Constantinople. In 1887, he became a Privatgelehrter in Berlin, edited the
journal Fortschritte der Physik, and entered the company Siemens & Halske as a physicist
in 1892. In 1911, he retired and moved to Feldafing, near Lake Starnberg, since he had been
advised by his physicians to live at an altitude of at least 600m [Laue 1921, Werner 1921].
9In [Norton 1992, pp. 309–310] this textbook is cited as evidence for the argument that
Grossmann’s generalization of the term ‘tensor’ in [Einstein and Grossmann 1913] was an
original development.
10[Budde 1914b], [Budde 1914c].
The interesting part of Einstein’s response follows after this negative com-
ment. Einstein continued to relate that he himself had put his hopes on a
different aspect, namely that “due to the lensing effect” the distant star
would appear with an “immensely increased intensity,” and that he initially
had thought that this would provide an explanation of the “new stars.” He
went on to list three reasons why he had given up this hope after more
careful consideration. First, the temporal development of the intensity of a
nova is asymmetric. The luminosity increases much faster than it declines
again. Second, the color of the novae usually changes towards the red and,
in general, its spectral character changes in a distinct and characteristic way.
Third, the phenomenon would be very unlikely for the same reasons that the
observation of an Einstein ring would be unlikely.
In the beginning of his letter, Einstein pointed out that Budde’s idea con-
cerned the same thing that “about half a year ago” (“vor etwa einem halben
Jahre”) had put him into “joyous excitement” (“freudige Aufregung”). At
the end of the letter, he again wrote that the joy had been “just as short as
it had been great.” Counting back six months from the date of Einstein’s
letter, 22 May 1916, takes us to the 22nd of November 1915, which is just
the time of the final formulation of general relativity. It is also just another
six weeks or so away from the date of his letter to Zangger of early October,
in which he wrote about the very same subject of the possible explanation
of novae as a phenomenon of gravitational lensing.
2 The lensing calculations in the scratch note-
In light of this new letter, let us briefly reexamine the calculations in the
Scratch Notebook that had been dated to April 1912.11 Stellar gravitational
lensing is an implicit consequence of a law of the deflection of light rays in
a gravitational field. Such a law had been obtained by Einstein in 1911 as
a direct consequence of the equivalence hypothesis. The angle of deflection
11The following brief recapitulation refers to [CPAE3, 585–586], or
http://www.alberteinstein.info/db/ViewImage.do?DocumentID=34432&Page=23 and
· · ·&Page=26. For a complete and detailed paraphrase of Einstein’s notes, see the
Appendix below.
http://www.alberteinstein.info/db/ViewImage.do?DocumentID=34432
Figure 1: The geometric constellation for stellar gravitational lensing as
sketched in Einstein’s Scratch Notebook. From [CPAE3, p. 585].
α̃12 was found to be
where k is the gravitational constant, M the mass of the lensing star, c
the speed of light, and ∆ the distance of closest approach of the light ray
measured from the center of the massive star.13 On [p. 43] of the Scratch
Notebook we find the sketch shown in Fig. (1) and underneath it the lensing
equation
r = ρ
R +R′
where R denotes the distance between the light emitting distant star and
the massive star that is acting as a lens, R′ the distance between the lensing
star and the position of a terrestial observer who is located a distance r away
from the line connecting light source and lensing star. ρ is the distance of
closest approach of a light ray emitted by the star and seen by the observer.
α = 2kM/c2 is a typical length (later known as the Schwarzschild radius)
that depends on the mass of the light deflecting star and that determines
12I am using the notation α̃ instead of α (as in [Einstein 1911]) in order to distinguish
this angle from the quantity α (effectively the Schwarzschild radius) in Einstein’s scratch
notebook.
13[Einstein 1911, p. 908]. Qualitatively, Einstein had already derived the consequence of
light bending in a gravitational field when he first formulated his equivalence hypothesis
[Einstein 1907, p. 461]. In the final theory of general relativity, the same relation is
obtained with an additional factor of 2, as observed explicitly in [Einstein 1915c, p. 834].
Incidentally, the relevant formula was printed incorrectly by a factor of 2 in (the first
printing of) Einstein’s 1916 review paper of general relativity [Einstein 1916, p. 822], see
[CPAE6, Doc. 30, n. 36] and also Einstein’s response to Carl Runge, 8 November 1920
[CPAE10, Doc. 195].
the angle of deflection to be α
. The lensing equation can be written in
dimensionless variables as
r0 = ρ0 −
, (1)
after defining r0 and ρ0 as
r0 = r
R′(R +R′)α
ρ0 = ρ
R +R′
. (2)
The fact that equation (1) is a quadratic equation for ρ0 entails that there are
two solutions which correspond to two light rays that can reach an observer,
along either side of the lensing star,14 and hence that a terrestial observer
will see a double image of the distant star. For perfect alignment, the double
image will turn into a ring shaped image, an “Einstein-ring” whose diameter
0 = ρ
= 1 also follows immediately from the lensing equation.
In light of Einstein’s letters to Zangger and Budde, it is interesting that
Einstein went on to compute also the apparent magnification, obtaining the
following expression:
Htot = H
. (3)
Here Htot is the total intensity received by the observer, and H the intensity
of the star light at distance R. ρ1,2 denote the two roots of the quadratic
equation (1). The term in brackets gives the relative brightness, reducing
to 1 if no lensing takes place. Finally, some order of magnitude calculations
on these pages showed that the probability of observing this effect would be
given by the probability of having two stars within a solid angle that would
cover 10−15 of the sky, which is highly improbable given that the number of
known stars at the time was of the order of 106.15
Equations that are entirely equivalent to these were published much later,
in 1936, in Einstein’s note to Science.16
14Since only three points are given, the problem is intrinsically a planar one, as long as
the three points are not in perfect alignment.
15See the discussion in the appendix.
16[Renn, Sauer, and Stachel 1997].
The dating of the lensing notes in the scratch notebook to Einstein’s visit
in Berlin in April 1912 was based on other evidence in the notebook. Most
importantly, p. [36] lists Einstein’s appointments during his Berlin visit. In
addition, pp. [38] and [39] recapitulate very specifically the equations of Ein-
stein’s two papers on the theory of the static gravitational field of February
and March 1912, respectively.17 The calculations that deal with the lensing
problem then appear on pp. [43]-[48], and on pp. [51] and [52] of the note-
book. The sheet containing pp. [44] and [45] is a loose sheet inserted between
p. [43] and p. [45]. After p. [53], three pages have been torn out, and then
follow 37 blank pages, with some pages torn out in between. The remainder
of the notebook contains entries that begin at the other end of the notebook
which was turned upside down. Except for some apparently unrelated and
undated entires on pp. [49], [50],18 and [54], the lensing calculations hence
are at the end of a more or less continuous flow of entries. These physical
characteristics of the notebook lead to an important consequence. All infor-
mation that was pointing to a date of the lensing calculations in the year
1912 preceded the actual lensing calculations. Reexaming pp. [51] and [52]
of the notebook in light of the letters to Zangger and to Budde in fact reveals
that at least these entries were not written in 1912, but rather most likely
at the time of the letter to Zangger, in early October 1915. There are two
reasons for this. First, at the top of p. [51], Einstein wrote down the title
of a book published only in 1914.19 Therefore, the following calculations are
almost certainly to be dated later than the publication of this book. Second,
at the bottom of p. [52], Einstein explicitly refers to the “apparent diameter
of a Nova st[ar].” The calculations on pp. [51] and [52] in fact are a calcula-
tion of the apparent brightness and diameter of a star. We conclude that, in
all probability, the calculations on pp. [51] and [52] were written at the time
of Einstein’s letter to Zangger, early October 1915.
Does the dating of pp. [51] and [52] to October 1915 also compel us to
17[Einstein 1912a, Einstein 1912b].
18On the bottom half of p. [49] there is a sketch of Pascal’s and Brianchon’s Theorems,
which deal with hexagons inscribed in or circumscribed on a conical section. I wish to
thank Jesper Lützen for this identification. Other entries on pp. [49] and [50] also appear
to deal with problems from projective geometry. There is also a sketch of a vessel filled
with a liquid and the words “eau glyceriné” and what appears to be sketch of a magnetic
moment in a sinusoidal magnetic field.
19[Fernau 1914]. Could it be that the book was mentioned to Einstein when he met
with Romain Rolland in Geneva in September 1915, see [CPAE8, Doc. 118]?
revise our dating of the other lensing calculations in the notebook? To answer
this question, we need to consider the broader historical context of the notes.
But before doing so, we first observe that pp. [49] and [50] contain entries
that appear unrelated to the lensing problem. As shown by the detailed
paraphrase given in the appendix, the calculations on pp. [43] to [48] on
the other hand represent a coherent train of thought, as do the calculations
of pp. [51] and [52]. We also note that Einstein used a slightly different
notation on pp. [43]ff. and on pp. [51]-[52]. In the first set, he denoted the
distances between light source and lens and between lens and observer as
R and R′, respectively. On pp. [51]-[52] he used the notation R1 and R2,
respectively. He also reversed the roles of r and ρ. We conclude that there
is a discontinuity between the first set of lensing calculations on pp. [43] to
[48] and the second set on pp. [51] and p. [52].
3 The context of Einstein’s early lensing cal-
culations
From Einstein’s letter to Budde we learn that he had investigated the idea
that stellar lensing might explain the phenomenon of the “new stars,” and
that he had given up this idea after looking more closely into the character-
istic features of novae, especially their light curves and the changes in their
spectral characteristics. Let us therefore briefly look into the astronomical
knowledge about novae at the time.
The observation of a new star is an event that, in the early twentieth
century, occurred only every few years. Between 1900 and 1915, eight novae
were observed:20 Nova Persei 1901 (GK Per), Nova Geminorum (1) 1903 (DM
Gem), Nova Aquilae 1905 (V604 Aql), Nova Vela 1905 (CN Vel), Nova Arae
1910 (OY Ara), Nova Lacertae 1910 (DI Lac), Nova Sagittarii 1910 (V999
Sgr), and Nova Geminorum (2) 1912 (DN Gem) with maximum brightness
of 0.2, 4.8, 8.2, 10.2, 6.0, 4.6, 8.0, 3.5 magnitudes, respectively. At the time,
“the two most interesting Novae of the present century,” [Campbell 1914,
p. 493], were Nova Persei of 1901 and Nova Geminorum of 1912. The next
spectacular nova to occur was the very bright Nova Aquilae 1918 (V603 Aql)
with a maximum brightness of −1.1 mag.
Nova Geminorum (2) was discovered on March 12, 1912, by the as-
20For the following, see [Duerbeck 1987].
Figure 2: The light curve of Nova Geminorum 1912 for the first three
months after its appearance, as put together by Fischer-Petersen on the
basis of 253 individual observations. The points are the magnitudes re-
ported by the individual observers, the solid line is to guide the eye. From
[Fischer-Petersen 1912, p.429].
tronomer Sigurd Enebo at Dombaas, Norway [Pickering 1912]. On a pho-
tographic plate taken at Harvard College Observatory on March 10, showing
stars of magnitude 10.5, it was not visible, but it was visible as a magni-
tude 5 star in the constellation Gemini on a Harvard plate of March 11. On
March 13, a cablegram was received at Harvard and distributed throughout
the United States. In the following days all major observatories as well as
many amateur astronomers pointed their instruments towards the new star.
The maximum brightness of mag 3.5 was reached on March 14 (Einstein’s
33rd birthday!) [Fischer-Petersen 1912]. By March 16, the brightness was
down to a magnitude of 5.5 and in the following weeks it decreased further,
with distinct oscillations. By mid-April 1912, most observers registered a
brightness of mag 6 ≈ 7, see Fig. (2). We now know that the DN Gem is a
fast nova with a t3-time of 37d. Its light curve is type Bb in the classification
of [Duerbeck 1987], i.e. it declines with major fluctuations.
Like all classical novae, Nova Geminorum is, in fact, a binary system of
a white dwarf and main sequence star, where hydrogen-rich matter is being
accreted onto the white dwarf. Recent observations have even determined the
binary period [Retter et al. 1999]. The eruption of a classical nova occurs
when a hydrogen-rich envelope of the white dwarf suffers a thermonuclear
runaway.21 This explanation of classical novae also entails that they display
the same sequence of spectral behaviour as the luminosity decreases, see also
Fig. (3) below. However, our current understanding of classical novae was
suggested only in the fifties.22
The temporal proximity of the appearance of Nova Geminorum 1912 with
Einstein’s Berlin visit during the week of April 15–22, suggests that this
astronomical event was discussed also when Einstein met with Freundlich
for the first time.23 We know that the observatory in Potsdam took a
number of photographs of the new star between March 15 and April 12
[Furuhjelm 1912, Ludendorff 1912], and that Freundlich, among others, was
charged with photometric observations of the nova [Fischer-Petersen 1912,
p. 429]. Einstein and Freundlich had earlier corresponded about the possib-
lity of observing gravitational light deflection through the gravitational field
of the sun.24 The purpose of their meeting was to discuss possible astro-
nomical tests of Einstein’s emerging relativistic theory of gravitation. The
recent observation of the brightest nova since 1901 must have been on Fre-
undlich’s mind, and it seems more than likely that the idea of explaining
the phenomenon in terms of gravitational lensing therefore came up in the
course of their conversation. We conclude that our earlier dating of the first
set of calculations of the lensing problem in the Scratch Notebook to the
time of Einstein’s encounter with Freundlich in April 1912 is the most likely
possibility.
In fact, the context of the observation of Nova Geminorum 1912 provides
an answer to the question as to why Einstein would have done the calculations
at all and, in particular, why he would not have been content at the time
with a calculation of the lensing equation, the separation of the double star
image and, perhaps, the radius of the Einstein ring. Without this context
it might seem a rather ingenious move on Einstein’s part to go ahead and
immediately compute the apparent magnification of the lensed star as well.
But this answer to the question of motivation for the specific details of the
21For a review, see [Shara 1989].
22For a historical overview of previous theories, see [Duerbeck 2007].
23For evidence that Einstein met with Freundlich, see his letter to Michele Besso, 26
March 1912, in which he mentions planned discussions (“Besprechungen”) with Nernst,
Planck, Rubens, Warburg, Haber, and “an astronomer”—presumably Freundlich [CPAE5,
Doc. 377].
24Einstein to Freundlich, 1 September 1911, 21 September 1911, and 8 January 1912
[CPAE5, Docs. 281, 287, 336].
Figure 3: Changes in the spectrum of Nova Geminorum 1912, March 22 to
August 19, 1912. From [Adams and Kohlschütter 1912].
calculations in the Scratch Notebook, immediately raises another question.
Assuming that the first set of lensing calculations were done in spring
1912, why do we have no evidence that this idea was followed up by either
Einstein or by Freundlich until the fall of 1915? To answer this question,
it should first be observed that no summarizing results and analyses of the
observations of Nova Geminorum 1912 were published before the end of the
summer.
Let us briefly recall Einstein’s intellectual preoccupations after his visit
to Berlin in April 1912.25 Shortly before his trip to Berlin he had submitted
his two papers on a theory of the static gravitational field.26 After his return
to Prague in April 1912, Einstein was preparing for his move to Zurich. The
two papers were published in the 23 May issue of the Annalen der Physik.
Einstein wrote an addendum at proof stage to the second one, in which
he showed that the equations of motion could be written in a variational
form, adding that this would give us “an idea about how the equations of
motion of the material point in a dynamic gravitational field are constructed”
[Einstein 1912b, p. 458]. He also entered into a published dispute with Max
Abraham on their respective theories of gravitation.27 At the end of July,
he departed Prague for Zurich. The next thing we know about his work on
gravitation comes from a letter to Ludwig Hopf, dated 16 August 1912, in
which he wrote:
The work on gravitation is going splendidly. Unless I am com-
pletely wrong, I have now found the most general equations.28
These most general equations are, in all probability, equations of motion
in a gravitational field, represented by a metric tensor. After his arrival
in Zurich, Einstein began a collaboration with his former classmate Marcel
Grossmann, now his colleague at the ETH. Their research on a generalized
25We will focus here on his work of gravitation yet for the sake of completeness it should
be noted that Einstein at the same time was also thinking about quantum theory, most
notably about the law of photochemical equivalence and about the problem of zero point
energy, see [CPAE4, Docs. 5, 6, 11, 12].
26[Einstein 1912a], [Einstein 1912b], were received by the Annalen der Physik on 26
February and 23 March, respectively.
27[Einstein 1912c] which was received by the Annalen on 4 July 1912 is a response to a
critique by Abraham.
28Einstein to Hopf, 16 August 1912 [CPAE5, Doc. 416].
theory of relativity is documented in Einstein’s so-called “Zurich Notebook”29
and culminates in the publication of the “Outline [Entwurf] of a generalized
theory of relativity and a theory of gravitation,” in early summer of 1913 co-
authored with Marcel Grossmann.30 This so-called Entwurf-theory contains
all the elements of the final theory of general relativity, except for generally
relativistic field equations. Einstein would hold onto this theory until his
final breakthrough to general relativity in the fall of 1915.
In conclusion, we observe that Einstein’s path toward the general theory
of relativity in 1912 took him deep into the unknown land of the mathematics
associated with the metric tensor, before there was a chance to reconsider
the lensing idea in light of the data for Nova Geminorum 1912. In any case,
he would have to rely on Freundlich or other professional astronomers for a
secure assessment of the possibilities of an observation of the lensing effect
at the time.
Freundlich, on the other hand, continued to think about ways to test
Einstein’s new theory of gravitation.31 But his focus was on observations
of light deflection during a solar eclipse.32 In August 1914, he led a first
(unsuccessful) expedition to the Crimea to observe the eclipse of 21 August
1914. Even these efforts were hampered by the lack of funding and, more
generally, by the difficulties of securing increased research time that would
have allowed Freundlich to freely pursue his collaboration with Einstein.
Given these circumstances, and the fact that order-of-magnitude calcu-
lations may have convinced Einstein already in 1912 that the phenomenon
would be rare, it seems plausible that the lensing idea was not pursued further
for some time after Einstein’s visit in Berlin in April 1912.
Let us finally reexamine the events of fall 1915. Einstein, in the meantime
had left Zurich in the spring of 1914, accepting an appointment as member
of the Prussian Academy in Berlin. In September 1915, Einstein spent a few
weeks in Switzerland where he met, among others, with Heinrich Zangger,
Michele Besso, and Romain Rolland. On 22 September 1915, he left Zurich33
but travelled via Eisenach where he was on the 24th of September.34 By the
29AEA 3-006, see [CPAE4, Doc. 10]. For a comprehensive discussion of this document,
including a facsimile, transcription, and detailed paraphrase, see [Janssen et al. 2007].
30[Einstein and Grossmann 1913].
31See [Hentschel 1994] and [Hentschel 1997].
32See his correspondence with Einstein in [CPAE5].
33[CPAE8, p. 998].
34[CPAE10, Doc. Vol. 8, 122a].
30th of September, at the latest, he was back in Berlin, and wrote a letter
to Freundlich:
I am writing you now about a scientific matter that electrifies me
enormously.35
It is clear from the letter, however, that the excitement indicated to Fre-
undlich is not about the idea of gravitational lensing. Rather, Einstein
had found an internal contradiction in his Entwurf theory that amounted
to the realization that Minkowski space-time in rotating Cartesian coordi-
nates would not be a solution of the Entwurf field equations.36 This insight
undermined his confidence in the validity of the Entwurf theory, and is later
mentioned as one of three arguments that induced Einstein to lose faith in
the Entwurf equations.37 The first of these arguments was the fact that a cal-
culation of the planetary perihelion advance in the framework of the Entwurf
theory did not produce the well-known anomaly that had been established
for Mercury. This problem had been known to Einstein for some time.38
The third argument was realized sometime in early October, a few days after
stumbling upon the problem with rotation, and concerned the mathematical
derivation of the Entwurf field equations in Einstein’s comprehensive review
of October 1914.39 In any case, we know that Einstein asked Freundlich to
look into the problem of the rotating metric, and that they met some time in
early October. This follows from a letter Einstein wrote to Otto Naumann,
35Einstein to Freundlich, 30 September 1915 [CPAE8, Doc. 123]. For a detailed discus-
sion of this letter and its significance for the reconstruction of Einstein’s final breakthrough
to general relativity, see [Janssen 1999].
36Interestingly, the Scratch Notebook contains an entry that is pertinent to this problem.
On p. [66], i.e. on the last page of the backward end of the notebook, Einstein considers the
case of rotation in a calculation that exactly matches corresponding calculations dating
from October 1915, see [Janssen 1999]. Janssen cautiously remarks that he believes this
calculation to date from 1913 [Janssen 1999, p. 139]. It seems possible, however, that these
entries as well as the immediately preceding ones on the perihelion advance (see note 38)
may well date from late 1915 as well.
37See Einstein to Arnold Sommerfeld, 28 November 1915, and to Hendrik A. Lorentz, 1
January 1916 [CPAE8, Docs. 153, 177].
38See [Earman and Janssen 1993] and [CPAE4, pp. 344–359]. The Scratch Notebook
contains some calculations related to the perihelion advance on pp. [61–66], i.e. in the
backward end of the notebook. On p. [61], Einstein there explicitly noted that the advance
of Mercury’s perihelion would be 17′′ which is the value that is obtained on the basis of
the Entwurf-theory. These calculations are undated, see note 36.
39[Einstein 1914].
dated after 1 October 1915, in which Einstein asked about possibilities to al-
low Freundlich more freedom to pursue independent research. In this letter,
Einstein mentioned that Freundlich had visited him “recently.”40
By 12 October, Einstein had realized the third problem with the Entwurf
theory, the unproven uniqueness of the Lagrangian for the Entwurf field equa-
tions, as he reported in a letter to Lorentz. In this letter, he neither men-
tioned the problem with the rotating metric nor the issue of gravitational
lensing.41
For our reconstruction of this episode, the precise date of Einstein’s letter
to Zangger in which he remarked that he had given up the hope of explaining
the “new stars” as a lensing phenomenon is relevant. It could have been
written either on the 8th or the 15th of October.42
The letter to Zangger suggests that they had talked about the idea earlier
since Einstein seems to presuppose that Zangger knew what he was talking
about and did not explain what he meant by “lens effect” (“Linsenwirkung”).
As mentioned before, Einstein had just recently met with Zangger, as well as
with Besso before returning to Berlin. The following scenario seems therefore
plausible:
Upon returning to Berlin some time after the 24th of September 1915,
Einstein realized the problem of the rotating metric solution and wrote to
Freundlich on the 30th, asking him to look into this issue. Shortly afterwards,
the two met in person. Most likely they discussed not only the rotation
problem, but also the lensing idea. Having found troubling indications of an
inner inconsistency in the very foundations of this theory, it would have been
a natural move for Einstein to go back and reconsider early arguments such as
one based safely on the equivalence hypothesis.43 After this meeting, Einstein
40“Letzter Tage war Herr Dr. Freundlich von der Sternwarte N bei mir.” [CPAE8,
Doc. 124].
41In a letter to Hilbert, dated 7 November 1915, Einstein wrote that he realized the flaw
in his proof “about four weeks ago” [CPAE8, Doc. 136].
42The editors of [CPAE8] dated this letter explicitly to the 15th of October. It seems,
however, that the 8th is also a possibility. The letter was written on a Friday between
September 30, when a fire and explosion took place in the comb factory Walter near Lake
Biel took place, mentioned in the letter, and October 22 when Einstein participated in
the first Academy session after the summer break. I see no reason why Einstein could not
have heard of the accidents from Zangger before October 8.
43It seems unlikely that Einstein at that time was already contemplating a quantitatively
different law of light deflection. Einstein first observed in [Einstein 1915c, p. 834] that an
additional factor of 2 would arise from the different first-order approximation for the
wrote to Naumann exploring possibilities to give Freundlich more research
freedom. By October 8, Einstein had convinced himself that gravitational
lensing cannot explain the “new stars.” On 12 October, he realized the third
problem of his mathematical derivation of the Entwurf field equation.
According to this reconstruction of the sequence of events, it is remarkable
that the “joyous excitement” about the lensing idea falls within days after
his being “electrified” about the realization of the rotation problem on 30
September, and his realization of the third problem of the mathematical
derivation of the Entwurf equation, on or before 12 October 1915.44
Some five weeks later, his excitement was even greater and his heart,
allegedly, skipped a beat when he found that he could derive the anomalous
advance of Mercury’s perihelion on the basis of his new field equations. And
after having submitted the last of his four November communications to the
Prussian Academy on 25 November which presented the final gravitational
field equations, the “Einstein equations,” he wrote to Sommerfeld:
You must not be cross with me that I am answering your kind
and interesting letter only today. But in the last month I had
one of the most exciting, exhausting times of my life, indeed also
one of the most successful. I could not think of writing.45
It is interesting to learn from Einstein’s letter to Budde that in addition to
the realization of the problems with the Entwurf theory and the eventual suc-
metric if the Newtonian limit is derived on the basis of generally covariant field equations
in which the Ricci tensor is directly set proportional to the energy-momentum tensor.
These latter equations were published in his second November memoir, presented on 11
November, under the assumption that the trace of the energy-momentum tensor vanish.
In his comment on the factor of 2, Einstein refers to this result as being in contrast to
“earlier calculations” where the hypothesis of vanishing energy-momentum had not yet
been made.
44For completeness, one should point one other intellectual activity of Einstein’s during
those days. In Einstein’s letter to Zangger of 8 or 15 October, he also mentioned that
he wrote “a supplementary paper to my last year’s analysis on general relativity.” The
last year’s analysis is, in all likelyhood [Einstein 1914]; the supplementary paper is, in all
likelihood, an early version of [Einstein 1916b], or, perhaps, an early version of Einstein’s
first November memoir [Einstein 1915a], see [CPAE8, Doc. 130, note 5] and [Janssen 1999,
note 51].
45“Sie dürfen mir nicht böse sein, dass ich erst heute auf Ihren freundlichen und in-
teressanten Brief antworte. Aber ich hatte im letzten Monat eine der aufregendsten,
anstrengendsten Zeiten meines Lebens, allerdings auch der erfolgreichsten.” Einstein to
Sommerfeld, 28 November 1915 [CPAE8, Doc. 153].
cess of his breakthrough to general relativity, an astronomical problem, the
idea of explaining novae in terms of gravitational lensing added to Einstein’s
excitement in the midst of what must indeed have been the most intense
period of intellectual turmoil in his life.
4 Concluding remarks
Einstein’s recollections of his thought concerning the explanation of the “new
stars” as a phenomenon of gravitational lensing in his letter to Budde add
two significant insights to our reconstruction of the genesis of general rel-
ativity. If our dating and context hypothesis of the lensing calculations in
the scratch notebook are correct, we learn that it was an astronomical ob-
servation that triggered the elaboration of a significant consequence of the
equivalence hypothesis and its consequence of gravitational light deflection.
It is also interesting that on his intellectual path from the Entwurf theory to
the final theory of general relativity, Einstein also took a detour in which he
explored further consequences of one of the solid pillars of general relativity,
the equivalence hypothesis.
Appendix: Einstein’s lensing calculations in
the Scratch Notebook AEA 3-013
The following is a self-contained line-by-line paraphrase of Einstein’s lensing
calculations in his scratch notebook, [CPAE3, pp. 585–589]. The pagination
in square brackets refers to the sequence of pages in the notebook.
The calculations start out on p. [43] with Fig. (1) and continue on the
facing page p. [46]. From the more explicit sketch in Fig. (4), we read off the
lensing equation:
r = ρ
R +R′
. (4)
Here R is the distance between the light emitting star S and the lensing star
L; R′ the distance between the massive star L and the projected position of
the observer O on the line connecting light source and lens; ρ is the distance
of closest approach of a light ray emitted from the distant star and seen by
an observer; r is the orthogonal distance of the terrestial observer to the
line connecting light source and lens. The first term in the lensing equation
Figure 4: The geometry of stellar lensing.
(4) is obtained from the similarity of triangles with baseline R and R + R′,
respectively, and the second term is the angle of deflection as given by the law
of gravitational light bending, where α is the Schwarzschild radius 2GM/c2.
If we want to write this equation in dimensionless variables, we need to
multiply it by a factor of
R′(R +R′)α
so that, when we define r0 and ρ0 as
r0 = r
R′(R +R′)α
ρ0 = ρ
R +R′
the lensing equation (4) turns into
r0 = ρ0 −
. (8)
This is a quadratic equation for r0, the two solutions of which correspond to
the two light rays passing above and below L. The observer O therefore sees
two images of S at positions S ′ and S ′′, respectively. To read off the radius
of an “Einstein ring,” obtained for perfect alignment of S, L, and O, one
only needs to set r0 ≡ 1.
In order to get an expression for the apparent magnification, Einstein
proceeded as follows. He first took the square of eq. (8) as
2 + r2 = ρ2 +
. (9)
If we multiply this equation by π and denote the areas of the circles corre-
sponding to the radii r and ρ as f = πr2 and ϕ = πρ2, respectively, we can
write this equation as
2π + f = ϕ+
. (10)
We are not interested in the full circle corresponding to these radii but in the
differential area element associated with these radii. More precisely, we are
interested in the change of the differential area element df associated with f
when we change the differential area element dϕ associated with ϕ. Hence,
Einstein wrote
dϕ. (11)
The intensity H of the brightness received at r is related to the intensity H
of the brightness at ρ by
Hdf = ±Hdϕ, (12)
where the plus and minus signs refer to the two solutions of the quadratic
equation. Since we have from (11)
, (13)
we get
H = ± H
. (14)
or, inserting the explicit solutions, we can write the total brightness at r as
Htot = H
. (15)
As Einstein remarked, the term in brackets gives the relative brightness, if
we take the value for r → ∞ to be 1.46 This result is equation number (3)
46“Klammer gibt relative Helligkeit”
in Einstein’s notes, and most of the following material on pp. [47] and [48],
as well as on the loose sheet containing pp. [44] and [45], will be a discussion
of this expression for the relative brightness.
On p. [47], Einstein first rewrote the reduced lensing equation as
− x, (16)
and then the terms in brackets as
1− x41
x42 − 1
. (17)
The next step is to bring the two terms to a common denominator47
x41 − x42
(1− x41)(1− x42)
. (18)
If one squares the lensing equation (16) twice, one obtains
− 2 + (2 + r2)2 = 1
+ x4. (19)
If we now introduce new variables A and u via
2A = −2 + (2 + r2)2, (20)
A = −1 +
(2 + r2)2 = 1 + 2r2 +
r4, (21)
u = x4, (22)
we can write the quadrupled equation (19) as
2A = u+
. (23)
Multiplication by u and adding A2 on each side gives
u2 − 2Au+ A2 = −1 + A2, (24)
47In the notes, Einstein refers to this step as “Rationalisierung”.
from which one can immediately read off the two solutions of eq. (23) as
u = −A±
A2 − 1. (25)
Given (18), the difference between the two roots,
u1 − u2 = 2
A2 − 1, (26)
provides an expression for the nominator of Hr in (18). With the two roots,
we can also rewrite the quadratic equation in the form
u2 − 2Au+ 1 = (u− u1)(u− u2), (27)
and if we now set u = 1, we obtain
2(1−A) = (1− u1)(1− u2), (28)
which gives us an expression for the denominator of Hr in (18). Combining
the two expressions, as Einstein did on p. [48], we obtain
1 + 1
) , (30)
where we have inserted (21) to obtain the second line.
We now have an explicit expression for the relative brightness as a func-
tion of the dimensionless variable r. We now evidently see that Hr → 1/r
for r → 0, and that Hr approaches 1 asymptotically from above for large r,
see Fig. (5).
Let us now reconstruct Einstein’s order-of-magnitude estimate for the
expected frequency of the phenomenon on p. [45]. The explicit expression
for the relative brightness gives us a measure of the maximal distance r for
which significant magnification is obtained. We can look at specific values of
Hr(r). For instance, for r0 = 12 we find
1 + 1
5 ≈ 2. (31)
0 1 2 3
H (r)
Figure 5: A plot of the expression (30) for the relative brightness Hr as a
function of r. The inset is from [CPAE3, p. 587].
Hence, Einstein concluded that up to a distance of r0 =
one would obtain
an increase of the intensity by a factor of 2. In other words, if we write the
intensity Hr0 asymptotically for small r0 and R′ ≫ R as
R′(R +R′)α
, (33)
we see that for a lensing star at a distance of R, the relative increase in
intensity is given by
= tg ᾱ. (34)
Here ᾱ is the angle that determines how well the distant star has to be aligned
with the lensing star and the observer to produce appreciable magnification.
In order to get an order-of-magnitude estimate for this angle, one needs an
order-of-magnitude estimate for
. In order to obtain such an estimate,
Einstein notes that the ratio of the solar Schwarschild radius α to the solar
equatorial radius Rs is given approximately by
= 3 · 10−6. (35)
The radius of the sun is 2 light seconds, and the distance of the nearest stars
is of the order of 10 light years, or
105 · 365 · 10 ≈ 4 · 108 lightseconds. (36)
It follows that α
for a star of 1 solar mass 10 lightyears away is
≈ 10−14 or
≈ 10−7. (37)
To see the distant star with double intensity, we therefore have
tg ᾱ
, (38)
so that the angle ᾱ is of order 10−7. A linear angle corresponds to a solid
angle roughly by taking its square. Thus, the angular size of the region
where the distant star needs to be found behind a massive star in order to
be magnified in the lens is of order 10−14. In angular units, the total sky has
an area of 4π ≈ 10, so that the angular size of the region in question covers a
fraction of 10−15 of the total sky. This has to be contrasted with the average
density of stellar population in the sky. The Bonner Durchmusterung listed
of the order of 3 · 105 stars to ninth magnitude for the northern hemisphere,
so a reasonable average density of the number of stars would be 1 star per
10−5 of the sky.48
On the back of the loose sheet [p. 44] we find a few more calculations
related to order-of-magnitude estimates that start from (32). Einstein here
again goes back to the definition of r0 and ρ0 in terms of R, R
′, and α.49
Again, he observes that r0 =
would give twice the usual intensity, and
rewrites (6) for this case:
R′(R +R′)α
. (39)
The latter equation for R′ ≫ R turns into
, (40)
and for R ≪ R′ into
αR′. (41)
Einstein concluded that the smaller of the two distances R and R′ determines
the angle r
. In the top right corner of the page, Einstein jotted down another
order-of-magnitude calculation, which I do not fully understand. Apparently,
he computed the distance of 100 lightyears in terms of centimeters
3 · 1010 · 3 · 107 · 102 [cm] ≈ 1020 cm (42)
48On the relevant page under discussion here, we also find a little sketch by Einstein of
a circle and the angle of its radius for a point some distance away. The precise meaning
of this sketch is unclear but the numbers written next to it suggest that Einstein was
considering the order of magnitude for the angular size of the moon. The radius of the
moon is seen under an angle of 15′ from the earth, and the mean distance between the
earth and the moon in units of the lunar radius is about 200, which translates to an angle
of 50o.
49One can see here that Einstein corrected an error in his earlier calculations on [p. 43],
where he had erroneously written the second term of the lensing equation (4) with R
instead of R′, which resulted in a confusion of the factors of R and R′ in expressions (5)
and (6).
Figure 6: A sketch in Einstein’s scratch notebook to obtain eq. (43). From
[CPAE3, p.585].
He also computed the angle x under which the star at distance R′ and
the star at distance R+R′ would be seen by an observer at distance r away
from the connecting line between the two stars if no lensing took place:
x = r
R +R′
R′(R +R′)
. (43)
The first equation can be read off from a little sketch of the geometry of light
source, lensing star, and observer, at the bottom of the page, see Fig. (6).
Let us finally comment on the calculations on pp. [51] and [52]. As men-
tioned in the main text of this article, Einstein here introduced a change
of notation. On p. [51], he sketched again the geometry for stellar lensing.
Here, the geometry has been turned by 90 degrees, and the notation changed
so that R and R′ become R1 and R2, and ρ and r are interchanged to be-
come r and ρ, respectively. This change of notation is reflected in the lensing
equation, written down on p. [52] as
ρ = r +R1
, (44)
where tanw = r/R2. Einstein then immediately proceeded to compute the
magnification by taking the square of the lensing equation and then comput-
ing the derivative as
d(ρ2)
d(r2)
(R1α)
= A ·
. (45)
Instead of pursueing this calculation further, Einstein instead wrote “appar-
ent diameter of a Nova star,” and wrote down the solution of eq. (44) for
ρ = 0, as to obtain the diameter of an Einstein ring:
R1R2α
R1 +R2
. (46)
He computed the angle w0 as
R2(R1 +R2)
. (47)
The calculation ends with an attempt at a numerical order-of-magnitude
estimation which seems to proceed along the same lines as in eqs. (35,36).
The calculation, however, was broken off, and the whole page was struck
through.
Acknowledgments
I wish to thank Diana Buchwald for a critical reading of an earlier version of
this paper, and Hilmar Duerbeck for some helpful comments. Unpublished
correspondence in the Albert Einstein Archives is quoted by kind permission.
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	Einstein's letter to Emil Budde
	The lensing calculations in the scratch notebook
	The context of Einstein's early lensing calculations
	Concluding remarks
