The Formative Years of Relativity and a Prejudice on Poincaré’s Conventionalism

The purpose of this piece is to review Hanoch Gutfreund’s and Jürgen Renn’s new book The Formative Years of Relativity: The History and Meaning of Einstein’s Princeton Lectures, Princeton University Press and Oxford University Press. I find two problems in the book the first of which is Poincaré’s influence on Einstein. This is the first part of the review which deals with Poincaré’s influence on Einstein. Since the book has mistakes and also errors in English and Jürgen Renn is considered a notable scholar, I assume that Gutfreund is probably responsible for the mistakes and for the errors in English.

Let us begin with page 26 of the book The Formative Years of Relativity. Gutfreund is apparently much attracted by “the painstaking analysis by the philosopher of science Yemima Ben-Menachem” in her book Conventionalism: From Poincaré to Quine. Cambridge University Press, Cambridge (2006):

Yemima 1921

Gutfreund writes that “Until 1921, Einstein did not mention Poincaré explicitly”.

Einstein obviously mentioned Poincaré before 1921. For instance, after the first Solvay congress in 1911, Einstein wrote to Heinrich Zangger (see the Collected Papers of Albert Einstein, CPAE):




Einstein says that Poincaré was in general simply antagonistic and for all his acuity showed little understanding of the situation.

There was then great excitement among philosophers and historians of science when they discovered, as Gutfreund writes on page 26 above that “We know that he [Einstein] read Science and Hypothesis with his friends in the Akademie Olympia in 1902″.

Poincaré’s publisher Flammarion published La science et l’hypothèse (Science and Hypothesis) in Paris in 1902. How do we know that Einstein read this book in 1902 and not in 1903 or 1904? Take a look at Gutfreund’s words: “We know that he [Einstein] read Science and Hypothesis with his friends in the Akademie Olympia in 1902″. It means that Einstein rushed to the local bookstore in Bern the day the book was out, dodged people in the crowd waiting outside the bookstore and found the first French edition of Poincaré’s book. But maybe Einstein read the 1904 German translation of Poincaré’s 1902 book? This could be quite different from the original 1902 French edition.

Subsequently,  on page 26 Gutfreund writes: “Einstein’s biographer Abraham Pais quotes one of the members, Maurice Solovine, as saying: ‘This book profoundly impressed us and kept us breathless for weeks on end'”:


Gutfreund simply takes the Pais paragraph from Yemima Ben-Menahem’s book, Conventionalism, see footnote 80 below (page 134):


This is a mistake: We cannot cite Einstein’s biographer Abraham Pais quoting one of the members of the Akademie Olympia (Olympia Academy). The biography of Pais is not a primary source. We have to check a primary source and see whether Maurice Solovine himself said: “This book profoundly impressed us and kept us breathless for weeks on end”.

Here is the original primary source:


Lettres à Maurice Solovine. Paris: Gauthier-Villars, 1956.

Here luck plays an important role because in the above book Solovine writes in French that Poincaré’s book “profoundly impressed us and kept us breathless for many weeks”. One should, however, check the original quote in French.

On page 30 Gutfreund tells the story of Einstein who explored “a famous example that goes back Poincaré”. There are several typos in the book.

Poincare typo

It is interesting, however, to look at the following sentence, several sentences below the above one on page 30:

He typo

“Without the distinction between axiomatic Euclidean geometry and practical rigid-body geometry, we arrive at the view advanced by Poincaré”. And then Gutfreund adds an end-note 15: “For an extensive analysis of Poincaré’s conventionalism, see Yemima Ben-Menachem, […]” Her book Conventionalism.

Yemima 1921-2

You might, of course, be tempted to suppose that Ben-Menahem has said the above words in her book. But this is by no means the case. Einstein says this in his 1921 talk, “Geometry and Experience” (see CPAE):

geometry and experience

After the words: “Without the distinction between axiomatic […]” Gutfreund writes: “He suggested that […]”

He typo

Who is “He”? Einstein or Poincaré?

Let us then examine Yemima Ben-Menahem’s book, Conventionalism.

I am quoting from Yemima Ben-Menahem’s book Conventionalism, page 84:

“… as both GR [general relativity] and the special theory of relativity originated in insights about equivalence, an element of conventionality might seem to be built right into the theory.  It is important to recognize, however, that Einstein’s use of equivalence arguments differs fundamentally from that of the conventionalist”.

And on page 134 Ben-Menahem writes: “The preceding discussion should alert us to the traces of Poincaré’s equivalence argument in Einstein’s work on GR as well. […] The centrality of equivalence arguments and their geometric implications is too obvious in Science and Hypothesis to be missed by a reader such as Einstein, who, we know, was familiar with the book. Beginning with the hypothesis of equivalence in 1907, Einstein makes use not only of the general idea of equivalent descriptions, but also of the types of examples Poincaré used”.


and on page 135, Yemima Ben Menahem argues that “Einstein was deeply influenced by the idea of equivalence, and to that extent could concede that Poincaré was right”:

page 135

I have found no historical evidence (primary documents, i.e. correspondence of Einstein with others, manuscripts, and also interviews with Einstein) supporting the claim that Einstein makes use of Poincaré’s equivalent descriptions.

In the 1920 unpublished draft of a paper for Nature magazine, “Fundamental Ideas and Methods of the Theory of Relativity, Presented in Their Development”, Einstein explained how he arrived at the principle of equivalence (see CPAE):

happiest happiest2

(Original in German). “When I (in Y. 1907) [in Bern] was busy with a comprehensive summary of my work on the special theory relativity for the ‘Jahrbuch für Radioaktivität und Elektronik’, I also had to try to modify Newton’s theory of gravitation in such a way that its laws fitted into the theory. Attempts in this direction showed the feasibility of this enterprise, but did not satisfy me, because they had to be based upon unfounded physical hypotheses. Then there came to me the happiest thought of my life in the following form:

The gravitational field is considered in the same way and has only a relative existence like the electric field generated by magneto-electric induction. Because for an observer freely falling from the roof of a house there is during the fall – at least in his immediate vicinity – no gravitational field. Namely, if the observer lets go of any bodies, they remain relative to him, in a state of rest or uniform motion, regardless of their particular chemical and physical nature. The observer is therefore justified in interpreting his state as being ‘at rest’.

The extremely strange experimental law that all bodies fall in the same gravitational field with the same acceleration, immediately receives through this idea a deep physical meaning. If there were just one single thing that fell differently in a gravitational field from the others, the observer could recognize with its help that he was in a gravitational field and that he was falling in the latter. But if such a thing does not exist – as experience has shown with great precision – then there is no objective reason for the observer to regard himself as falling in a gravitational field. Rather, he has the right to consider his state at rest with respect to gravitation, and his environment as field-free.

The experimental fact of independence of the material of acceleration, therefore, is a powerful argument for the extension of the relativity postulate to coordinate systems moving nonuniformly relative to each other”.

Isaac Newton had already recognized that Galileo’s law of free fall was connected with the equality of the inertial and gravitational mass. In approximately 1685, Newton realized that there was an (empirical) equality between inertial and gravitational mass (Newton 1726, Book I, 9). For Newton, however, this connection was accidental. Einstein, on the other hand, said that Galileo’s law of free fall could be viewed as Newton’s equality between inertial and gravitational mass, but for him the connection was not accidental.

Hence, Einstein made use of Newton’s equality (accidental equivalence) between inertial and gravitational mass and Galileo’s law of free fall and in his 1907 paper, “On the Relativity Principle and the Conclusions Drawn from It”, he invoked a new principle, the equivalence principle or hypothesis. He assumed the complete physical equivalence of a homogeneous gravitational field and a corresponding (uniform) acceleration of the reference system. Acceleration in a space free of homogeneous gravitational fields is equivalent to being at rest in a homogeneous gravitational field.

Ernst Mach criticized Newton’s bucket experiment. He said that we cannot know which of the two, the water or the sky, are rotating; both cases produce the same centrifugal force. Mach thus expressed a kind of equivalence principle: Both explanations lead to the same observable effect. Einstein could have been influenced by Mach’s idea that we cannot know which of the two, the water or the sky, are rotating. Indeed Charles Nordmann interviewed Einstein and wrote: “Perhaps even more than Poincaré, Einstein admits to have been influenced by the famous Viennese physicist Mach”.

On page 31, Gutfreund writes in his book:


“Had he [Einstein] instead accepted the conventionalist position […]” and then Gutfreund writes: “This in fact is exactly the situation in which Einstein introduced the mental model of a rotating disk, which he used as early as 1912 to show that the new theory of ravitation requires a new framework for space and time”.

Another typo: it should be the new theory of gravitation.

The rotating disk story starts with a problem in special relativity, with Max Born’s notion of rigidity and not with Poincaré! Einstein never mentioned any influence Poincaré had had on him when inventing the disk thought experiment.

At the annual eighty-first meeting of the German Society of Scientists and Physicians in Salzburg on 21-25 September 1909, Born first analyzed the rigid body problem and showed the existence of a class of rigid motions in special relativity.

John Stachel describes this state of affairs in his seminal paper of 1980: “The Rigidly Rotating Disk as a ‘Missing Link’ in the History of General Relativity”. It seems that Gutfreund is unacquainted with Stachel’s paper.

On September 29, 1909 the Physikalische Zeitschrift received a short note from Paul Ehrenfest. In his note Ehrenfest demonstrated that according to Born’s notion of rigidity, one cannot bring a rigid body from a state of rest into uniform rotation about a fixed axis. Ehrenfest had pointed out that a uniformly rotating rigid disk would be a paradoxical object in special relativity; since, on setting it into motion its circumference would undergo a contraction whereas its radius would remain uncontracted.

Born noted: “Mr. Ehrenfest shows that the rigid body at rest can never be brought into uniform rotation; I have discussed the same fact with Mr. Einstein in the meeting of natural scientists in Salzburg”. Born discussed the subject with Einstein and they were puzzled about how the rigid body at rest could never be brought into uniform motion. Born and Einstein discovered in that discussion that setting a rigid disk into rotation would give rise to a paradox: the rim becomes Lorenz-contracted, whereas the radius remains invariant. This problem was discussed almost simultaneously by Ehrenfest in the above short note.

Later in 1919, Einstein explained to Joseph Petzoldt why it was impossible for a rigid disk in a state of rest to gradually set into rotation around its axis:


On page 32 Gutfreund mentions the 10th German edition of Einstein’s popular book Relativity the Special and General Theory. He says that in a copy of this book there is a sheet of paper in the handwriting of Einstein’s stepdaughter containing a remark:


As you can see this remark is quite similar to Einstein’s letter to Petzoldt. Thus, it is preferable to quote Einstein’s own words, his letter to Petzhold. It seems that Gutfreund is unacquainted with the history of the rotating disk, because according to his book he is unaware of Stachel’s paper and the letter to Petzhold.

At the end of October 1909 Born submitted an extended version of his Salzburg talk to Physikalische Zeitschrift. In December 1909 Gustav Herglotz published a paper in which he noted that according to Born’s notion of rigidity, a “rigid” body with a fixed point can only rotate uniformly about an axis that goes through it, like an ordinary rigid body. Several months later, Einstein mentioned Born’s and Herglotz’s papers in a letter from March 1910 to Jakob Laub, in which he said that he was very much interested in their then recent investigations on the rigid body and the theory of relativity.  A month later, in conversations with Vladimir Varičak Einstein explained that the great difficulty lies in bringing the “rigid” body from a state of rest into rotation. In this case, each material element of the rotating body must Lorentz contract. See my new book Einstein’s Pathway to the Special Theory of Relativity 2Ed for full details.

In his paper from February 1912, Einstein considered a system K with coordinates x, y, z in a state of uniform rotation (disk) in the direction of its x-coordinate and referred to it from a non-accelerated system. Einstein wrote that K‘s uniform rotation is uniform “in Born’s sense”, namely, he considered a rotating disk already in a state of uniform rotation observed from an inertial system and reproduced his conversations with Varičak. Einstein then extended the 1907–1911 equivalence principle to uniformly rotating systems as promised in conversations with Sommerfeld in 1909.

All we know according to primary sources is that the origin of the rotating disk story is in a problem in special relativity, Max Born’s notion of rigidity and Ehrenfest’s paradox, which Einstein mentioned many times before 1912. Einstein never mentioned any influence Poincaré had had on him when inventing the disk thought experiment. Writing that Einstein was influenced by Poincaré’s conventionalism and equivalent arguments is speculating about the influence of the later on the former.

Gutfreund’s mistake about Poincaré’s influence on Einstein and Einstein’s so-called failure to acknowledge Poincaré’s work in connection with the equivalence principle and the rotating disk thought experiment in general relativity comes from Yemima Ben-Menahem’s book, Conventionalism. However, this misconception or prejudice on the part of Ben-Menahem comes from my PhD thesis which was submitted to the Hebrew University of Jerusalem back in 1998. I was a PhD student in the program for the history and philosophy of science and Yemima Ben-Menahem was a professor there. Here for example are several paragraphs from my PhD thesis:




And Poincaré’s disk thought experiment:








I have thus written in my thesis about Poincaré’s disk thought experiment and the equivalence of Euclidean and non-Euclidean geometries. I then mentioned Einstein’s rotating disk thought experiment and said that we eliminate absolute motion of the disk by assuming the equivalence of gravity and inertia. I then spoke about conventionalism and Einstein’s equivalence principle.

After the PhD I corrected and edited my PhD but then I was horrified to discover what looked like a magnification of the prejudice of Poincaré’s conventionalism and equivalence argument and his disk thought experiment influencing Einstein when creating general relativity: In 2006 Yemima Ben-Menahem said exactly the same thing in her book, Conventionalism. You might say that it is even a more unfortunate instance to write about Poincaré’s influence on Einstein in connection with the equivalence principle and the disk thought experiment in general relativity over and over again in a single book… (see now for instance her book, pages 64-65):






Going on to Hanoch Gutfreund, in his new book of 2017, The Formative Years of Relativity, he has simply brought this incidence to the surface when he told the whole story of this prejudice all over again.








100 GTR: Uniformly Rotating Disk and the Hole Argument

The “Ehrenfest paradox“: Ehrenfest imagined a rigid cylinder set in motion from rest and rotating around its axis of symmetry. Consider an observer at rest measuring the circumference and radius of the rotating cylinder. The observer arrives at two contradictory requirements relating to the cylinder’s radius:

  1. Every point in the circumference of the cylinder moves with radial velocity ωR, and thus, the circumference of the cylinder  should appear Lorentz contracted to a smaller value than at rest, by the usual “relativistic” factor γ: 2πR‘ < 2πR.
  2. The radius R’ is always perpendicular to its motion and suffers no contraction at all; it should therefore be equal to its value R: R’ = R.

Einstein wrote to Vladimir Varićak either in 1909 or in 1910 (Febuary 28): “The rotation of the rigid body is the most interesting problem currently provided by the theory of relativity, because the only thing that causes the contradiction is the Lorentz contraction”.  CPAE 5, Doc. 197b.

Ehrenfest imagined a rigid cylinder gradually set into rotation (from rest) around its axis until it reaches a state of constant rotation.

In 1919 Einstein explained why this was impossible: (CPAE 9, Doc. 93)

“One must take into account that a rigid circular disk at rest would have to snap when set into rotation, because of the Lorentz shortening of the tangential fibers and the non-shortening of the radial ones. Similarly, a rigid disk in rotation (made by casting) would have to shatter as a result of the inverse changes in length if one attempts to bring it to the state of rest. If you take these facts fully into consideration, your paradox disappears”.

Assuming that the cylinder does not expand or contract, its radius stays the same. But measuring rods laid out along the circumference 2πR should be Lorentz-contracted to a smaller value than at rest, by the usual factor γ. This leads to the paradox that the rigid measuring rods would have to separate from one another due to Lorentz contraction; the discrepancy noted by Ehrenfest seems to suggest that a rotated Born rigid disk should shatter. According to special relativity an object cannot be spun up from a non-rotating state while maintaining Born rigidity, but once it has achieved a constant nonzero angular velocity it does maintain Born rigidity without violating special relativity, and then (as Einstein showed in 1912) a disk riding observer will measure a circumference.

Hence, in 1912, Einstein discussed what came to be known as the uniformly rotating disk thought experiment in general relativity. Thinking about Ehrenfest’s paradox and taking into consideration the principle of equivalence, Einstein considered a disk (already) in a state of uniform rotation observed from an inertial system.

We take a great number of small measuring rods (all equal to each other) and place them end-to-end across the diameter 2R and circumference 2πR of the uniformly rotating disk. From the point of view of a system at rest all the measuring rods on the circumference are subject to the Lorentz contraction. Since measuring rods aligned along the periphery and moving with it should appear contracted, more would fit around the circumference, which would thus measure greater than 2πR. An observer in the system at rest concludes that in the uniformly rotating disk the ratio of the circumference to the diameter is different from π:

circumference/diameter = 2π(Lorentz contracted by a factor…)/2R = π (Lorentz contracted by a factor….).

According to the equivalence principle the disk system is equivalent to a system at rest in which there exists a certain kind of static gravitational field. Einstein thus arrived at the conclusion that a system in a static gravitational field has non-Euclidean geometry.

Soon afterwards, from 1912 onwards, Einstein adopted the metric tensor as the mathematical respresentation of gravitation.

Indeed Einstein’s first mention of the rotating disk in print was in his paper dealing with the static gravitational fields of 1912; and after the 1912 paper, the rotating-disk thought experiment occurred in Einstein’s writings only in a 1916 review article on general relativity: “The Foundation of the General Theory of Relativity”.

He now understood that in the general theory of relativity the method of laying coordinates in the space-time continuum (in a definite manner) breaks down, and one cannot adapt coordinate systems to the four-dimensional space.


My new paper deals with Einstein’s 1912 and 1916 rotating disk problem, Einstein’s hole argument, the 1916 point coincidence argument and Mach’s principle; a combined-into-one deal (academic paper) for the readers of this blog.


Sitting: Sir Arthur Stanley Eddington and Hendrik Antoon Lorentz. Standing: Albert Einstein, Paul Ehrenfest and Willem de Sitter. September 26, 1923.

Further reading: Ehrenfest paradox

My new paper on Einstein and Schwarzschild

My new paper on General Relativity: Einstein and Schwarzschild.

Sometime in October 1915 Einstein dropped the Einstein-Grossman theory. Starting on November 4, 1915, Einstein gradually expanded the range of the covariance of his field equations. On November 11, 1915 Einstein was able to write the field equations of gravitation in a general covariant form, but there was a coordinate condition (there are no equations here so I cannot write it down here).

On November 18, 1915, Einstein presented to the Prussian Academy his paper, “Explanation of the Perihelion Motion of Mercury from the General Theory of Relativity”. Einstein reported in this talk that the perihelion motion of Mercury is explained by his theory. In this paper, Einstein tried to find approximate solutions to his November 11, 1915 field equations. He intended to obtain a solution, without considering the question whether or not the solution was the only possible unique solution.

Einstein’s field equations are non-linear partial differential equations of the second rank. This complicated system of equations cannot be solved in the general case, but can be solved in particular simple situations. The first to offer an exact solution to Einstein’s November 18, 1915 field equations was Karl Schwarzschild, the director of the Astrophysical Observatory in Potsdam. On December 22, 1915 Schwarzschild wrote Einstein from the Russian front. Schwarzschild set out to rework Einstein’s calculation in his November 18 1915 paper of the Mercury perihelion problem. He first responded to Einstein’s solution for the first order approximation from his November 18, 1915 paper, and found another first-order approximate solution. Schwarzschild told Einstein that the problem would be then physically undetermined if there were a few approximate solutions. Subsequently, Schwarzschild presented a complete solution. He said he realized that there was only one line element, which satisfied the conditions imposed by Einstein on the gravitational field of the sun, as well as Einstein’s field equations from the November 18 1915 paper.

“Raffiniert ist der Herrgott, aber boshaft ist er nicht” (Einstein might have already said….), because the problem with Schwarzschild’s line element was that a mathematical singularity was seen to occur at the origin! Oh my, Einstein abhorred singularities.

Actually, Schwarzschild “committed another crime”: he did not satisfy the coordinate condition from Einstein’s November 11 or November 18, 1915 paper. Schwarzschild admitted that his coordinates were not “allowed” coordinates, with which the field equations could be formed, because these spherical coordinates did not have determinant 1. Schwarzschild chose then the non-“allowed” coordinates, and in addition, a mathematical singularity was seen to occur in his solution. But Schwarzschild told Einstein: Don’t worry, “The equation of [Mercury’s] orbit remains exactly as you obtained in the first approximation”! See my paper from 2012.

Einstein replied to Schwarzschild on December 29, 1915 and told him that his calculation proving uniqueness proof for the problem is very interesting. “I hope you publish the idea soon! I would not have thought that the strict treatment of the point- problem was so simple”. Subsequently Schwarzschild sent Einstein a manuscript, in which he derived his solution of Einstein’s November 18, 1915 field equations for the field of a single mass. Einstein received the manuscript by the beginning of January 1916, and he examined it “with great interest”. He told Schwarzschild that he “did not expect that one could formulate so easily the rigorous solution to the problem”. On January 13, 1916, Einstein delivered Schwarzschild’s paper before the Prussian Academy with a few words of explanation. Schwarzschild’s paper, “On the Gravitational Field of a Point-Mass according to Einstein’s Theory” was published a month later.


Karl Schwarzschild

In March 1916 Einstein submitted to the Annalen der Physik a review article on the general theory of relativity, “The Foundation of the General Theory of Relativity”. The paper was published two months later, in May 1916. The 1916 review article was written after Schwarzschild had found the complete exact solution to Einstein’s November 18, 1915 field equations. Even so, in his 1916 paper, Einstein preferred NOT to base himself on Schwarzschild’s exact solution, and he returned to his first order approximate solution from his November 18, 1915 paper.

A comment regarding Einstein’s calculations in his November 18, 1915 paper of the Mercury perihelion problem and Einstein’s 1916 paper. In his early works on GTR, in order to obtain the Newtonian results, Einstein used the special relativistic limit and the weak field approximation, and assumed that space was flat (see my paper). Already in 1914 Einstein had reasoned that in the general case, the gravitational field was characterized by ten space-time functions of the metric tensor. g were functions of the coordinates. In the case of special relativity this reduces to g44 = c2, where c denotes a constant. Einstein took for granted that the same degeneration occurs in the static gravitational field, except that in the latter case, this reduces to a single potential, where g44 = c2 is a function of spatial coordinates, x1, x2, x3. 

Later that year David Hilbert (with a vengeance from 1915?…) arrived at a line-element similar to Schwarzschild’s one, and he concluded that the singularity disappears only if we accept a world without electricity. Such an empty space was inacceptable by Einstein who was apparently much attracted by Mach’s ideas! (later termed by Einstein “Mach’s Principle”). Okay, Einstein, said Hilbert: If there is matter then another singularity exists, or as Hilbert puts it: “there are places where the metric proves to be irregular”…. (See my paper from 2012).








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