My new paper on Einstein and general relativity

didMy new paper on Einstein and the general theory of relativity:



אודיסאת איינשטיין ליחסות הכללית Einstein’s Odyssey to General Relativity

מאמר שלי על דרכו של איינשטיין לתורת היחסות הכללית: אודיסאת איינשטיין ליחסות הכללית

סיינטיפיק אמריקן ישראל

“Einstein’s Odyssey to General Relativity”, Scientific American Israel

את המונח “אודיסאה” ליחסות הכללית טבע פרופ’ ג’ון סטצ’ל מאוניברסיטת בוסטון והוא מייצג את המסע המפרך של איינשטיין בדרכו ליחסות הכללית. ראו המאמר של סטצ’ל למטה

Odyssey to general relativity is John Stachel’s memorable phraseology. See:

Stachel, John (1979). “Einstein’s Odyssey: His Journey from Special to General Relativity”. In Einstein from B to Z, 2002.

I am sorry but this piece is in Hebrew. You can read my book General Relativity Conflict and Rivalries, my papers on Einstein and general relativity and a short summary below.


מפייסבוק: מארחים את ד”ר גלי וינשטיין לדבר על איינשטיין


My drawing of Einstein:      האיור שלי של איינשטיין

איינשטיין צעיר

And the original (I tried as hard as I could to draw a young Einstein…):       המקור


The article discusses the following topics:

1907. The Happiest thought of my life.


1907-1911. The equivalence principle and elevator experiments.


1911. Deflection of light and explaining deflection of light using an elevator thought experiment.


1911-1912 (1916). The disk thought experiment, gravitational time dilation and gravitational redshift.


1912. The disk thought experiment and non-Euclidean geometry.


1912. Einstein to Marcel Grossmann: “Grossmann, you must help me or else I’ll go crazy!”. Grossmann searched the literature, and brought the works of Bernhard Riemann, Gregorio Curbastro-Ricci, Tullio Levi-Civita and Elwin Bruno Christoffel to Einstein’s attention. With Grossmann’s help Einstein searched for gravitational field equations for the metric tensor in the Zurich Notebook.


1913-1914. The Entwurf theory. In 1913, Einstein and Michele Besso both tried to solve the new Entwurf field equations to find the perihelion advance of Mercury.

2October 1915. Einstein realizes there are problems with his 1914 Entwurf theory. November 1915. Einstein’s competition with David Hilbert.


November 1915. Four ground-breaking papers: Einstein presents the field equations of general relativity, finds the advance of the perihelion of Mercury and predicts that a ray of light passing near the Sun would undergo a deflection of amount 1.7 arc seconds.

Some of the topics discussed in my new book General Relativity Conflict and Rivalries


The back cover of my book:

General Relativity Conflict and Rivalries: Einstein’s Polemics with Physicists:


This book focuses on Albert Einstein and his interactions with, and responses to, various scientists, both famous and lesser-known. It takes as its starting point that the discussions between Einstein and other scientists all represented a contribution to the edifice of general relativity and relativistic cosmology. These scientists with whom Einstein implicitly or explicitly interacted form a complicated web of collaboration, which this study explores, focusing on their implicit and explicit responses to Einstein’s work.

This analysis uncovers latent undercurrents, indiscernible to other approaches to tracking the intellectual pathway of Einstein to his general theory of relativity. The interconnections and interactions presented here reveal the central figures who influenced Einstein during this intellectual period. Despite current approaches to history presupposing that the efforts of scientists such as Max Abraham and Gunnar Nordström,  which differed from Einstein’s own views, be relegated to the background, this book shows that they all had an impact on the development of Einstein’s theories, stressing the limits of approaches focusing solely on Einstein. As such, General Relativity Conflict and Rivalries proves that the general theory of relativity was not developed as a single, coherent construction by an isolated, brooding individual, but, rather, that it came to fruition through Einstein’s conflicts and interactions with other scientists, and was consolidated by his creative processes during these exchanges.


Grossmann and Einstein


From Zurich to Berlin

  1. Einstein and Heinrich Zangger (Einstein and Michele Besso)
  2. From Zurich to Prague.
  3. Back to Zurich (Einstein and Marcel Grossmann).
  4. From Zurich to Berlin (Einstein and Max Planck, Erwin Freundlich and others in Berlin) .

General Relativity between 1912 and 1916

  1. The Equivalence Principle.
  2. Einstein’s 1912 Polemic with Max Abraham: Static Gravitational Field
  3. Einstein’s 1912 Polemic with Gunnar Nordström: Static Gravitational Field
  4. Einstein’s 1912-1913 Collaboration with Marcel Grossmann: Zurich Notebook to Entwurf Theory.
  5. Einstein’s 1913-1914 Polemic with Nordström: Scalar Theory versus Tensor Theory (Einstein and Adriaan Fokker).
  6. Einstein’s Polemic with Gustav Mie: Matter and Gravitation.
  7. 1914 Collaboration with Grossmann and Final Entwurf Theory.
  8. Einstein’s Polemic with Tullio Levi-Civita on the Entwurf Theory.
  9. Einstein’s 1915 Competition with David Hilbert and General Relativity
  10. Einstein Answers Paul Ehrenfest‘s Queries: 1916 General Relativity.
  11. The Third Prediction of General Relativity: Gravitational RedShift
  12. Erich Kretschmann‘s Critiques of Einstein’s Point Coincidence Argument (Einstein and Élie Cartan).
  13. Einstein and Mach‘s Ideas.
  14. Einstein’s Reaction to Karl Schwarzschild‘s Solution (Einstein and Nathan Rosen and Leopold Infeld and others in Princeton).
  15. The Fourth Classic Test of General Relativity: Light Delay.

General Relativity after 1916

  1. Einstein’s 1916 Polemic with Willem de Sitter, Levi-Civita and Nordström on Gravitational Waves (Einstein and Nathan Rosen and Leopold Infeld and Howard Percy Robertson).
  2. Einstein’s Polemic with de Sitter: Matter World and Empty World.
  3. Bending of Light and Gravitational Lens: Einstein and Arthur Stanley Eddington
  4. Einstein’s Interaction with Hermann Weyl and the Cosmological Constant
  5. Einstein’s 1920 Matter World, Mach’s Ether and the Dark Matter
  6. Einstein’s 1920 Polemic with Eddington on de Sitter’s World.
  7. Einstein’s Reaction to the Aleksandr Friedmann Solution.
  8. Einstein’s Reaction to the Georges Lemaître Solution.
  9. Edwin Hubble‘s Experimental Results.
  10. The Lemaître-Eddington Model
  11. Einstein and the Matter World: the Steady State Solution.
  12. Einstein’s Collaboration with de Sitter.
  13. Einstein’s Reaction to Lemaître‘s Big Bang Model
  14. Einstein’s Interaction with George Gamow: Cosmological Constant is the Biggest Blunder
  15. Einstein, Gödel and Backward Time Travel



Einstein 1916

People ask many questions about Einstein’s general theory of relativity. For instance: How did Einstein come up with the theory of general relativity? What did he invent? What is the theory of general relativity? How did Einstein discover general relativity? How did he derive his theory? Why was Einstein the first to arrive at generally covariant field equations even though many lesser-known scientists worked on the gravitational problem?

Did David Hilbert publish the field equations of general relativity before Einstein? Was Einstein’s theory of relativity revolutionary for scientists of his day? How did the scientific community receive Einstein’s theory of general relativity when he published it? What were the initial reaction in the scientific community after Einstein had published his paper on relativity?

Why did Einstein object so fiercely to Schwarzschild’s’ singularity (black holes)? Why did Einstein introduce the cosmological constant? Was it his biggest blunder? Why did Einstein suggest Mach’s principle? Is Mach’s principle wrong? And so forth.


In my book General Relativity Conflict and Rivalries: Einstein’s Polemics with Physicists I try to answer these and many other questions.


Between 1905 and 1907, Einstein first tried to extend the special theory of relativity and explain gravitational phenomena. This was the most natural and simplest path to be taken. These investigations did not fit in with Galileo’s law of free fall. This law, which may also be formulated as the law of the equality of inertial and gravitational mass, was illuminating Einstein, and he suspected that in it must lie the key to a deeper understanding of inertia and gravitation. He found “the happiest thought of my life”. He imagined an observer freely falling from the roof of a house; for the observer there is during the fall – at least in his immediate vicinity – no gravitational field. 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”. Newton realized that Galileo’s law of free fall is connected with the equality of the inertial and gravitational mass; however, this connection was accidental. Einstein said that Galileo’s law of free fall can be viewed as Newton’s equality between inertial and gravitational mass, but for him the connection was not accidental. Einstein’s 1907 breakthrough was to consider Galileo’s law of free fall as a powerful argument in favor of expanding the principle of relativity to systems moving nonuniformly relative to each other. Einstein realized that he might be able to generalize the principle of relativity when guided by Galileo’s law of free fall; for if one body fell differently from all others in the gravitational field, then with the help of this body an observer in free fall (with all other bodies) could find out that he was falling in a gravitational field.

In June 1911, Einstein published his paper, “On the Influence of Gravitation on the Propagation of Light”. An important conclusion of this paper is that the velocity of light in a gravitational field is a function of the place. In December 1911, Max Abraham published a paper on gravitation at the basis of which was Einstein’s 1911 conclusion about a relationship between the variable velocity of light and the gravitational potential. In February 1912, Einstein published his work on static gravitational fields theory, which was based on his 1911 June theory. In March 1912, Einstein corrected his static gravitational fields paper, but Abraham claimed that Einstein borrowed his equations; however, it was actually Abraham who needed Einstein’s ideas and not the other way round. Einstein thought that Abraham converted to his theory of static fields while Abraham presumed exactly the opposite. Einstein then moved to Zurich and switched to new mathematical tools, the metric tensor as representing the gravitational potential. He examined various candidates for generally covariant field equations, and already considered the field equations of his general theory of relativity about three years before he published them in November 1915. However, he discarded these equations only to return to them more than three years later. Einstein’s 1912 theory of static fields finally led him to reject the generally covariant field equations and to develop limited generally covariant field equations.


Max Abraham

The Finnish physicist Gunnar Nordström developed a competing theory of gravitation to Einstein’s 1912-1913 gravitation theory. The equivalence principle was valid in his theory and it also satisfied red shift of the spectral lines from the sun. However, it was unable to supply the advance of the Perihelion of Mercury, such as Einstein’s theory; it led to a Perihelion like the one predicted by Newton’s law of gravity, and, it could not explain the deflection of light near the sun, because in Nordström’s theory the velocity of light was constant. Einstein’s 1913-1914 Entwurf theory of gravitation, the field equations of which were not generally covariant, remained without empirical support. Thus a decision in favor of one or the other theory – Einstein’s or Nordström’s – was impossible on empirical grounds. Einstein began to study Nordström’s theory from the theoretical point of view and he developed his own Einstein-Nordström theory on the basis of his conception of the natural interval. Eventually, in a joint 1914 paper with Lorentz’s student Adrian Fokker, Einstein showed that a generally covariant formalism is presented from which Nordström’s theory follows if a single assumption is made that it is possible to choose preferred systems of reference in such a way that the velocity of light is constant; and this was done after Einstein had failed to develop a generally covariant formulation for his own Entwurf theory.


Gunnar Nordström

After arriving back to Zurich in summer 1912, Einstein was looking for his old student friend Marcel Grossmann, who had meanwhile become a professor of mathematics in the Swiss Federal Polytechnic institute. He was immediately caught in the fire. So he arrived and he was indeed happy to collaborate on the problem of gravitation. Einstein’s collaboration with Marcel Grossmann led to two joint papers, the first entitled, “Entwurf einer verallgemeinerten Relativitätstheorie und einer Theorie der Gravitation” (“Outline of a Generalized Theory of Relativity and of a Theory of Gravitation”) is called by scholars the Entwurf paper.


Marcel Grossmann

The Entwurf theory was already very close to Einstein’s general theory of relativity that he published in November 1915. The gravitational field is represented by a metric tensor, the mathematical apparatus of the theory is based on the work of Riemann, Christoffel, Ricci and Levi-Civita on differential covariants, and the action of gravity on other physical processes is represented by generally covariant equations (that is, in a form which remained unchanged under all coordinate transformations). However, there was a difference between the two theories, the Entwurf and general relativity. The Entwurf theory contained different field equations that represented the gravitational field, and these were not generally covariant.


Elwin Bruno Christoffel


Gregorio Curbastro Ricci


Bernhard Riemann

Indeed at first though – when Einstein first collaborated with Grossmann – he considered (in what scholars call the “Zurich Notebook) field equations that were very close to the ones he would eventually choose in November 1915:



(See visual explanation of the Zurich Notebook on John Norton’s website).

In 1913, Einstein thought for a while – or persuaded himself – that generally covariant field equations were not permissible; one must restrict the covariance of the equations. He introduced an ingenious argument – the Hole Argument – to demonstrate that generally covariant field equations were not permissible. The Hole Argument seemed to cause Einstein great satisfaction, or else he persuaded himself that he was satisfied. Having found the Hole argument, Einstein spent two years after 1913 looking for a non-generally covariant formulation of gravitational field equations.

Einstein’s collaboration with his close friends included Michele Besso as well. During a visit by Besso to Einstein in Zurich in June 1913 they both tried to solve the Entwurf field equations to find the perihelion advance of Mercury in the field of a static sun in what is known by the name, the Einstein-Besso manuscript. Besso was inducted by Einstein into the necessary calculations. The Entwurf theory predicted a perihelion advance of about 18” per century instead of 43” per century.


Michele Besso

Towards the end of 1915 Einstein abandoned the Entwurf theory, and with his new theory got the correct precession so quickly because he was able to apply the methods he had already worked out two years earlier with Besso. Einstein though did not acknowledge his earlier work with Besso.

Levi-Civita (2)

Tulio Levi-Civita

Tullio Levi-Civita from Padua, one of the founders of tensor calculus, objected to a major problematic element in the Entwurf theory, which reflected its global problem: its field equations were restricted to an adapted coordinate system. Einstein proved that his gravitational tensor was a covariant tensor for adapted coordinate systems. In an exchange of letters and postcards that began in March 1915 and ended in May 1915, Levi-Civita presented his objections to Einstein’s above proof. Einstein tried to find ways to save his proof, and found it hard to give it up. Finally, Levi-Civita convinced Einstein about a fault in his arguments. Einstein realized that his Entwurf field equations of gravitation were entirely untenable. He discovered that Mercury’s perihelion’s motion was too small. In addition, he found that the equations were not covariant for transformations which corresponded to a uniform rotation of the reference system. Thus he came to the conviction that introducing the adapted coordinate system was a wrong path and that a more far-reaching covariance, preferably a general covariance, must be demanded.


Sometime in October 1915, Einstein dropped the Einstein-Grossman Entwurf theory. He adopted the postulate that his field equations were covariant with respect to arbitrary transformations of a determinant equal to 1 (unimodular transformations), and on November 4, 1915, he presented to the Prussian Academy of Sciences these new field equations. Einstein gradually expanded the range of the covariance of the field equations until November 25, 1915. On that day, Einstein presented to the Prussian Academy his final version to the gravitational field equations.

Einstein’s biographer Albrecht Fölsing explained: Einstein presented his field equations on November 25, 1915, but six days earlier, on November 20, Hilbert had derived the identical field equations for which Einstein had been searching such a long time. On November 18 Hilbert had sent Einstein a letter with a certain draft, and Fölsing asked about this possible draft: “Could Einstein, casting his eye over this paper, have discovered the term which was still lacking in his own equations, and thus ‘nostrified’ Hilbert?” Historical evidence support a scenario according to which Einstein discovered his final field equations by “casting his eye over” his own previous works. In November 4, 1915 Einstein wrote the components of the gravitational field and showed that a material point in a gravitational field moves on a geodesic line in space-time, the equation of which is written in terms of the Christoffel symbols. Einstein found it advantageous to use for the components of the gravitational field the Christoffel symbols. Einstein had already basically possessed the field equations in 1912, but had not recognized the formal importance of the Christoffel symbols as the components of the gravitational field. Einstein probably found the final form of the generally covariant field equations by manipulating his own (November 4, 1915) equations. Other historians’ findings seem to support the scenario according to which Einstein did not “nostrify” Hilbert.

Einstein and Hilbert


David Hilbert

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. In this paper Einstein presented a comprehensive general theory of relativity. In addition, in this paper Einstein presented the disk thought experiment. 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: 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.

Further, Einstein avoided the Hole Argument quite naturally by the Point Coincidence Argument. The point being made in the 1916 Point-Coincidence Argument is, briefly, that unlike general relativity, in special relativity coordinates of space and time have direct physical meaning. Since all our physical experience can be ultimately reduced to such point coincidences, there is no immediate reason for preferring certain systems of coordinates to others, i.e., we arrive at the requirement of general covariance. In 1918 Einstein saw the need to define the principles on which general relativity was based. In his paper, “Principles of the General Theory of Relativity”, he wrote that his theory rests on three principles, which are not independent of each other. He formulated the principle of relativity in terms of the Point Coincidence Argument and added Mach’s principle.

On November 18, 1915 Einstein reported to the Prussian Academy that the perihelion motion of Mercury is explained by his new General Theory of Relativity: Einstein found approximate solutions to his November 11, 1915 field equations. Einstein’s field equations cannot be solved in the general case, but can be solved in particular situations. The first to offer such an exact solution was Karl Schwarzschild. Schwarzschild found 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. On December 22, 1915 Schwarzschild told Einstein that he reworked the calculation in his November 18 1915 paper of the Mercury perihelion. Subsequently Schwarzschild sent Einstein a manuscript, in which he derived his exact solution of Einstein’s field equations. On January 13, 1916, Einstein delivered Schwarzschild’s paper before the Prussian Academy, and a month later the paper was published. Einstein though objected to the Schwarzschild singularity in Schwarzschild’s solution.


Karl Schwarzschild

In 1917 Einstein introduced into his 1915 field equations a cosmological term having the cosmological constant as a coefficient, in order that the theory should yield a static universe. Einstein desired to eliminate absolute space from physics according to “Mach’s ideas”.


Ernst Mach

Willem De Sitter objected to the “world-matter” in Einstein’s world, and proposed a vacuum solution of Einstein’s field equations with the cosmological constant and with no “world-matter”. In 1920 the world-matter of Einstein’s world was equivalent to “Mach’s Ether”, a carrier of the effects of inertia.


Einstein, Paul Ehrenfest and De Sitter; Eddington and Hendrik Lorentz. Location: office of W. de Sitter in Leiden (The Netherlands). Date: 26 Sept. 1923

De Sitter’s 1917 solution predicted a spectral shift effect. In 1923 Arthur Stanley Eddington and Hermann Weyl adopted De Sitter’s model and studied this effect. Einstein objected to this “cosmological problem”. In 1922-1927, Alexander Friedmann and Georges Lemaitre published dynamical universe models.


Friedmann’s model with cosmological constant equal to zero was the simplest general relativity universe. Einstein was willing to accept the mathematics, but not the physics of a dynamical universe.


In 1929 Edwin Hubble announced the discovery that the actual universe is apparently expanding. In 1931 Einstein accepted Friedmann’s model with a cosmological constant equal to zero, which he previously abhorred; he claimed that one did not need the cosmological term anymore. It was very typical to Einstein that he used to do a theoretical work and he cared about experiments and observations.




Einstein in America.

In Princeton in 1949, Kurt Gödel found an exact solution to Einstein’s field equations. Gödel’s solution was a static and not an expanding universe. Gödel’s universe allows the existence of closed timelike curves (CTCs), paths through spacetime that, if followed, allow a time traveler to interact with his/her former self.



The book also discusses other topics: for instance,  gravitational lensing, gravitational waves, Einstein-Rosen bridge, unified field theory and so forth.

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My new book on Einstein and the history of the general theory of relativity

Here is the dust jacket of my new scholarly book on the history of general relativity, to be released on… my Birthday:

General Relativity Conflict and Rivalries: Einstein’s polemics with Physicists.


The book is illustrated by me and discusses the history of general relativity, gravitational waves, relativistic cosmology and unified field theory between 1905 and 1955:

The development of general relativity (1905-1916), “low water mark” period and several results during the “renaissance of general relativity” (1960-1980).


Conversations I have had more than a decade ago with my PhD supervisor, the late Prof. Mara Beller (from the Hebrew University in Jerusalem), comprise major parts of the preface and the general setting of the book. However, the book presents the current state of research and many new findings in history of general relativity.


My first book:

Einstein’s Pathway to the Special Theory of Relativity (April, 2015)

includes a wide variety of topics including also the early history of general relativity.


Review of Arch and Scaffold Physics Today

Arch and scaffold: How Einstein found his field equations” by Michel Janssen and Jürgen Renn. Physics Today 68(11), 30 (2015). The article is published in November 2015, which marks the centenary of the Einstein field equations. (Renn co-authored with Gutfreund The Road to Relativity, Princeton Press)

This is a very good article. However, I would like to comment on several historical interpretations. . Michel Janssen and Jürgen Renn ask: Why did Einstein reject the field equations of the first November paper (scholars call them the “November tensor”) when he and Marcel Grossmann first considered them in 1912–13 in the Zurich notebook?

They offer the following explanation: In 1912 Albert Einstein gave up the November tensor (derived from the Ricci tensor) because the rotation metric (metric of Minkowski spacetime in rotating coordinates) did not satisfy the Hertz restriction (the vanishing of the four-divergence of the metric). Einstein wanted the rotation metric to be a solution of the field equations in the absence of matter (vacuum field equations) so that he could interpret the inertial forces in a rotating frame of reference as gravitational forces (i.e. so that the equivalence principle would be fulfilled in his theory).

However, the above question – why did Einstein reject the November tensor in 1912-1913, only to come back to it in November 1915 – apparently has several answers. It also seems that the answer is Einstein’s inability to properly take the Newtonian limit.

Einstein’s 1912 earlier work on static gravitational fields (in Prague) led him to conclude that in the weak-field approximation, the spatial metric of a static gravitational field must be flat. This statement appears to have led him to reject the Ricci tensor, and fall into the trap of Entwurf limited generally covariant field equations. Or as Einstein later put it, he abandoned the generally covariant field equations with heavy heart and began to search for non-generally covariant field equations. Einstein thought that the Ricci tensor should reduce in the limit to his static gravitational field theory from 1912 and then to the Newtonian limit, if the static spatial metric is flat. This prevented the Ricci tensor from representing the gravitational potential of any distribution of matter, static or otherwise. Later in the 1920s, it was demonstrated that the spatial metric can go to a flat Newtonian limit, while the Newtonian connection remains non-flat without violating the compatibility conditions between metric and (affine) connection (See John Stachel).


Phys. Today 68, 11, 30 (2015).

As to the “archs and scaffolds” metaphor. Michel Janssen and Jürgen Renn demonstrate that the Lagrangian for the Entwurf field equations has the same structure as the Lagrangian for the source-free Maxwell equations: It is essentially the square of the gravitational field, defined as minus the gradient of the metric. Since the metric plays the role of the gravitational potential in the theory, it was only natural to define the gravitational field as minus its gradient. This is part of the Entwurf scaffold. The authors emphasize the analogy between gravity and electromagnetism, on which Einstein relied so heavily in his work on the Entwurf theory.

However, I am not sure whether in 1912-1913 Einstein was absolutely aware of this formal analogy when developing the Entwurf field equations. He first found the Entwurf equations, starting from energy-momentum considerations, and then this analogy (regarding the Lagrangian) lent support to his Entwurf field equations. Anyway, I don’t think that this metaphor (analogy between gravity and electromagnetism) persisted beyond 1914. Of course Einstein came back to electrodynamics-gravity, but I think that he discovered his 1915 field equations in a way which is unrelated to Maxwell’s equations (apart from the 1911 generally covariant field equations, influenced by Hilbert’s electromagnetic-gravitational unified theory, but this is out of the scope of this post and of course unrelated to the above metaphor).


As to the November 4, 1915 field equations of Einstein’s general theory of relativity: When all was done after November 25, 1915, Albert Einstein said that the redefinition of the components of the gravitational field in terms of Christoffel symbols had been the “key to the solution”. Michel Janssen and Jürgen Renn demonstrate that if the components of the gravitational field – the Christoffel symbols – are inserted into the 1914 Entwurf Lagrangian, then the resulting field equations (using variational principle) are the November tensor. In their account, then, Einstein found his way back to the equations of the first November paper (November 4, 1915) through considerations of physics. Hence this is the interpretation to Einstein’s above “key to the solution”.

I agree that Einstein found his way back to the equations of the first November paper through considerations of physics and not through considerations of mathematics. Mathematics would later serve as heuristic guide in searching for the equations of his unified field theory. However, it seems to me that Michel Janssen and Jürgen Renn actually iterate Einstein’s November 4, 1915 variational method. In November 4, 1915, Einstein inserted the Christoffel symbols into his 1914 Entwurf Lagrangian and obtained the November 4, 1915 field equations (the November tensor). See explanation in my book, General Relativity Conflict and Rivalries, pp. 139-140.

Indeed Janssen and Renn write: There is no conclusive evidence to determine which came first, the redefinition of the gravitational field (in terms of the Christoffel symbols) or the return to the Riemann tensor.

Hence, in October 1915 Einstein could have first returned to the November tensor in his Zurich Notebook (restricted to unimodular transformations) and only afterwards in November 1915, could he redefine the gravitational field components in terms of the Christoffel symbols. Subsequently, this led him to a redefinition of the Entwurf Lagrangian and, by variational method, to a re-derivation of the 1912 November tensor.


Van Gogh had nostrified Hilbert (Hilbert visited Van Gogh, closed time-like loops…. ….)

Finally, Michel Janssen and Jürgen Renn write: Despite Einstein’s efforts to hide the Entwurf scaffold, the arch unveiled in the first November paper (November 4, 1915) still shows clear traces of it.

I don’t think that Einstein tried to hide the Entwurf scaffold. Although later he wrote Arnold Sommerfeld: “Unfortunately, I have immortalized the last error in this struggle in the Academy-papers, which I can send to you soon”, in his first November paper Einstein had explicitly demonstrated equations exchange between 1914 Entwurf and new covariant November ones, restricted to unimodular transformations.

Stay tuned for my book release, forthcoming soon (out by the end of 2015) on the history of general relativity, relativistic cosmology and unified field theory between 1907 and 1955.

Einstein’s pathway to his General Theory of Relativity

Einstein thought that when dealing with gravity high velocities are not so important. So in 1912 he thought about gravity in terms of the principle of relativity and not in terms of the constant-speed-of-light postulate (special relativity). But then he engaged in a dispute with other scholars who claimed that he gave up the central postulate of his special theory of relativity. x

File:Max abraham.png

Max Abraham

File:Gunnar Nordström.jpg

Gunnar Nordström

Einstein’s Pathway to his Equivalence Principle 1905-1907


1912 – 1913 Static Gravitational Field Theory


1913 – 1914 “Entwurf” theory


Berlin “Entwurf” theory 1914


The Einstein-Nordström Theory


Dawn of “Entwarf theory”


1915 Relativity Theory


1916 General Theory of Relativity


שנים לטנסור המטרי 100 Years of Metric Tensor

 לפני מאה שנים ב-2012 איינשטיין לראשונה אימץ את הטנסור המטרי. באחד ממאמריו איינשטיין סיפר לקורא שלו:”הדרך שבה אני עצמי עברתי, הייתה עקיפה ופתלתלה למדי, כי אחרת אינני יכול לקוות שהוא יתעניין יותר מידי בתוצאה של סוף המסע”, תורת היחסות הכללית. למטה מובאת רשימה של ספרים ומאמרים שנכתבו על ידי חוקרים מהעולם על דרכו של איינשטיין לתורת היחסות הכללית והפילוסופיה של היחסות הכללית.

A hundred years ago, in 1912, Einstein adopted the metric tensor. In one of his later papers Einstein told his reader about “the road that I have myself travelled, rather an indirect and bumpy road, because otherwise I cannot hope that he will take much interest in the result at the end of the journey”: the General theory of Relativity

Relatively speaking

The following  list of books and papers discussing Einstein’s Pathway to General Relativity and Philosophy of General Relativity covers the period 1912-1916, and beyond

Corry, Leo, Renn, Jürgen and John Stachel, “Belated Decision in the Hilbert-Einstein Priority Dispute”, 1997, in Stachel 2002, pp. 339-346

Corry, Leo, Renn, Jürgen and John Stachel, “Response to F. Winterberg ‘On Belated Decision in the Hilbert-Einstein Priority Dispute'”, 2004, Z. Naturforsen 59a, pp. 715-719

Earman, John, Janis, Allen, I., Massey, Gerald. I., and Rescher, Nicholas, Philosophical Problems of the Inertial and External Worlds, Essays of the philosophy of Adolf Grünbaum, 1993, University of Pittbsbutgh/ Universitätswerlag Konstanz

Earman, John, Janssen, Michel, Norton, John (ed), The Attraction of Gravitation, New Studies in the History of General Relativity, Einstein Studies Vol 5, 1993, Boston: Birkhäuser

Earman, John and Janssen, Michel, “Einstein’s Explanation of the Motion of Mercury’s Perihelion”, in Earman, Janssen, and Norton, John, 1993, pp. 129-172

Goener, Hubert, Renn Jürgen, Ritter, Jim, Sauer, Tilman (ed), The Expanding Worlds of General Relativity, 1999, Boston: Brikhäser

Howard, Don, “Point Coincidences and Pointer Coincidences: Einstein on Invariant Structure in Spacetime Theories”, in Goener et al, 1999, pp. 463-500

Howard, Don and Norton, John, “Out of the Labyrinth? Einstein, Hertz, and the Göttingen Answer to the Hole Argument”, in Earman, Janssen, Norton (ed), 1993, pp. 30-61

Howard, Don and Stachel, John (eds.), Einstein and the History of General Relativity: Einstein Studies, Volume 1, 1989, New York: Birkhauser

Janssen, Michel, “The Einstein-De Sitter Debate and its Aftermath”, lecture, pp. 1-8, based on “The Einstein-De Sitter-Weyl-Klein Debate” in CPAE, Vol. 8, 1998, pp. 351-357

Janssen, Michel, “Rotation as the Nemesis of Einstein’s Entwurf Theory”, in Goener, Renn, Ritter and Sauer (ed), 1999, pp. 127-157

Janssen, Michel, “The Einstein-Besso Manuscript: A Glimpse Behind the Certain of a Wizard”, Freshman Colloquium: “Introduction to the Arts and Sciences”, Fall 2002

Janssen, Michel, “Of Pots and Holes: Einstein’s Bumpy Road to General Relativity”, in Renn, 2005, pp-58-85; reprinted as “Einstein’s First Systematic Exposition of General Relativity”, pp, 1-39

Janssen Michel and Renn, Jürgen, “Untying the Knot: How Einstein Found His Way Back to Field Equations Discarded in the Zurich Notebook”, in Renn et all, Vol. 1, 2007, pp. 839-925

Janssen, Michel, “What did Einstein know and When did he Know It?” in Renn et all, Vol. 2, 2007, pp. 786-837

Janssen, Michel, “‘No Success Like Failure’: Einstein’s Quest for General Relativity”, The Cambridge Companion to Einstein, 2009

Alfred Eisenstaedt, Einstein Life Magazine

Norton, John, “How Einstein Found His Field Equations: 1912-1915”, Historical Studies in the Physical Sciences 14, 1984, pp. 253-315. Reprinted in Howard, Don and Stachel, John (eds.), 1989, pp 101-159

Norton, John, “General Covariance and the Foundations of General Relativity: Eight Decades of Dispute,” Reports on Progress in Physics 56, 1993, pp.791-858

Norton, John, “Einstein and Nordström: Some Lesser-Known Thought Experiments in Gravitation”, Archive for History of Exact Sciences 45, 1993, pp.17-94

Norton, John, “Nature in the Realization of the Simplest Conceivable Mathematical Ideas: Einstein and the Canon of Mathematical Simplicity”, Studies in the History and Philosophy of Modern Physics 31, 2000, pp.135-170

Norton, John, “Einstein, Nordström and the early Demise of Lorentz-covariant, Scalar Theories of Gravitation,” in Renn et all, Vol. 3, 2007, pp. 413-487

Renn, Jürgen and Tilman Sauer, “Heuristics and mathematical Representation in Einstein Search for a Gravitational Field Equation”, Preprint 62, Max Planck Institute for the History of Science, 1997

Renn, Jürgen, (ed.) Einstein’s Annalen Papers. The Complete Collection 1901-1922, 2005, Germany: Wiley-VCH Verlag GmbH & Co

Renn, Jürgen, “The Summit Almost Scaled: Max Abraham as a Pioneer of a Relativistic Theory of Gravitation”, in Renn et all, Vol.3, 2007, pp. 305-330

Renn, Jürgen, Norton, John, Janssen, Michel and Stachel John, ed., The Genesis of General Relativity. 4 Vols., 2007, New York, Berlin: Springer

Renn, Jürgen and Stachel, John, “Hilbert’s Foundation of Physics: From a Theory of Everything to a Constituent of General Relativity”, in Renn et all, Vol. 4, 2007, pp. 857-974

Time Magazine Photo

Stachel, John, “The Genesis of General Relativity”, Physics 100, 1979, pp. 428-442; reprinted in Stachel, 2002, pp. 233-244

Stachel, John, “The Rigidity Rotating Disk as the ‘Missing Link’ in the History of General Relativity”, General Relativity and Gravitation one Hundred Years After the Birth of Albert Einstein, Vol. 1, 1980, pp. 1-15; reprinted in Stachel, 2002, pp. 245-260

Stachel, John, “‘Subtle is the Lord'”… The Science and Life of Albert Einstein” by Abraham Pais”, Science 218, 1982, pp. 989-990; reprinted in Stachel, 2002, pp. 551-554

Stachel, John, “Albert Einstein: The Man beyond the Myth”, Bostonia Magazine 56, 1982, pp. 8–17; reprinted in Stachel, 2002, pp. 3-12

Stachel, John, “Einstein and the ‘Research Passion'”, Talk given at the Annual Meeting of the American Associates for the Advancement of Science in Detroit, May 1983; reprinted in Stachel, 2002, pp. 87-94

Stachel, John, “The Generally Covariant Form of Maxwell’s Equations”, in J.C. Maxwell, the Sesquicentennial Symposium, M.S. Berger (ed), 1984, Elsevier: Amsterdam, pp. 23-37.

Stachel, John, “What a Physicist Can Learn From the Discovery of General Relativity”, Proceedings of the Fourth Marcel Grossmann Meeting on General relativity, ed. R. Ruffini, Elsevier: Amsterdam, 1986, pp.1857-1862

Stachel, John, “How Einstein Discovered General Relativity: A Historical Tale With Some Contemporary Morals”, in MacCallum, M.A.H., General Relativity and Gravitation Proceedings of the 11th International Conference on General Relativity and Gravitation, 1987, pp. 200-209, reprinted in Satchel, 2002, pp. 293-300

Stachel, John, “Einstein’s Search for General Covariance 1912-1915”, in Howard, Don and Stachel John (eds), 1989, pp. 63-100; reprinted in Stachel, 2002, pp. 301-338

Stachel, John, “Albert Einstein (1897-1955), The Blackwell Companion to Jewish Culture, ed Glenda Abramson, Oxford: Blackwell, pp. 198-199; reprinted in Stachel, 2002, pp. 19-20

Stachel, John, “the Meaning of General Covariance. The Hole Story”, in Earman, John, Janis, Allen, et all, 1993, pp. 129-160.

Stachel, John, “The Other Einstein: Einstein Contra Field Theory”, Science in Context 6, 1993, pp. 275-290; reprinted in Stachel, 2002, pp. 141-154

Stachel, John, “Changes in the Concept of Space and Time Brought About by Relativity”, Artifacts, Representations and Social Practice: Essays for Marx Wartofsky, eds Carol Gould and Robert S. Cohen., Dordrecht/Boston/London: Kluwer Academic, pp. 141-162

Stachel, John, “History of relativity,” in Brown, Laurie M., Pais, Abraham, and Sir Pippard, Brian (eds.), Twentieth century physics, 1995, New York: American Institute of Physics Press, pp. 249-356

Stachel, John, “Albert Einstein: A Biography by Albert Fölsing”, Review of Albrecht Fölsing, Albert Einstein: A Biography, in Physics Today, January, 1998; reprinted in Stachel, 2002, pp. 555-556

Stachel, John, “New Light on the Einstein-Hilbert Priority Question”, Journal of Astrophysics 20, 1999, pp. 90-91; reprinted in Stachel, 2002, pp. 353-364

Stachel, John, “Einstein and Infeld: Seen through Their Correspondence”, Acta Physica Polonica B 30, 1999, pp. 2879–2904; reprinted in Stachel, 2002, pp. 477–497

Stachel, John, “The First-two Acts”, in Renn et all, 2007, Vol. 1, pp. 81-112; appeared first in Stachel, 2002, pp. 261-292

Stachel, John, Einstein from ‘B’ to ‘Z’, 2002, Washington D.C.: Birkhauser

Stachel, John, “Albert Einstein”, The 2005 Mastermind Lecture, Proceedings of the British Academy 151, 2007, pp.423-458

Stachel, John, “Where is Creativity? The example of Albert Einstein”, Invited Lecture at the Congresso International de Filosophia, Pessoa & Sociadade (Person and Society), Braga, 17-19 November 2005, to appear in 2012

Stachel, John, “Einstein and Hilbert”, invited lecture in March 21, 2005 Session: Einstein and Friends, American Physical Society, Los Angeles; and response to Questions from FAZ on Hilbert and Einstein

Stachel, John, “Einstein’s Intuition and the Post-Newtonian Approximation”, World Scientific, 2006, pp. 1-15

Stachel, John, “The Story of Newstein: Or is Gravity Just Another Pretty Force?”, in Renn, et all, Vol. 4, 2007, pp. 1041-1078

Stachel, John, “A world Without Time: the Forgotten Legacy of Gödel and Einstein”, Notices of the American Mathematical Society 54, 2007, pp. 861-868/1-8

Stachel, John, “Albert Einstein”, The New Dictionary of Scientific Biography, Vol. 2, Gale 2008, pp. 363-373

Stachel, John, “The Hole Argument”, Living Reviews in Relativity, June 25, 2010, to appear in 2012

Stachel, John, “The Rise and Fall of Einstein’s Hole Argument”, pp. 1-15 to appear in 2012

Stachel, John, “The Scientific Side of the Einstein-Besso Relationship”, to appear in 2012

Torretti, Roberto, Relativity and Geometry, 1983/1996, Ney-York: Dover

Prime Minister of Israel Ben Gurion visits Albert Einstein in Princeton. Here

Einstein’s Pathway to the Special Theory of Relativity and the General Theory of Relativity

Einstein’s Pathway to the Special Theory of Relativity

Einstein’s Pathway to the Special Theory of Relativity

Einstein’s discovery of Special Relativity

Einstein Believes in the Ether

Einstein Chases a Light Beam

Einstein recounts the Aarau thought experiment in his Autobiographical Notes, 1949

Magnet and Conductor Thought Experiment, Faraday’s magneto-electric induction

Föppl’s book on Maxwell’s theory

Ether drift and Michelson and Morley’s experiment

The Role of the Michelson-Morley Experiment on the Discovery of Relativity

The Dayton Miller Experiments

Emission theory and ether drift experiments

Paul Ehrenfest and Walter Ritz. Ritz’s Emission Theory

“The Step”

Einstein defined distant simultaneity physically; relativity of simultaneity

The Kyoto lecture notes – Einstein could have visited and consulted his close friend Michele Besso, whom he thanked at the end of his relativity paper. The Patent Office brought them together – their conversations on the way home. Besso was always eager to discuss the subjects of which he knew a great deal – sociology, medicine, mathematics, physics and philosophy – Einstein initiated him into his discovery

Joseph Sauter – Before any other theoretical consideration, Einstein pointed out the necessity of a new definition of synchronization of two identical clocks distant from one another; to fix these ideas, he told him, “suppose one of the clocks is on a tower at Bern and the other on a tower at Muri (the ancient aristocratic annex of Bern)” – synchronization of clocks by light signals.

Did Poincaré have an Effect on Einstein’s Pathway toward the Special Theory of Relativity? Einstein’s reply to Carl Seelig


Einstein’s pathway to the General Theory of Relativity

Entwurf theory – Einstein-Grossmann theory, Hole argument, field equations and the Einstein-Besso manuscript


Gunnar Nordström develops a competing theory of gravitation to Einstein’s 1912-1913 gravitation theory. Einstein begins to study Nordström’s theory and develops his own Einstein-Nordström theory. In a joint 1914 paper with Lorentz’s student Adrian Fokker – a generally covariant formalism is presented from which Nordström’s theory follows if the velocity of light is constant Here


The three problems that led to the fall of the entwurf theory –

The gravitational field on a uniformly rotating system does not satisfy the field equations.

Covariance with respect to adapted coordinate system was a flop.

In the Entwurf theory the motion of Mercury’s perihelion came to 180 rather than 450 per century

The General Theory of Relativity – 1915

David Hilbert Enters the Game, the priority dispute – Einstein and Hilbert

In November 18 1915 Einstein calculated rapidly the precession of Mercury’s


Geodesic Equation. Metric tensor. Einstein’s November 4, 11, and 25 field equations.The Riemann-Christoffel Tensor; the Ricci tensor; the Einstein tensor

von Deinem zufriedenen aber ziemlich kaputen

General Theory of Relativity – 1916

Mid December to Mid January 1915: Exchange of letters between Einstein and Ehrenfest

The disk thought experiment; coordinates have no direct physical meaning Euclidean Geometry breaks down; two Globes Thought Experiment; Mach’s Principle; the principle of general relativity; the Equivalence Principle; the principle of general covariance

The Summation Convention

Motion of the Perihelion of the Planetary Orbit; Redshift; Deflection of light in a gravitational field of the sun

Einstein in the Patent Office:

Michele Besso, Joseph Sauter, and Lucian Chavan – Patent Office, Maurice Solovine and Conrad Habicht – the Olympian Academy

Annus mirabilis papers

On a Heuristic Viewpoint Concerning the Generation and Transformation of Light – It argues a heuristic manner for the existence of light quanta and derives the photoelectric law

On a New Determination of Molecular Dimensions – doctoral thesis submitted to the mathematical and natural science branch of Zürich University

On the Movement of Particles Suspended in Fluids at Rest, as Postulated by the Molecular Theory of Heat. The Brownian motion paper

On the Electrodynamics of Moving Bodies. The Relativity Paper

Does the Inertia of a Body Depend on its Energy Content? The first derivation of the mass energy equivalence

German Scientists Responded to Einstein’s Relativity Paper – Max Planck wrote Einstein. Max von Laue met Einstein

Einstein teaches his 3 friends from the Patent Office at the University of Bern

Finally Einstein leaves the Patent Office to his first post in the University of Zürich

Further reading: Stachel, John, Einstein from ‘B’ to ‘Z’, 2002, Washington D.C.: Birkhauser