## General Relativity without the Equivalence principle?

I have skimmed through this book Handbook of Spacetime:

The following represents my impressions formulated after reading the sections about the equivalence principle.

However, Einstein did not write this wonderful passage in the letter to Robert Lawson. Here is the letter to Lawson (Einstein to Lawson, 22 January 1920):

Einstein writes to Lawson in the above letter: “The article for Nature is almost finished, but it has unfortunately become so long that I very much doubt whether it could appear in Nature“. Indeed, in a 1920 unpublished draft of a paper for Nature, “Fundamental Ideas and Methods of the Theory of Relativity, Presented in Their Development”, Einstein wrote the above long paragraph describing him in 1907 sitting in the Patent Office. He was brooding on special relativity, and suddenly there came to him the happiest thought of his life:

Let us analyze this passage. The man in free fall (elevator experiments): Special relativity is incorporated into general relativity as a model of space-time experienced by an observer in free fall, over short times and distances (locally):

Between 1905 and 1907, Einstein tried to extend the special theory of relativity so that it would explain gravitational phenomena. He reasoned that the most natural and simplest path to be taken was to correct the Newtonian gravitational field equation. Einstein also tried to adapt the Newtonian law of motion of the mass point in a gravitational field to the special theory of relativity. However, he found a contradiction with Galileo’s law of free fall, which states that all bodies are accelerated in the gravitational field in the same way (as long as air resistance is neglected). Einstein was sitting on a chair in my patent office in Bern and then suddenly a thought struck him: If a man falls freely, he would not feel his weight. This was the happiest thought of his 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 (locally) at rest. Einstein’s 1907 breakthrough was to consider Galileo’s law of free fall as a powerful argument in favor of expanding the special principle of relativity to systems moving non-uniformly relative to each other. Einstein realized that he might be able to generalize and extend special relativity when guided by Galileo’s law of free fall. The Galilean law of free fall (or inertial mass is equal to gravitational mass) became known as the weak principle of equivalence.

Lewis Ryder explains: “Some writers distinguish two versions of the equivalence principle: the weak equivalence principle, which refers only to free fall in a gravitational field and is stated… as The worldline of a freely falling test body is independent of its composition or structure; and the strong equivalence principle, according to which no experiment in any area of physics should be able, locally, to distinguish a gravitational field from an accelerating frame”.

There are several formulations of the weak and the strong principles of equivalence in the literature. By far the most frequently used formulation of the strong principle of equivalence is Einstein’s 1912 local principle of equivalence: In a local free falling system special relativity is valid. (See my book General Relativity Conflict and Rivalries. Einstein’s Polemics with Physicists, 2015, for further details).

Nick Woodhouse explains:

in the chapter:

Hence Joshi says:

in the chapter:

Lewis Ryder

writes in the above paper:

(i.e. Einstein 1911 paper: “On the Influence of Gravitation on the Propagation of Light”). He formulates the equivalence principle in the following way: “In a freely falling (non-rotating) laboratory occupying a small region of spacetime, the local reference frames are inertial and the laws of physics are consistent with special relativity”. He then writes:

The equivalence principle enables us to find just one component g00 – of the metric tensor gmn. All components can be found (at least in principle) from the Einstein field equations. Ryder thus concludes that the equivalence principle is dispensable. I don’t quite agree with Ryder.

In my 2012 paper, “From the Berlin ‘Entwurf’ Field equations to the Einstein Tensor III: March 1916”, ArXiv: 1201.5358v1 [physics.hist-ph], 25 January, 2012 and also in my 2014 paper,  “Einstein, Schwarzschild, the Perihelion Motion of Mercury and the Rotating Disk Story”, ArXiv: 1411.7370v [physics.hist-ph], 26 Nov, 2014, I demonstrate the following:  On November 18, 1915, Einstein found approximate solutions to his November 11, 1915 field equations and explained the motion of the perihelion of Mercury. Einstein’s field equations cannot be solved in the general case, but can be solved in particular situations. Indeed, the first to offer 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 11, 1915 paper. Schwarzschild sent Einstein a manuscript, in which he derived his exact solution of Einstein’s field equations. In January, 1916, Einstein delivered Schwarzschild’s paper before the Prussian Academy, and a month later the paper was published. In March 1916 Einstein submitted to the Annalen der Physik a review article, “The Foundation of the General Theory of Relativity”, on 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, Einstein preferred not to base himself on Schwarzschild’s exact solution, and he returned to his first order approximate solution from November 18, 1915. In the final part of the 1916 review paper Einstein demonstrated that a gravitational field changes spatial dimensions and the clock period:

This equation is further explained in my 2012 paper (page. 56):

Neither did Einstein use the Schwarzschild solution nor was he guided by the  equivalence principle. He was rather using an approximate solution and the metric, the line element to arrive at the same factor he had obtained by assuming the heuristic equivalence principle. He thus demonstrated that the equivalence principle was a fundamental principle of his theory, because in 1912 he formulated an equivalence principle valid only locally  (see my book: General Relativity Conflict and Rivalries. Einstein’s Polemics with Physicists, 2015, p. 184). I further explain it below.

Ryder then explains: The equivalence principle is local (a complete cancelation of a gravitational field by an accelerating frame holds locally). However, over longer distances two objects in free fall at different places in a realistic gravitational field move toward each other and this does not happen in an accelerating elevator. The cancelation of the gravitational field by an accelerating field is thus not complete. According to general relativity this effect (tidal effect) is a consequence of the curvature of space-time:

Although the equivalence principle might have been a heuristic guide to Einstein in his route to the fully developed theory of general relativity, Ryder holds that it is now irrelevant.

I don’t agree with Ryder’s conclusion which resembles that of John Lighton Synge (and Hermann Bondi). Indeed the equivalence principle is not valid globally (i.e. for tidal effects). Although the strong equivalence principle can at best be valid locally, it is still crucial for the general theory of relativity:

1. Einstein formulated an equivalence principle which is valid only locally. Special relativity is valid locally and space-time is locally the Minkowski space-time.
2. The principle of equivalence is fundamental for a metric theory and for our understanding of curved space-time: Freely falling test bodies move along geodesic lines under the influence of gravity alone, they are subject to an inertio-gravitational field . The metric determines the single inertio-gravitational field (affine connection), and there is breakup into inertia and gravitation relative to the acceleration. According to the equivalence principle, the components of the affine connection vanish in local frames. John Stachel quotes a passage from Einstein’s letter to Max von Laue:

Stachel, John, “How Einstein Discovered General Relativity: A Historical Tale with Some Contemporary Morals”, Einstein B to Z, 2002.

Indeed Ryder quotes J. L. Synge :

Einstein’s equivalence principle was criticized by Synge:

Synge, J. L. (1960). Relativity: The General Theory (Amsterdam, The Netherlands: North Holland Publishing Co).

And Hermann Bondi reacted to Einstein’s principle of equivalence:

Bondi also said (‘NO SUCCESS LIKE FAILURE …’: EINSTEIN’S QUEST FOR GENERAL RELATIVITY, 1907–1920, Michel Janssen):

Other authors contributing to the Handbook of Spacetime write the following:

Graham S. Hall in his paper:

writes the following:

“The choice of a geodesic path (Einstein’s principle of equivalence) reflects the results of the experiments of Eötvös and others, which suggest that the path of a particle in a pure gravitational field is determined by its initial position and initial velocity”. This is not Einstein’s equivalence principle. This is the Galilean principle of equivalence or the weak equivalence principle.

And according to Vesselin Petkov:

the geodesic line is indeed a manifestation of Galileo’s free fall law:

Ryder presents tests for the equivalence principle. The operation of the global positioning system, the GPS, is a remarkable verification of the time dilation. The GPS system consists of an array of 24 satellites, which describe an orbit round the earth of radius 27,ooo km, and are 7000 km apart, and every 12 hours travel at about 4km/s.  Each satellite carries an atomic clock, and the purpose is to locate any point on the earth’s surface. This is done by sensing radio signals between the satellites and the receiver on the earth, with the times of transmission and reception recorded. The distances are then calculated. Only three satellites are needed to pinpoint the position of the receiver on the earth. Relativistic effects must be taken into account arising both from special relativity (time dilation: moving clocks on the satellites run slower than clocks at rest on the surface of the earth) and from general relativity (gravitational time dilation/gravitational frequency shift: when viewed from the surface of the Earth, clocks on the satellites appear to run faster than identical clocks on the surface of the earth). The combined effect (the special relativistic correction and the general relativistic correction) is that the clocks on the satellites run faster than identical clocks on the surface of the earth by 38.4 microseconds per day. The clocks thus need to be adjusted by about 4 x 10-10s per day. If this factor is not taken into account, the GPS system ceases to function after several hours. This provides a stunning verification of relativity, both special and general.

Neil Ashby dedicates his paper to the GPS:

and gives a critical reason why the equivalence principle is indeed relevant. Consider again the GPS (global positioning system) or generally, Global navigation satellite systems (GNNS). For the GPS or GNNS, the only gravitational potential of significance is that of the earth itself. The earth and the satellites fall freely in the gravitational field of the sun (and external bodies in the solar system). Hence, according to the equivalence principle one can define a reference system which is locally very nearly inertial (with origin at the earth’s center of mass). In this locally inertial coordinate system (ECI) clocks can be synchronized using constancy of the speed of light (remember that special relativity is incorporated into general relativity as a model of space-time experienced locally by an observer in free fall):

One writes an approximate solution to Einstein’s field equation and obtains that clocks at rest on earth

run slow compared to clocks at rest at infinity by about seven parts in 1010.

Unless relativistic effects on clocks [clock synchronization; time dilation, the apparent slowing of moving clocks (STR); frequency shifts due to gravitation, gravitational redshift(GTR)] are taken into account, GPS will not work. GPS is thus a huge and remarkable laboratory for applications of the concepts of special and general relativity. In addition, Shapiro signal propagation delay (an additional general relativistic effect) and spatial curvature effects are significant and must be considered at the level of accuracy of 100 ps of delay. Ashby mentions another effect on earth that is exactly cancelled:

Wesson in this paper:

presents the standard explanation one would find in most recent textbooks on general relativity:

The Christoffel symbols are also used to define the Riemann tensor, which encodes all the relevant information about the gravitational field. However, the Riemann tensor has 20 independent components, and to obtain field equations to solve for the 10 elements of the metric tensor requires an object with the same number of components. This is provided by the contracted Ricci tensor. This is again contracted (taking its product with the metric tensor) to obtain the Ricci curvature scalar.  This gives a kind of measure of the average intensity of the gravitational field at a point in space-time. The combination of the Ricci tensor and the Ricci scalar is the Einstein tensor and it comprises the left hand-side of Einstein’s field equations.

At every space-time point there exist locally inertial reference frames, corresponding to locally flat coordinates carried by freely falling observers, in which the physics of general relativity is locally indistinguishable from that of special relativity. In physics textbooks this is indeed called the strong equivalence principle and it makes general relativity an extension of special relativity to a curved space-time.

Wesson then writes that general relativity is a theory of accelerations rather than forces and refers to the weak equivalence principle:

As said above, Einstein noted that if an observer in free fall 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. This is the weak principle of equivalence: The worldline of a freely falling test body is independent of its composition or structure. The test body moves along a geodesic line. The geodesic equation is independent of the mass of the particle. No experiment whatsoever is able, locally, to distinguish a gravitational field from an accelerating system – the strong principle of equivalence (see Ryder above). A freely falling body is moving along a geodesic line. However, globally space-time is curved and this causes the body’s path to deviate from a geodesic line and to move along a non-geodesic line. Hence we speak of geodesics, manifolds, curvature of space-time, rather than forces.

José G. Pereira explains the difference between curvature and torsion (and force) (see paper here):

General relativity is based on the equivalence principle and geometry (curvature) replaces the concept of force. Trajectories are determined not by force equations but by geodesics:

How do we know that the equivalence principle is so fundamental?  Gravitational and inertial effects are mixed and cannot be separated in classical general relativity and the energy-momentum density of the gravitational field is a pseudo-tensor (and not a tensor):

General relativity is grounded on the equivalence principle. It includes the energy-momentum of both inertia and gravitation:

In 1928 Einstein proposed a geometrized unified field theory of gravitation and electromagnetism and invented teleparallelism. Einstein’s teleparallelism was a generalization of Elie Cartan’s 1922 idea.

According to Pereira et al: “In the general relativistic description of gravitation, geometry replaces the concept of force. This is possible because of the universal character of free fall, and would break down in its absence. On the other hand, the teleparallel version of general relativity is a gauge theory for the translation group and, as such, describes the gravitational interaction by a force similar to the Lorentz force of electromagnetism, a non-universal interaction. Relying on this analogy it is shown that, although the geometric description of general relativity necessarily requires the existence of the equivalence principle, the teleparallel gauge approach remains a consistent theory for gravitation in its absence”.

See his paper with R. Aldrovandi and K. H. Vu: “Gravitation Without the Equivalence Principle”, General Relativity and Gravitation 36, 2004, 101-110.

Petkov explains in his paper: (see further above)

the following:

The bottom line is that classical general relativity is fundamentally based on the equivalence principle. One cannot reject Einstein’s route to the theory of general relativity.

## The Genesis and ‘Renaissance’ of General Relativity: Jürgen Renn’s talk

All the Berlin Century of General Relativity and MPIWG conference talks and discussions are on the web site of the Max Planck Institute for the History of Science in Berlin.

I shall begin by presenting the opening remarks of prof. Jürgen Renn’s talk, “The ‘Renaissance’ of General Relativity: Social and Epistemic Factors”, and thereafter I shall list my various comments on this talk. I shall present my views at the end regarding the term “Renaissance” describing the period from 1955 to the end of 1964.

Prof. Jürgen Renn:

“This talk is about explaining historical change; how we preliminary see the various stages of the history of general relativity: there was the genesis of general relativity [the period from 1907 to 1916] where Einstein worked not quite but almost alone with a few helpers. There were the formative years [the period from 1916 to 1925], in which the theory was discussed among a group of experts; there was then what Jean Eisenstaedt has epically termed the low-water mark period [the period from 1925 to 1955]; and then comes the renaissance [the period from 1955 to 1964]. Clifford Will gave the name to this period; and for us historians then came the golden ages [the period from 1964 to the mid 1970s] and today and the future, but I will not talk about that.

So the main challenge is to explain how development went from the low-water mark period to the renaissance; how what we today see as the basic theory of cosmology and astrophysics came back into the mainstream of physics after the low-water mark period. There were many factors of course and we have concentrated on several scenarios, examining them in preliminary ways to try to come up with what we think as a convincing explanation.

Let me start to outline the thesis. We think that the renaissance was mainly due to two factors:

One was the discovery of the untapped potential of general relativity as it has been created as a tool for theoretical physics. Hidden secrets that were discovered in this period, hidden potential of application, these would not have been explored and actively developed had it not been within a community that was just forming in this period.

[The second factor] We think that the renaissance is also very much history of a community, in which for the first time a real community of relativists and cosmologists emerged. We have kind of loosely been talking of relativists and cosmologists, even when referring to the twenties and the thirties [1920s and 1930s]. This is a somewhat anachronistic use of terms, because the real community only emerged as we see it during the period of the renaissance.

So you see already that the general approach is one that combines epistemological aspects with sociological aspects, and that is very much the spirit of our thinking. Robert Schulmann [a speaker in the conference] used the terminology of internalist and externalist. So we see this very much as a development that can only be understood if you combine the cognitive, the epistemological side, with the sociological developments parallel to it”.

Prof. Jürgen Renn says: “There was the genesis of general relativity where Einstein worked not quite but almost alone with a few helpers. There were the formative years, in which the theory was discussed among a group of experts; there was then what Jean Eisenstaedt has epically termed the low-water mark period”.

I think that even during the genesis of general relativity, between 1912 and 1916, Einstein’s interaction with and response to eminent and non-eminent scientists and his ongoing discussions with other scientists contributed to the formation of general relativity. The efforts invested by physicists like Max Abraham, Gunnar Nordström, Gustav Mie, David Hilbert and others, which presented differing outlooks and discussions revolving around the theory of gravitation, were relegated to the background. Those works that did not embrace Einstein’s overall conceptual concerns – these primarily included the heuristic equivalence principle and Mach’s ideas (later called Mach’s principle) – were rejected, and authors focused on Einstein’s prodigious scientific achievements. However, one cannot discard Einstein’s response to the works of Abraham, Nordström, Mie, Tullio Levi-Civita, Hilbert and others. On the contrary, between 1912 and 1915 (the so-called genesis of general relativity) Einstein’s response to these works, and corrections made by these scientists to his work, constitutes a dynamic interaction that assisted him in his development (so-called genesis) of the general theory of relativity. In my book, General Relativity Conflict and Rivalries I show that general relativity was not developed as a single, coherent construction by an isolated individual, brooding alone. Instead, general relativity was developed through Einstein’s conflicts and interactions with other scientists, and was consolidated by his creative process during these exchanges.

Indeed, performing a historical research and also applying comparative research in a sociological context, we can emphasize the limits and associated problems tracing Einstein’s “odyssey”, i.e. intellectual road to the general theory of relativity. An intellectual approach emphasizes the simplistic hero worship narrative. In addition, some philosophers embrace externalist theories of justification and others embrace internalist theories of justification. There are many different versions of internalism and externalism, and philosophers offered different conceptions of internalism and externalism.

I would like to comment on the “formative years” and the “low-water mark” period. I show in my book General Relativity Conflict and Rivalries that between 1916 and 1955, Einstein was usually trusted as the authority on scientific matters. His authority in physics is revealed even on first-rate mathematicians and physicists. For example, in 1918 Felix Klein demonstrated to Einstein that the singularity in the de Sitter solution to the general relativity field equations was an artefact of the way in which the time coordinate was introduced. Einstein failed to appreciate that Klein’s analysis of the de Sitter solution showed that the singularity could be transformed away. In his response to Klein, Einstein simply reiterated the argument of his critical note on the de Sitter solution. In 1917-1918 the physicist-mathematician Hermann Weyl’s position corresponded exactly to Einstein’s when he criticized de Sitter’s solution; Weyl’s criticism revealed the influence of Einstein’s authority in physics even on first-rate mathematicians like Weyl.

Two other examples from the introduction of my book:

In March 1918, before publishing the book Space-Time-Matter, Weyl instructed his publisher to send Einstein the proofs of his book. In the same month, Weyl also instructed his publisher to send David Hilbert the proofs of his book. Hilbert looked carefully at the proofs of Weyl’s book but noticed that the latter did not even mention his first Göttingen paper from November 20, 1915, “Foundations of Physics”. Though Weyl mentioned profusely Einstein’s works on general relativity, no mention was made of Hilbert’s paper. Einstein received the proofs page-by-page from the publisher and read them with much delight and was very impressed. However, Einstein, an initial admirer of the beauty of Weyl’s theory, now raised serious objections against Weyl’s field theory. Einstein’s objection to Weyl’s field theory was Weyl’s attempt to unify gravitation and electromagnetism by giving up the invariance of the line element of general relativity. Weyl persistently held to his view for several years and only later finally dropped it.

Einstein also seemed to influence Sir Arthur Stanley Eddington when he objected to what later became known as “black holes”. In his controversy during the Royal Astronomical Society meeting of 1935 with Subrahmanyan Chandrasekhar, Eddington argued that various accidents may intervene to save a star from contracting into a diameter of a few kilometres. This possibility, according to Eddington, was a reductio ad absurdum of the relativistic degeneracy formula. Chandrasekhar later said that gravitational collapse leading to black holes is discernible even to the most casual observer. He, therefore, found it hard to understand why Eddington, who was one of the earliest and staunchest supporters of the general theory of relativity, should have found the conclusion that black holes may form during the natural course of the evolution of stars, so unacceptable. However, it is very reasonable that Eddington, who was one of the earliest and staunchest supporters of Einstein’s classical general relativity, found the conclusion that “black holes” were so unacceptable, because he was probably influenced by Einstein’s objection to the Schwarzschild singularity.

Prof. Jürgen Renn explained in his talk: “We think that the renaissance is also very much a history of a community, in which for the first time a real community of relativists and cosmologists emerged. We have kind of loosely been talked of relativists and cosmologists, even when referring to the twenties and the thirties. This is a somewhat anachronistic use of terms, because the real community only emerged as we see it during the period of the renaissance”.

I would like to comment on the proposal to call the period from 1955 to 1964 the “renaissance” of general relativity. Clearly, in the atmosphere of cinquecento Florence or Milan, a scientist and artist needed a community. Actually for the first time in history, in renaissance Italy there were concentrations of scientists, writers, artists and patrons in close communities and courts. There were communication and interaction between those communities. Historians have reconstructed what went in these communities and how these communities functioned. Hence renaissance is a suitable historical term to describe the period from 1955 to 1964.

Updated July 2016, another talk by Jürgen Renn: the sixth biennial Francis Bacon Conference, “General Relativity at 100”, Bacon Award Public Lecture: (I did not attend this conference but the talk was uploaded to YouTube):

In 1907 in the patent office, Einstein was sitting down to write a review article in which he reviewed all the phenomena of physics in order to adapt them to the new framework of space and time established by special relativity. It was a routine task and dealing with gravitation in that context was also a routine task, so it seemed at the beginning.

[My comment: Einstein said in 1933, in the Glasgow lecture: “I came a step closer to the solution of the problem for the first time, when I attempted to treat the law of gravity within the framework of the special theory of relativity.” Apparently, sometime between September 1905 and September 1907 Einstein had already started to deal with the law of gravity within the framework of the special theory of relativity. When did he exactly start his work on the problem? Einstein did not mention any specific date, but in the Glasgow lecture he did describe the stages of his work presumably prior to 1907].

But then Einstein had to deal with the principle that was already established by Galileo hundreds of years ago, namely, the universality of free fall, the fact that all bodies fall with the same acceleration. How can these two be reconciled? When two people, one on the moving train and one on the platform each drop a stone, will the two stones hit the ground simultaneously? According to classical, Newtonian physics yes, but according to special relativity no. That is surprising. Was there any way to impose the universality of free fall and maintain Galileo’s principle?

The following is what inspired Einstein. Later he recalled in 1920: “Then came to me the happiest thought of my life in the following form. In an example worth considering the gravitational field only has a relative existence in a manner similar to the electric field generated by electromagnetic induction. Because for an observer in free-fall from the roof of a house, there is during the fall – at least in his immediate vicinity – no gravitational field”. Now the great thing is that this allowed Einstein to simulate gravity by acceleration and he could now treat accelerated frames of reference with the help of relativity, so he had a handler in the framework of special relativity, but in a new way that preserved universality of free fall. He could now predict that light bends in a gravitational field. The principle of equivalence (elevator experiments – linear acceleration, a heuristic guide).

Where did the idea of a generalization of the special relativity principle to accelerated motion actually come from? Einstein was particularly fascinated by Ernst Mach’s historical critical analysis of mechanics. What could Einstein learn from Mach? Mach had reconsidered Newton’s bucket experiment. Why does the water rise when it stars rotating? Newton’s answer was the following: because it moves with respect to absolute space. Mach’s answer was different: he claimed that it moves because it moves with respect to the fixed stars. That would make it a relative motion, an inertia, an interaction of bodies in relative motion with respect to each other: the water and the stars.

Rotation – heuristic guide

Besides Mach’s critique of mechanics, there is another element which can be identified in what Einstein called the happiest thought of his life. Einstein said (1920): “Then came to me the happiest thought of my life in the following form. In an example worth considering the gravitational field only has a relative existence in a manner similar to the electric field generated by electromagnetic induction. Because for an observer in free-fall from the roof of a house, there is during the fall – at least in his immediate vicinity – no gravitational field” (not italicized in the original). Use electromagnetic field theory as a mental model. We can see that it was the analogy of the gravitational field theory with the well-known theory of the electromagnetic field, on which Einstein of course was a specialist, that inspired him to the happiest thought as well, alongside the influence of Mach.

Now with the help of Mach, Einstein was in the position to complete the analogy between electromagnetic theory on the one hand and gravitational theory on the other hand by conceiving gravitation and inertia together as corresponding to the electromagnetic field. This analogy will guide him all these years from 1907 till 1915 to the completion of general relativity.

Let us look at the milestones of the genesis of general relativity. Usually this is portrayed as a drama in three acts:

[My comment: John Stachel wrote in his paper “The First Two Acts”, Einstein from B to Z, p. 261, that in 1920 Einstein himself wrote a short list of “my most important scientific ideas” in a letter to Robert Lawson (April 22, 1920):

1907 Basic idea for the general theory of relativity

1912 Recognition of the non-Euclidean nature and its physical determination by gravitation

1915 Field equations of gravitation. Explanation of the perihelion motion of Mercury.

Einstein’s words provide the warrant for comparing the development of general relativity to a three-act drama]:

According to Jürgen Renn: The problem here is that this portrait, this drama here leaves out the villain in this story, what is usually considered a villain, namely a theory on which Einstein worked between 1913 and 1915, in Zurich mostly but later also in Berlin, where he discarded it. It is called the preliminary or the draft and in German, the Entwurf theory. Now the point is what for other accounts is the villain of the story, a theory that was discarded, in my account, my book [? not yet published?] is the actual hero.

[My comment: I see what Jürgen Renn means, but I don’t think that for other historical accounts the Entwurf theory is a so-called villain of the story, a theory that is discarded in accounts of historians. Jürgen Renn even mentions these historians in his talk – Michel Janssen and John Stachel. What he jokingly calls the “villain”, the Entwurf theory, is a major part of my book, General Relativity Conflict and Rivalries, December 2015 and it is spread over many pages of it].

But let us proceed in order. At the begging of 1911 Einstein became the chair of physics at the German university of Prague, his first full professorship. Parague: What did Einstein achieve in 1912? He knew that the field had to be a combination of gravitation and inertia, but he did not know how to represent it mathematically. Fortunately the problem had two parts: equation of motion – the field tells matter how to move, and field equation – matter tells the field how to behave. He thus first tried to solve the problem of the equation of motion. And in particular the simple case how does a body move when no other forces, other than gravitation and inertia, act on it. Again the analogy with electromagnetism came to his rescue. It made sense that other special cases of dynamic gravitational fields such as the forces acting in a rotating frame of reference… [?] It should be possible to consider such a system at rest and the centrifugal forces acting there as dynamical gravitational forces. That’s what Einstein took from Mach. But the clue is a combination of gravity and inertia. So what could he learn?

Let us look at a rotating disk and try to measure its circumference with little roods. The disk is set into motion. Because of the length contraction predicted by special relativity, the rods would be shrunk, so that we need more of them to cover the circumference. In other words, the ratio between the circumference and the diameter would be larger than pi. This simple thought experiment gave Einstein the idea that, to describe general dynamical gravitational fields one needs to go beyond Euclidean geometry.

Non-Euclidean geometries were known. Einstein himself was not too familiar with non-Euclidean geometries. He had some courses at the ETH, he had skipped some courses at the ETH, but he did know that a straight line in such a geometry corresponds to a straightest line, or a geodesic. That solved for him the problem of the equation of motion, because when no other forces act in such a geometry, a particle would just follow the straightest possible line. Now the program of the new theory was clear. (John A, Wheeler: “matter tells space-time how to curve; curved space-time tells matter how to move”).

In the summer of 1912, Einstein returned to Zurich. He knew that he could describe a curved space-time by the metric tensor. He knew that he could define the deviation of space-time from Euclidean flat geometry in terms of the metric tensor.  A metric tensor is a complicated object:

The metric tensor replaces the one Newtonian gravitational potential with ten gravitational potentials. He could even write down the equation of motion in terms of the metric tensor, he achieved that relatively quickly. But he had no clue as how to find a field equation for this complicated object, the metric tensor, these ten gravitational potentials.

One of the most important sources for our story is a notebook in which Einstein entered his calculations, in the winter of 1912-1913, the so-called Zurich Notebook.

The following fraternity of scholars provided historical-critical-mathematical-physical interpretation of the Zurich Notebook (it took them 10 years to interpret Einstein’s calculations):

Einstein’s first attempts to deal with the mathematics of the metric tensor look rather pedestrian. He tried to bring together the metric tensor with what he knew about the gravitational field equation, which was also relatively little.

When Einstein was desperate he called his old friend Marcel Grossmann: “Grossman you have got to help me or I will go crazy!” Grossman had helped Einstein to survive his exams and he got him his job at the patent office. Now he helped him master the problem of gravitation. Indeed, one immediately recognizes Grossman’s intervention in the notebook. His name appears next to the Riemann tensor, the crucial object for building a relativistic field equation. Einstein immediately used it to form what he considered a candidate to the left hand-side of the field equation.

Einstein and Grossmann found that these field equations do not match their physical expectations. It turned out to be difficult to reconcile Einstein’s physical expectations with the new formalism. Groping in the dark, Einstein and Grossmann essentially hit upon the correct field equations, in the winter of 1912-1913, three years before the final paper, in the weak field limit:

But Einstein and Grossmann found that these field equations do not match their physical expectations. Eventually they had to learn how to adapt these physical expectations to the implications of the new formalism: we are talking about a learning experience that took place between a mathematical formalism and new physical concepts that were being shaped during the process.

Jürgen Renn then gives the following metaphor [a machine] to explain the mechanism behind a field equation (the Einstein-Grossmann Entwurf gravitational field equation): Einstein started out with a hand made mathematical formalism, at least before Grossmann came into the game, but it did the job. Extracting from the source of the field (mass and energy) a gravitational field. That’s what a field equation is all about. Of course it was most crucial that the familiar special case of Newtonian gravity would also come out in the appropriate circumstances. And Einstein had to make sure that this machine was firmly grounded in basic physical principles and in particular in the conservation of energy and momentum. But he also wanted to generalize the principle of relativity to accelerated motions and he looked for a machine that worked in more general coordinate systems, but at that point he didn’t know quite at which. The later point was unclear to Einstein.

The problem was that it was not clear whether that machine would actually deliver the requested physical results. With Grossmann came the dream for a much more sophisticated machine, a machine that worked for all coordinate systems because it was generally covariant:

Here the starting point was a sophisticated mathematical formalism based on the Riemann tensor, and then of course the machine had to work in the same way: the source makes a field, but does the Newtonian limit come out right? And is the machine firmly grounded in the principles of energy and momentum conservation? It certainly doesn’t look that quite way. In any way it was generally covariant, working in all coordinate systems. The problem was that it was not entirely clear whether that machine would actually deliver the requested physical results and what kind of tweaking it would take to get them. In short, the mechanism was great but the output was uncertain. Given this situation, Einstein could now peruse two different strategies:

*physical requirements: the Newtonian limit and the energy momentum conservation.

In the winter of 1912-1913, Einstein and with him Grossmann constantly oscillated between these two strategies. At the end of the winter he decided for one of them: the physical strategy. Well not quite, but rather a physical strategy tweaked and adapted to match the requirements of energy-momentum conservation and a generalized principle of relativity. So it was a home made extension of the original machine.

[My comment: The “physical strategy” and the “mathematical strategy” and the “oscillation” between them are memorable phraseology of Jürgen Renn. These had already been invented by Jürgen Renn several years ago. You can find it in many papers by Jürgen Renn. For instance Renn, Jürgen and Sauer, Tilman, “Pathways out of Classical Physics”, The Genesis of General Relativity 1, 2007, 113-312].

The result of the collaboration between Einstein and Grossmann in the winter of 1912-1913 was a hybrid theory, the so-called Entwurf or draft theory, the villain or hero mentioned before. Why was the theory, “Draft of a Generalized Theory of Relativity and a Theory of Gravitation” a hybrid theory? It was a hybrid theory because the equation of motion (the field tells matter how to move) is generally covariant (retaining its form in all coordinate systems), but the field equation (matter tells the field how to behave) is not generally covariant. It was not even clear in which coordinate system the field equation would be covariant. Nevertheless, Einstein was quite proud. To his future wife Elsa he wrote: “I finally solved the problem a few weeks ago. It is a bold extension of the theory of relativity together with the theory of gravitation. Now I must give myself some rest, otherwise I will go kaput”. But was it worth the effort? Wasn’t the Entwurf theory just a blind alley and a waste of time (for more than two and a half years)? Most accounts say yes and speak of a comedy of errors.

[My comment: As I mentioned previously, I am afraid I don’t really agree with this conclusion. Other historians don’t say yes. In my account, for instance, in my book General Relativity Conflict and Rivalries, I demonstrate that the Entwurf theory plays a crucial role in Einstein’s development of the November 1915 theory. I refrain, however, from using the terms “bridges”, “scaffolding” and other metaphors Einstein did not use (see below) because I think that, these terms and metaphors embed constraints that impact the understanding of the historical narrative. I rather prefer Einstein’s own terms, for instance, “heuristic guide”. Hence, Einstein was guided by the 1913 calculation of the perihelion of Mercury. So was he guided by the 1914 variational principle, formalism].

But if the Entwurf theory was important, what was its role? What function did it have for the creation of the general theory of relativity, if it turned out to be a wrong theory at the end? To answer that question we shall use another metaphor: the Entwurf theory as a scaffolding for building an arch or a bridge between physics and mathematics (Michel Janssen’s metaphor). The building of a bridge between physics and mathematics. On its basis, Einstein first calculated the Mercury perihelion motion, and working out its mathematical structure, and by “its”, meaning the preliminary Entwurf draft theory. Einstein worked out its mathematical structure and set up a variational formalism for it.

The Mercury calculation eventually helped him to solve the problem of the Newtonian limit and the variational formalism helped him to solve the other problem he had encountered with the mathematical strategy, that is, the conservation of energy and momentum. All this prepared the situation of November 1915, when it eventually came to a situation when the scaffolding was torn down.

Let us look at the first issue, the issue of the Mercury perihelion motion (a problem studied in detail by Michel Janssen). In 1913 Einstein together with Michele Besso calculated the Mercury perihelion motion on the basis of the Entwurf theory and the value which they miscalculated came out too small.  The theory predicated only 18″per century. This did not shatter, however, Einstein’s confidence in that theory. Einstein never mentioned this unsatisfactory result until 1915 but he reused this method developed under the auspices of the Entwurf theory in November 1915. The method of calculation,  the scaffolding, could be used in the final November theory. The creation of general relativity was a team effort. Besso had a role in the perihelion calculation in building a scaffolding for the transition to the final theory. The Mercury calculation helped Einstein understand the problem of Newtonian limit and accept the field equation he had earlier discarded. The Mercury (perihelion) calculation of 1913 really did act as a scaffolding for what Einstein achieved in November 1915, when he redid this calculation now on the basis of the correct theory.

Einstein’s colleagues were amazed how quickly he could calculate. David Hilbert wrote in a postcard just a day after Einstein had submitted the paper: “Congratulations on conquering the perihelion motion. If I could calculate as fast as you can, the electron would be forced to surrender to my equations and the hydrogen atom would have to bring a note from home to be excused for not radiating”. However, all Einstein had to do is to redo the calculations for the perihelion motion in the Entwurf theory that he had done with Besso in 1913 but never published. Einstein did not bother to tell Hilbert about this earlier work. Apparently he wanted to give Hilbert a dose of his own medicine, seeing Hilbert as somebody who gave the impression of being superhuman by obfuscating his methods (Einstein to Ehrenfest, May 24, 1916).

In November 1915 the building of a scaffolding (Besso’s assistance) helped Einstein to overcome his earlier problems with extracting the Newtonian limit from the field equation found along the mathematical strategy.

But what about the second problem? The conservation laws of energy and momentum? Again the Entwurf theory served as a scaffolding. In 1914 Einstein and Grossmann set up a variational formalism for the Entwurf theory from which it was easy to derive the conservation laws. That formalism was general enough to allow the derivation of the conservation laws also for other theories, including the ones Einstein had discarded in the winter of 1912-1913.  You can use the variational formalism to make a candidate field equation for the physical strategy or you can change the settings, and then you get a different machine, a candidate field equation for the mathematical strategy.

Having constructed such a formalism, the variational formalism, is what allowed Einstein to switch from the Entwurf theory to the theory he presented on November 4, 1915.

Einstein had resolved his two major problems that had prevented him in Zurich to accept candidate field equations along the mathematical strategy; namely the requirement to get out the Newtonian limit in the special case and the conservation laws. That confronts us with a puzzle: Why did Einstein not come back to the mathematical strategy right away once he had resolved these problems at the end of 1914?  He firmly believed in the Entwurf theory and had concocted all kinds of arguments in its favor, for instance the hole argument.

[My comment: I am afraid I don’t agree with this conclusion: I don’t think that at the end of 1914 Einstein had already resolved his two major problems (Newtonian limit and conservation laws). He was only able to resolve them in November 1915].

But by October 1915, his perspective was gradually changing because problems with the Entwurf theory were gradually accumulated:

1) It did not explain the perihelion problem well, we have seen that. Einstein could live with it. He just put it under the rug and did not mention it in his publications.

2) It did not allow him to conceive rotation at rest. It was a major blow, considering his Machian vision.

3) And, it did not follow uniquely, as he had hoped, from the variational formalism that he had set up. But this failure was actually a blessing in disguise. It meant that the formalism was actually more general and not just tailor-made for the Entwurf theory. So he could use it.

So these problems were the prelude to the drama of November 1915 when Einstein published week after week his four conclusive publications on general relativity. On the 4th of November that is the transition from the Entwurf theory to the new mathematical objects; with an addendum on the 11th of November; the Mercury paper on the 18th of November, and the final field equations on the 25th. The first paper contains an interesting hint at what Einstein considered the “fatal prejudice” that had hindered him so far and also what was the key to the solution. The subsequent papers successively straighten out a logical structure of the theory, show that now the Mercury problem works and the final paper completes the logical structure of the theory.

In order to understand what the fatal prejudice was and what the key to the solution was, we have to once more time look at the mechanism at work here:

There is indeed not just the source and the field. There is also the gravitational potential represented by the metric tensor and its connection with the gravitational field. That connection is expressed by a differential operator. One way to express this differential operator turned out to be a fatal prejudice, the other a key to the solution. As it turned out, it was essentially sufficient to change one element in the variational formalism developed for the Entwurf field equations, in order to get the theory from November the 4th, 1915. Namely, redefine the gravitational field.

Here are two ways in which Einstein expressed the connection between the field and the potential: One in the 1913 Entwurf theory and the other in the theory of November 4th. In the paper itself he speaks of the first way as a fatal prejudice and in a letter to Sommerfield he characterizes the second option as the key to the solution:

In the end the transition from the Entwurf theory, based on the physical strategy, to the November 1915 theory, based on the sophisticated math of the Riemann tensor, seems to have been a rather simple step. But what made this step possible was the scaffolding represented by the variational formalism Einstein had built for the Entwurf theory.

[My comment: In my book, General Relativity Conflict and Rivalries, I demonstrate an additional element. Einstein wrote to Sommerfeld the following:

“scalar derived from the energy tensor of matter, for which I write T in the following”.

In my book I show how one can derive the second term on the right-hand side of the above November 25, 1915 equation on the basis of the variational formalism and on the basis of Einstein’s 1914 Entwurf theory and November 4th, 1915 paper. I connect between this latter derivation and the derivation of the November 4, 1915 field equation from the 1914 variation principle].

A variational formalism is a machine for making machines: It made it simpler to pass from a machine based on the fatal prejudice:

to a machine that represented the key to the solution:

Einstein had to revise the architecture of his theory step by step and that is what happened in the final publication of November 1915. To Arnold Sommerfeld he wrote: “unfortunately I have immortalized my final errors in the academy papers” (November 28, 1915). And to his friend Paul Ehrenfest he wrote: “It is convenient with that fellow Einstein: every year he retracts what he wrote the year before” (December 26, 1915).

In Einstein’s defense one had to remember that what contributed to the drama was that the mathematical David Hilbert was or at least seemed to have been hot on Einstein’s trail. Hilbert presented his field equations in Göttingen on November 20, 1915, five days before Einstein. He used the Riemann curvature scalar in his variational formalism. He did not explicitly write down the field equation but he could have easily calculated it of course. In the late 1990s, page proofs of Hilbert paper, which itself was not published until March 1916, turned up. These page proofs carry a date of December 6, 1915, after Einstein’s publication.

Did Hilbert beat Einstein to the punch? His page proofs show that the original version of Hilbert’s theory was conceptually closer to the Entwurf theory than to Einstein’s final version. Hilbert’s theory of December 6, 1915 was just as the Entwurf theory, a hybrid theory with extra conditions on the coordinate systems. This restriction on the coordinate systems was later dropped in the published version appearing in March 1916. Hence, the moral is: Einstein could have taken his time in November 1915 and need not have worried about Hilbert stealing his thunder. Hilbert did not build a bridge between mathematics and physics as Einstein had done. In fact he didn’t worry about these problems of Newtonian limit and energy momentum conservation in the way that Einstein had done. So on November 25, 1915, the edifice of general relativity seemed complete.

This is the first part of Jürgen Renn’s lecture, it deals with the genesis of general relativity (until approximately 49 minutes after the YouTube video start time). The second part of the lecture, not given here, is quite disappointing compared to the first part. In the first part of the lecture, Jürgen Renn is inspired by great scholars, notably John Stachel. Unfortunately, the second part of the lecture lacks the inspiration and sensation of the greatness of the fraternity of Einstein scholars (see photo further above).

## The centenary of Einstein’s General Theory of Relativity

Einstein’s first big project on Gravitation in Berlin was to complete by October 1914 a summarizing long review article of his Einstein-Grossmann theory. The paper was published in November 1914. This version of the theory was an organized and extended version of his works with Marcel Grossmann, the most fully and comprehensive theory of gravitation; a masterpiece of what would finally be discovered as faulty field equations.

On November 4, 1915 Einstein wrote his elder son Hans Albert Einstein, “In the last days I completed one of the finest papers of my life; when you are older I’ll tell you about it”. The day this letter was written Einstein presented this paper to the Prussian Academy of Sciences. The paper was the first out of four papers that corrected his November 1914 review paper. Einstein’s work on this paper was so intense during October 1915 that he told Hans Albert in the same letter, “I am often so in my work, that I forget lunch”.

In the first November 4 1915 paper, Einstein gradually expanded the range of the covariance of his field equations. Every week he expanded the covariance a little further until he arrived on November 25 1915 to fully generally covariant field equations. Einstein’s explained to Moritz Schlick that, through the general covariance of the field equations, “time and space lose the last remnant of physical reality. All that remains is that the world is to be conceived as a four-dimensional (hyperbolic) continuum of four dimensions” (Einstein to Schlick, December 14, 1915, CPAE 8, Doc 165) John Stachel explains the meaning of this revolution in space and time, in his book: Stachel, John, Einstein from ‘B’ to ‘Z’, 2002; see p. 323).

These are a few of my papers on Einstein’s pathway to General Relativity:

http://xxx.tau.ac.il/abs/1201.5352

http://xxx.tau.ac.il/abs/1201.5353

http://xxx.tau.ac.il/abs/1201.5358

http://xxx.tau.ac.il/abs/1202.2791

http://xxx.tau.ac.il/abs/1202.4305

http://xxx.tau.ac.il/abs/1204.3386

http://xxx.tau.ac.il/abs/1309.6590

http://xxx.tau.ac.il/abs/1310.1033

http://xxx.tau.ac.il/abs/1205.5966

http://xxx.tau.ac.il/abs/1310.2890

http://xxx.tau.ac.il/abs/1310.6541

Stay tuned for my next centenary of GTR post!