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”.


My comments on this talk:

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 scholars

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).







Einstein’s cosmological constant

The year 2013 is Israel’s “Space Year”. Here

Read my new Paper discussing Einstein’s cosmological model and the cosmological constant:

The Mythical Snake which Swallows its Tail: Einstein’s matter world

In 1917 Einstein introduced into his field equations a cosmological term having the cosmological constant as a coefficient; he invented a finite and spatially closed static universe, bounded in space, according to the idea of inertia having its origin in an interaction between the mass under consideration and all of the other masses in the universe (Mach’s Principle).

In 1931 new experimental findings led Einstein to drop his cosmological constant.

We usually characterize Einstein’s renouncement of the cosmological constant and coming up with new ideas as Einstein’s mistake. Perhaps we rather say that Einstein’s old and new ideas link up with the same good old Mach’s principle that brought him to introduce the cosmological constant.

Later cosmological models of Einstein are either compatible or incompatible with Einstein’s understanding of Mach’s principle.

In 1931 Einstein dropped the cosmological constant and later also dropped Mach’s principle.


Einstein and Willem de Sitter in 1932

מסיבת התה של הכובען המטורף: הקבוע הקוסמולוגי של איינשטיין

ב-1917 אלברט איינשטיין החליט להוסיף איבר למשוואות השדה של תורת היחסות הכללית שלו. הוא הכניס את האיבר הקוסמולוגי בעל המקדם שקרוי “הקבוע הקוסמולוגי”, כדי שתורת היחסות הכללית תניב יקום סטטי. איינשטיין טען שהאיבר הקוסמולוגי לא ישנה את הקוואריינטיות של משוואות השדה וגם לא את שאר ניבויי התיאוריה.

באותו הזמן תורת היחסות של איינשטיין עדיין לא אומתה ניסויית, אבל איינשטיין היה נחוש בדעתו להשמיט את שאריות המרחב המוחלט שלכאורה אולי נותרו בתורתו. לשם כך הוא המציא “טירה יפיפה שתלוי באוויר”, כפי שאיינשטיין עצמו תאר זאת, עולם סופי וסגור בממדיו המרחביים ובייחוד עולם סטטי. עולם זה תאם לרעיונות של מאך, לפיהם האינרציה מקורה באינטראקציה שבין המסה לשאר המסות ביקום. שנה אחר כך איינשטיין היה כה בטוח ברעיונותיו של מאך עד כי הוא קרא לרעיון זה עקרון מאך.

מבחינה פיזיקאלית, הקבוע הקוסמולוגי בהיותו גדול מאפס פירושו היה הקיום של דחייה קוסמית וכך היקום הסטטי של איינשטיין הוא כזה שבו הדחייה בכל מקום מאזנת את משיכת הכבידה.

והנה ידידו של איינשטיין מלידן, וילהם דה סיטר, הגה פיתרון לאותן משוואות שדה של איינשטיין עם הקבוע הקוסמולוגי, אבל שמניבות יקום ריק לחלוטין. היקום של דה סיטר היה כדורי בממדיו המרחביים, אבל פתוח לאינסוף כאילו היה היפרבולואיד. דה סיטר שמע על עבודתו הניסויית של וסטו סליפר שחקר את המהירויות של 25 ערפיליות ספיראליות (מה שיותר מאוחר כונה גלקסיות). דה סיטר גילה אפקט הסחה לאדום בעולם ההיפרבולואידי שלו.

דה סיטר החליט להשוות בין העולם שלו לעולם של איינשטיין. כדי לעשות זאת הוא ביצע לעולם שלו טרנספורמציה לצורה סטטית, כך שעתה שני העולמות, שלו ושל איינשטיין, היו בעלי עקמומיות חיובית; עולם דה סיטר היה האנלוגיה הארבע-ממדית של העולם התלת-ממדי של איינשטיין. אבל בעולם של דה סיטר הזמן הוא לגמרי יחסי ושווה-ערך במעמדו לשלושת הקואורדינאטות המרחביות ואילו בעולם של איינשטיין הזמן באינסוף היה שונה כאילו היה זה זמן דמוי-מוחלט. לפיכך, המערכת של איינשטיין מספקת את עקרון היחסות רק אם פוסטולט זה תקף לשלושת ממדי המרחב ולא לממד הזמן. מכאן, טען דה-סיטר, איינשטיין השיב במו-ידיו את המרחב המוחלט של ניוטון, אותו חלל מוחלט שהוא כה התאמץ לגרש!

אבל איינשטיין לא השתכנע מהטיעונים של דה סיטר; ולא זאת בלבד, איינשטיין טען שהעולם של דה סיטר מפר את עקרון מאך. איינשטיין ניסה במקום זאת להדגים שהפתרון של דה סיטר מכיל סינגולאריות בדיוק בקו המשווה. במקום הזה שבו מצויה הסינגולאריות מתחבא לו החומר הנעלם ולכן עולם דה סיטר אינו ריק כלל. הטיעון של איינשטיין היה כזה: לפי תורת היחסות הכללית, ככל ששעונים הם קרובים יותר למקור חומרי, כך הם נעים לאט יותר. מכיוון שהשעונים הולכים ומאטים ככל שמתקרבים ל”קו המשווה” בעולם דה סיטר בצורה הסטטית, כל החומר של עולם דה סיטר מרוכז שם בקו המשווה.

ארתור אדינגטון הגדיר זאת בצורה ציורית ב-1920: ביקום דה סיטר “כאשר אנחנו מגיעים למחצית הדרך לנקודה הנגדית, הזמן עומד מלכת. בדיוק כמו מסיבת התה של הכובען המטורף, השעה היא תמיד 6 אחר הצהריים; ושום דבר לא יכול בכלל להתרחש ולא משנה כמה נחכה”.

ולכן איינשטיין הסיק שבפתרון דה סיטר ישנה סינגולאריות אינהרנטית, שהיא חלק מהפיתרון עצמו; ואם כך הדבר, מתחבא לו חומר שם בקו המשווה.

איינשטיין התווכח עם דה סיטר ולא קיבל את עובדת קיום יקומו הריק שסותר את עקרון מאך; ואז נכנס לויכוח המתמטיקאי הדגול פליקס קליין. קליין הסביר לאיינשטיין שקו המשווה בצורה הסטטית של יקום דה סיטר היא תופעת לוואי של הצורה הסטטית. למעשה זו לגמרי מקריות שיקום דה סיטר יכול להיכתב בצורה סטטית. וזו הסיבה שאנחנו אף פעם לא יכולים להגיע לקו המשווה, בגלל שהוא אירוע שנמצא מחוץ להישג ידינו; מערכת הקואורדינאטות שבה העולם של דה סיטר הוא סטטי מכסה רק חלק ממרחב-זמן דה סיטר השלם. לכן הסינגולאריות בקו-המשווה היא סינגולאריות לא אינהרנטית.

איינשטיין בהתחלה התקשה לקבל את הטיעון, אבל בסוף הוא הסכים לקבל שפתרון דה סיטר הוא אכן פתרון למשוואות השדה שלו המתוקנות עם הקבוע הקוסמולוגי, יקום ריק מחומר שמפר את עקרון מאך. אבל הוא עדיין האמין שזהו לא פתרון אפשרי מבחינה פיזיקאלית, אין כזה יקום פיזיקאלי; איינשטיין האמין שכל מודל קוסמולוגי אפשרי צריך להיות סטטי והרי המודל של דה סיטר מבחינה גלובאלית הוא אינו סטטי.

ב-1922 אלכסנדר פרידמן וב-1927 ג’ורג’ למטר פרסמו באופן נפרד זה מזה מודלים דינמיים ליקום. פרידמן גילה מודלים לא-סטטיים מעניינים בעלי קבוע קוסמולוגי שהוא אינו אפס או שווה לאפס. הוא ניבה יקום מתפשט או מתכווץ, שהניב את העולמות של איינשטיין ודה סיטר כמקרה פרטי. המודל של פרידמן עם קבוע קוסמולוגי שווה לאפס היה היקום הפשוט ביותר במסגרת תורת היחסות הפרטית. אבל ב-1922 איינשטיין חשב שהוא מצא טעות בתוצאות של פרידמן, שאם תתוקן, תיתן את היקום הסטטי של איינשטיין. פרידמן שלח לאיינשטיין את החישובים שלו ואיינשטיין השתכנע שהתוצאות של פרידמן אכן נכונות מתמטית, אבל סירב לקבל את הפתרון של פרידמן כמודל פיזיקאלי אפשרי.

ב-1927 למטר פרסם פחות או יותר את אותו המודל כמו זה של פרידמן, כאשר המודל של למטר היה יותר אסטרונומי מאשר המודל המתמטי של פרידמן. אבל כאשר למטר פגש את איינשטיין בכנס סולביי ב-1927, תגובתו של איינשטיין לעבודתו של למטר לא הייתה שונה מתגובתו למודל של פרידמן. איינשטיין היה מוכן לקבל את המתמטיקה אבל לא את הפיזיקה של היקום הדינמי המתפשט.

ב-1929 אדווין האבל הכריז על תגליתו הניסויית לפיה דומה שהיקום למעשה מתפשט. בשנים שאחרי 1930 הנטייה של הקוסמולוגים הייתה לעבור מתמיכה במודלים סטטיים כמתארים את היקום למודלים דינמיים. הגילוי של האבל נחשב לגילוי מרעיש.

ב-1931 איינשטיין ביקר בפסדינה ובהר וילסון והאבל וד”ר אדמס ליוו אותו למצפה כדי שיצפה בשמיים באמצעות הטלסקופ. איינשטיין הביט בגרמי השמיים והתפעם ולא רצה לעזוב את המקום. הוא בחן את התצפיות של האבל ועדויות אחרות שאיששו שאכן היקום מתפשט. איינשטיין שמע מהאבל עצמו אודות התצפיות שלו שהובילו למסקנה שהיקום מתפשט.

בשובו לברלין איינשטיין החליט לנטוש את הקבוע הקוסמולוגי לטובת יקום פרידמן עם הקבוע הקוסמולוגי ששווה לאפס. איינשטיין שב למשוואות השדה שלו מ-1916 ללא הקבוע הקוסמולוגי. איינשטיין פרסם מאמר קצר ב-1931 בו הוא מציג מודל קוסמולוגי עם קבוע קסמולוגי ששווה לאפס. מיד אחר כך דה סיטר הציג מודל קוסמולוגי זהה וב-1932 איינשטיין ודה סיטר חברו יחד וכתבו מאמר משותף שבו הם הציגו את מודל איינשטיין-דה סיטר.

למטר נותר נאמן לקבוע הקוסמולוגי והציע ב-1933 את ההיסטוריה המודרנית הראשונה של העולם. אבל בגלל השפעתו העצומה של איינשטיין שויתר על הקבוע הקוסמולוגי, קוסמולוגים לא שמו לב בהתחלה לרעיונות של למטר.

למטר הניח שהקבוע הקוסמולוגי גדול מאפס. הוא גילה שעבור יקום הומוגני איזוטרופי מתפשט, בזמן אפס בהיסטוריה הייתה סינגולאריות (והרי אנחנו זוכרים שלאיינשטיין הייתה בעיה עם סינגולאריות). בעקבות הסינגולאריות הזו הייתה התפשטות. כאשר בוחרים את הערך של הקבוע הקוסמולוגי בצורה מתאימה, מתחילה התפשטות מואצת, תחת דחייה קוסמית שאחר כך מואטת על ידי כבידה-עצמית מגיעים לכמעט עצירה במצב של יקום איינשטיין סטטי, שהוא בלתי תלוי בזמן. לפי למטר היקום המוקדם מאוד היה אטום קדום, גרעין אטומי קוסמי, כאשר הוא התפרק רדיואקטיבית בצורה ספונטאנית בצורת המפץ הגדול. ולכן היקום המאוד קדום נשלט על ידי חלקיקים בעלי אנרגיה מאוד גבוהה שיצרו יקום קדום הומוגני. למטר הסיק את קיומן של הקרניים הקוסמיות, השריד הקדום ביותר מההתפרקות הזו, חלקיקים אנרגטיים המרכיבים קרינת רקע למודל.

סטודנט של למטר סיפר, שמרבית האסטרונומים בתקופתו חשדו בתורת המפץ הגדול של למטר ובייחוד איינשטיין חשד בה, כי מי שהציע אותה היה כומר קתולי ותמך בה קווייקר [זרם דתי נוצרי] אדוק (ארתור אדינגטון).

אחרי שהוא ויתר על הקבוע הקוסמולוגי, איינשטיין המבוגר גם ויתר על עקרון מאך; וכך הוא נותר בלי קבוע קוסמולוגי, בלי עקמומיות מרחבית ובלי עקרון מאך… ג