אודיסאת איינשטיין ליחסות הכללית Einstein’s Odyssey to General Relativity

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

סיינטיפיק אמריקן ישראל

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

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

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

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

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

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

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My drawing of Einstein:      האיור שלי של איינשטיין

איינשטיין צעיר

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

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The article discusses the following topics:

1907. The Happiest thought of my life.

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1907-1911. The equivalence principle and elevator experiments.

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

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

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

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

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

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

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

General Relativity without the Equivalence principle?

I have skimmed through this book Handbook of Spacetime:

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

I read this paper:

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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):

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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:

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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:

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in the chapter:

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Hence Joshi says:

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in the chapter:

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Lewis Ryder

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writes in the above paper:

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(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:

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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:

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

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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:

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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:

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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 :

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Einstein’s equivalence principle was criticized by Synge:

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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:

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

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

Graham S. Hall in his paper:

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writes the following:

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“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:

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

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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:

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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):

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

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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:

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Wesson in this paper:

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

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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:

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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):

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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:

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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):

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

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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. Picture20

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)

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the following:

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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.

 

 

 

 

 

 

 

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

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

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

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

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

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

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My first book:

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

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

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Review of Arch and Scaffold Physics Today

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

100 GTR: Uniformly Rotating Disk and the Hole Argument

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Further reading: Ehrenfest paradox

My new paper on Einstein and Schwarzschild

My new paper on General Relativity: Einstein and Schwarzschild.

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

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

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

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

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

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

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Karl Schwarzschild

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

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

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

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Strange Days at Blake Holsey High: a student is sucked into the black hole…

 

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.

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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”.

Einstein

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).

Albert Einstein as a Young Man

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!