Men Explain Einstein to Me

If you are a woman, a man has probably at some point explained to you something about the area of your expertise. Rebecca Solnit tells about her own experience with a man who tried to explain (mansplain) a book she wrote to her.

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Mansplainers usually assume you know less than you do because of your gender. Advices.

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Stories.

On March 14 the official Albert Einstein Facebook wall hosted Dr. Debbie Berebiches. This was a Facebook Live Event celebrating Einstein’s birthday. Dr. Berebiches answered questions about Einstein (I haven’t listened to her, I am sorry). Dr. Berebiches is the first Mexican woman to graduate with a Ph.D in physics from Stanford University. Yesterday an Israeli hi-tech veteran explained to her what was “the main contribution of general relativity to GPS”, a topic she clearly knows more about than him. She gave him a like and this was absolutely the right thing to do.

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He then explained things to me, about Einstein’s religion. I am terribly sorry. I may not be the number one expert in Einstein’s philosophy of religion and personal views… well not that he is a great expert in this topic. Subsequently he asked me: “Have you read anything Einstein wrote in German? How about the dice quote? How about general relativity? …” and gave me advices:

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In fact I read many papers and manuscripts in German and when I wake up in the middle of a dream at night, I think I was speaking German to Albert Einstein in my dream.

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I could have stumped him but I preferred not to do this and not to discuss these matters further with him. I don’t believe mansplainers will ever change. Hence, it is better not to stump them and simply ignore them and avoid arguing with them on the area of your expertise. A mansplainer’s advice is unsolicited so you are under no obligation to listen. Advice.

Einstein expressed his religious views in 1922:

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and he also expressed his views in 1930:

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This article is from the Einstein Encyclopedia:

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The prediction of gravitational waves emerged as early as 1913

On February 11, 2016, The Max Planck Institute for the History of Science in Berlin published the following announcement: “One Hundred Years of Gravitational Waves: the long road from prediction to observation”:

“Collaborative work on the historiography 20th century physics by the Einstein Papers Project at Caltech, the Hebrew University of Jerusalem, and the Max Planck Institute for the History of Science carried out over many years has recently shown that the prediction of gravitational waves emerged as early as February 1916 from an exchange of letters between Albert Einstein and the astronomer Karl Schwarzschild . In these letters Einstein expressed skepticism about their existence. It is remarkable that their significant physical and mathematical work was carried out in the midst of a devastating war, while Schwarzschild served on the Eastern Front”.

Collaborative work by experts on the physics of Einstein from the Einstein papers Project, from the Max Planck Institute for the History of Science in Berlin: Prof. Jürgen Renn, Roberto Lalli and Alex Blum; and from the Hebrew University of Jerusalem the only representative is Prof. Hanoch Gutfreund, the academic director of the Albert Einstein Archives. Their main finding is therefore:

The prediction of gravitational waves emerged as early as February 1916 from an exchange of letters between Albert Einstein and the astronomer Karl Schwarzschild. However, from a historical point of view this is not quite accurate because Einstein reached the main idea of gravitational waves three years earlier, as I demonstrate below. Any way the group published two summaries of the study.

A summary was published in German:

“Als Einstein dann seine abschließende Arbeit zur allgemeinen Relativitätstheorie am 25. November 1915 der Preussischen Akademie in Berlin vorlegte, war die Frage, ob solche Wellen tatsächlich aus seiner Theorie folgen, noch offen. Einstein erwähnte das Thema zum ersten Mal in einem Brief, den er am 19. Februar 1916 an Karl Schwarzschild schickte. Nach einigen obskuren technischen Bemerkungen, stellte er lakonisch fest: „Es gibt also keine Gravitationswellen, welche Lichtwellen analog wären”.”

“Gravitationswellen – verloren und wiedergefunden” von Diana K. Buchwald, Hanoch Gutfreund und Jürgen Renn.

and also in English:

“When Einstein presented his theory of general relativity on Nov. 25, 1915 in Berlin, the question of whether such waves would constitute a consequence of his theory remained untouched. Einstein mentioned gravitational waves for the first time in a letter of 19 February 1916 to Karl Schwarzschild, a pioneer of astrophysics. After some obscure technical remarks, he laconically stated: “There are hence no gravitational waves that would be analogous to light waves”.”

“Gravitational Waves: Ripples in the Fabric of Spacetime Lost and Found” by Hanoch Gutfreund, Diana K. Buchwald and Jürgen Renn.

And here as well.

Hence, according to the three above authors Einstein mentioned gravitational waves for the first time in a letter of 19 February 1916 to Karl Schwarzschild. However, this is wrong . Einstein reached the main idea of gravitational waves three years earlier, which is not when the above group of scholars had thought the gravitational waves were mentioned for the first time. As early as  1913, Einstein started to think about gravitational waves when he worked on his Entwurf gravitation theory.

In the discussion after Einstein’s 1913 Vienna talk on the Entwurf theory, Max Born asked Einstein about the speed of propagation of gravitation, whether the speed would be that of the velocity of light. Here is Einstein’s reply:

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In 1916, Einstein followed these steps and studied gravitational waves.

See my papers on gravitational waves (one and two) and my book for further information.

A Historical Note on Gravitational Waves

Dr. Roni Gross (press conference) holds Einstein’s general relativity paper from May 1916, “The Foundation of the General Theory of Relativity” (“Die Grundlage der allgemeinen Relativitätstheorie.” Annalen der Physik 49, 769-822). However, in this paper Einstein did not discover gravitational waves. Prof. Hanoch Gutfreund, the academic director of the Albert Einstein Archives, asked Dr. Rony Gross to present this document to the journalists.

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here.

Equations (52) and (53) from the original page on the right above:

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are Einstein’s field equations for systems in unimodular coordinates. There are no gravitational waves here!

In his 1916 general relativity paper, “The Foundation of the General Theory of Relativity”, Einstein imposed a restrictive condition on his field equations. This condition is called unimodular coordinates.

Einstein presented the gravitational waves later in 1916, in a paper published under the title, “Approximate Integration of the Field Equations of Gravitation” (“Näherungsweise Integration der Feldgleichungen der Gravitation.” Königlich Preußische Akademie der Wissenschaften (Berlin). Sitzungsberichte, 688–696).

After the 1916 general relativity paper, Einstein succeeded in relinquishing the restrictive unimodular coordinates condition and in his new gravitational waves paper his equations were not restricted to systems in unimodular coordinates.

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How did Einstein predict the existence of gravitational waves?

Einstein’s Discovery of Gravitational Waves 1916-1918

Einstein and Gravitational Waves 1936-1938

For further details on Einstein predicting gravitational waves read Chapter 3, section 1 in my new book: General Relativity Conflict and Rivalries, Einstein Polemics with physicists.

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I present and read several sections of my books on Einstein

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I present my two books and read several sections of my books out loud:

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Until February 29, 2016, you can all receive a generous discount when purchasing my book, General Relativity Conflict and Rivalries: Einstein’s Polemics with Physicists by Cambridge Scholars.

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Some of the topics discussed in my first book, Einstein’s Pathway to the Special Theory of Relativity

People ask questions about Einstein’s special theory of relativity: How did Einstein come up with the theory of special relativity? What did he invent? What is the theory of special relativity? How did Einstein discover special relativity? Was Einstein the first to arrive at special relativity? Was Einstein the first to invent E = mc2?

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

In my book, Einstein’s Pathway to the Special Theory of Relativity, I try to answer these and many other questions.The topics discussed in my book are the following:

I start with Einstein’s childhood and school days.

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I then discuss Einstein’s student days at the Zurich Polytechnic. Einstein the rebellious cannot take authority, the patent office, Annus Mirabilis, University of Bern and University of Zurich, Minkowski’s space-time formalism of special relativity.

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Young Einstein, Aarau Class 1896

Additional topics treeated in my book are the following: Fizeau’s water tube experiment, Fresnel’s formula (Fresnel’s dragging coefficient), stellar aberration, and the Michelson and Michelson-Morley Experiments.

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Albert Einstein at the Patent office

Mileva Marić and Einstein

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Eduard Tete, Mileva Marić and Hans Albert

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Einstein’s road to the special theory of relativity: Einstein first believes in the ether, he imagines the chasing a light beam thought experiment and the magnet and conductor thought experiment. Did Einstein respond to the Michelson and Morley experiment? Emission theory, Fizeau’s water tube experiment and ether drift experiments and Einstein’s path to special relativity; “The Step”.

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Henri Poincaré’s possible influence on Einstein’s road to the special theory of relativity.

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Einstein’s methodology and creativity, special principle of relativity and principle of constancy of the velocity of light, no signal moves beyond the speed of light, rigid body and special relativity, the meaning of distant simultaneity, clock synchronization, Lorentz contraction, challenges to Einstein’s connection of synchronisation and Lorentz contraction, Lorentz transformation with no light postulate, superluminal velocities, Laue’s derivation of Fresnel’s formula, the clock paradox and twin paradox, light quanta, mass-energy equivalence, variation of mass with velocity, Kaufmann’s experiments, the principles of relativity as heuristic principles, and Miller ether drift experiments.

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The book also briefly discusses general relativity: Einstein’s 1920 “Geometry and Experience” talk (Einstein’s notion of practical geometry), equivalence principle, equivalence of gravitational and inertial mass, Galileo’s free fall, generalized principle of relativity, gravitational time dilation, the Zurich Notebook, theory of static gravitational fields, the metric tensor, the Einstein-Besso manuscript, Einstein-Grossmann Entwurf theory and Entwurf field equations, the hole argument, the inertio-gravitational field, Einstein’s general relativity: November 1915 field equations, general covariance and generally covariant field equations, the advance of Mercury’s perihelion, Schwarzschild’s solution and singularity, Mach’s principle, Einstein’s 1920 suggestion: Mach’s ether, Einstein’s static universe, the cosmological constant, de Sitter’s universe, and other topics in general relativity and cosmology which lead directly to my second book, General Relativity Conflict and Rivalries.

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My books

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The Einstein Rosen bridge and the Einstein Podolsky Rosen paradox, ER and EPR: Wormhole and entanglement

The Einstein-Rosen Bridge and the Einstein Podolsky Rosen paradox. I demonstrate that the two-body problem in general relativity was a heuristic guide in Einstein’s and collaborators’1935 work on the Einstein-Rosen bridge and EPR paradox.

In 2013 Juan Maldacena and Leonard Susskind demonstrated that the Einstein Rosen bridge between two black holes is created by EPR-like correlations between the microstates of the two black holes. They call this the ER = EPR relation, a geometry–entanglement relationship: entangled particles are connected by a Schwarzschild wormhole. In other words, the ER bridge is a special kind of EPR correlation: Maldacena and Susskind’s conjecture was that these two concepts, ER and EPR, are related by more than a common publication date 1935. If any two particles are connected by entanglement, the physicists suggested, then they are effectively joined by a wormhole. And vice versa: the connection that physicists call a wormhole is equivalent to entanglement. They are different ways of describing the same underlying reality.

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This image was published in a Nature article, Ron Cowen, “The quantum source of space-time”,  explaining the geometry–entanglement relationship and Maldacena’s and Susskind’s ER = EPR idea: Quantum entanglement is linked to a wormhole from general relativity. This representation is deterministic and embodies the many worlds interpretation: Collapse of the wave function never takes place. Instead, interactions cause subsystems to become entangled. Each measurement causes the branches of the tree to decohere; the quantum superposition being replaced by classical probabilities. The observer follows a trajectory through a tree. The entire tree, i.e., the entire wave function, must be retained and the universe is the complicated network of entanglements, branches of the tree: the wormhole bifurcating throats and mouths in the universe. See Susskind’s new paper from April 2016. Hence general relativity and quantum mechanics are linked by ER = EPR and Einstein’s soul can rest in peace because god will not play dice.

Maldacena and Susskind explain that one cannot use EPR correlations to send information faster than the speed of light.  Similarly, Einstein Rosen bridges do not allow us to send a signal from one asymptotic region to the other, at least when suitable positive energy conditions are obeyed. This is sometimes stated as saying that (Schwarzschild) Lorentzian wormholes are not traversable.

I uploaded a paper to the ArXiv: “Two-body problem in general relativity: A heuristic guide for the Einstein-Rosen bridge and EPR paradox”

In this paper I discuss the possible historical link between the 1935 Einstein-Rosen bridge paper and the 1935 Einstein-Rosen-Podolsky paper. The paper is a first version and I intend to upload a second version.

Between 1935 and 1936, Einstein was occupied with the Schwarzschild solution and the singularity within it while working in Princeton on the unified field theory and, with his assistant Nathan Rosen, on the theory of the Einstein-Rosen bridges. He was also occupied with quantum theory. He believed that quantum theory was an incomplete representation of real things. Together with Rosen and Boris Podolsky he invented the EPR paradox. In this paper I demonstrate that the two-body problem in general relativity was a heuristic guide in Einstein’s and collaborators’1935 work on the Einstein-Rosen bridge and EPR paradox.

In 1935 Einstein explained that one of the imperfections of the general theory of relativity was that as a field theory it was not complete in the following sense: it represented the motion of particles by the geodesic equation. A mass point moves on a geodesic line under the influence of a gravitational field. However, a complete field theory implements only fields and not the concepts of particle and motion. These must not exist independently of the field but must be treated as part of it. Einstein wanted to demonstrate that the field equations for empty space are sufficient to determine the motion of mass points. In 1935 Einstein attempted to present a satisfactory treatment that accomplishes a unification of gravitation and electromagnetism. For this unification Einstein and Rosen needed a description of a particle without singularity. In 1935, they joined two Schwarzschild solutions at the Schwarzschild limit and omitted part of the space-time beyond the Schwarzschild singularity. They showed that it was possible to do this in a natural way and they proposed the Einstein-Rosen bridge solution.

In the Einstein-Rosen bridges paper of 1935, Einstein negated the possibility that particles were represented as singularities of the gravitational field because of his polemic with Ludwig Silberstein. Silberstein thought he had demonstrated that general relativity was problematic. He constructed, for the vacuum field equations for the two-body problem, an exact static solution with two singularity points that lie on the line connecting these two points. The singularities were located at the positions of the mass centers of the two material bodies. Silberstein concluded that this solution was inadmissible physically and contradicted experience. According to his equations the two bodies in his solution were at rest and were not accelerated towards each other; these were nonallowed results and therefore Silberstein thought that Einstein’s field equations should be modified together with his general theory of relativity. Before submitting his results as a paper to the Physical Review, Silberstein communicated them to Einstein. This prompted Einstein’s remark, in his paper with Rosen in 1935, that matter particles could not be represented as singularities in the field.

Einstein and Rosen were trying to permanently dismiss the Schwarzschild singularity and adhere to the fundamental principle that singularities of the field are to be excluded. Einstein explained that one of the imperfections of the general theory of relativity was that as a field theory it was not complete because it represented the motion of particles by the geodesic equation. Einstein also searched for complete descriptions of physical conditions in quantum mechanics. It seems that the two-body problem in general relativity was a heuristic guide in the search of a solution to the problem that the psi function cannot be interpreted as a complete description of a physical condition of one system. He thus proposed the EPR paradox with Rosen and Podolsky.

Updated 2016

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.

My book: Einstein’s Pathway to the Special Theory of Relativity

2015 marks several Albert Einstein anniversaries: 100 years since the publication of Einstein’s General Theory of Relativity, 110 years since the publication of the Special Theory of Relativity and 60 years since his passing.

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What is so special about this year that deserves celebrations? My new book on Einstein: Einstein’s Pathway to the Special Theory of Relativity has just been returned from the printers and I expect Amazon to have copies very shortly.

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The Publisher uploaded the contents and intro.

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I hope you like my drawing on the cover:

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Einstein, 1923: “Ohmmm, well… yes, I guess!”

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The book is dedicated to the late Prof. Mara Beller, my PhD supervisor from the Hebrew University of Jerusalem who passed away ten years ago and wrote the book: Quantum Dialogue (Chicago University Press, 1999):

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Have a very happy Einstein year!

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