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

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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…

 

שנים לטנסור המטרי 100 Years of Metric Tensor

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

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

Relatively speaking

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Alfred Eisenstaedt, Einstein Life Magazine

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

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

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

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

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

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

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

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

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

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

Time Magazine Photo

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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