My new paper on Einstein and the general theory of relativity:

# Tag Archives: gravitational field equations

# 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:

- 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.
- 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.

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.

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.

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 g_{44 }= c^{2}, 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 g_{44 }= c^{2} is a function of spatial coordinates, x_{1}, x_{2}, x_{3. }

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).

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

# Einstein, Gödel and Time Travel

In January 1940 Gödel left Austria and emigrated to America, to the newly founded Institute for Advanced Study, located in Princeton University’s Fine Hall. Princeton University mathematician, Oskar Morgenstein, told Bruno Kreisky on October 25 1965 that, Einstein has often told him that in the late years of his life he has continually sought Gödel’s company, in order to have discussions with him. Once he said to him that his own work no longer meant much that he came to the Institute merely to have the privilege to be able to walk home with Gödel (Wang, Hao* Reflections on Kurt **Gödel*, MIT Press, 1987, 31).

Image of Einstein and Gödel.

Palle Yourgrau writes in his book *A World Without Time*:

“Within a few years the deep footprints in intellectual history traced by Gödel and Einstein in their long walks home had disappeared, dispersed by the harsh winds of fashion and philosophical prejudice. A conspiracy of silence descended on the Einstein-Gödel friendship and its scientific consequences. An association no less remarkable than the friendship between Michelangelo and Leonardo – if such had occurred – has simply vanished from sight”. (Yourgrau, Palle* A World Without Time: The Forgotten Legacy of *Gödel* and Einstein*, Basic Books, 2005, 7).

In fact, cinquecento Florence the cradle of renaissance art, led to competition among the greatest artists, Leonardo di ser Piero da Vinci and Michelangelo di Lodovico Buonarroti Simoni. And in 1504 the two greatest artists of the Renaissance Leonardo and Michelangelo became direct rivals.

Scuola di Atene, Raphael. Plato (Leonardo) and Aristotle (not Michelangelo).

We can readily admit that, the comparison Yourgrau made between Leonardo’s “friendship” with Michelangelo and Gödel’s daily walk to and from the Institute of Advanced Studies with Einstein is a comparison between two very different things.

Picture of Gödel and Einstein

In Princeton, in 1949, Gödel found that Einstein’s theory of general relativity allows the existence of closed timelike curves (CTCs), paths through spacetime that, if followed, allow a time traveler to interact with his/her former self (Gödel, Kurt, “An example of a new type of cosmological solutions of Einstein’s field equations of gravitation, *Reviews of Modern Physics *21, 1949, 447-450).

He explained (Gödel, Kurt, “A remark about the relationship between relativity theory and idealistic philosophy, in: *Albert Einstein – Philosopher-Scientist*, ed. Paul . A. Schilpp, 1949, pp. 555-562; pp. 560-561):

“Namely, by making a round trip on a rocket ship in a sufficiently wide curve, it is possible in these worlds to travel into any region of the past, present, and future, and back again, exactly as it is possible in other worlds to travel to distant parts of space. This state of affairs seems to imply an absurdity. For it enables one e.g., to travel into the near past of those places where he has himself lived. There he would find a person who would be himself at some earlier period of his life. Now he could do something to this person which, by his memory, he knows has not happened to him. This and similar contradictions, however, in order to prove the impossibility of the worlds under consideration, presuppose the actual feasibility of the journey into one’s own past”.

Einstein replied to Gödel (in: *Albert Einstein – Philosopher-Scientist*, ed. Paul. A. Schilpp, 1949, pp.687-688):

“Kurt Gödel’s essay constitutes, in my opinion, an important contribution to the general theory of relativity, especially to the analysis of the concept of time. The problem here involved disturbed me already at the time of the building up of the general theory of relativity, without my having succeeded in clarifying it. Entirely aside from the relation of the theory of relativity to idealistic philosophy or to any philosophical formulation of questions, the problem presents itself as follows:

If *P* is a world-point, a ‘light-cone’ (ds^{2}= 0) belongs to it. We draw a ‘time-like’ world-line through P and on this line observe the close world-points *B* and *A*, separated by *P*. Does it make any sense to provide the world-line with an arrow, and to assert that *B* is before *P*, *A* after *P*?

Is what remains of temporal connection between world-points in the theory of relativity an asymmetrical relation, or would one be just as much justified, from the physical point of view, to indicate the arrow in the opposite direction and to assert that *A is before* *P*, *B after P?*

In the first instance the alternative is decided in the negative, if we are justified in saying: If it is possible to send (to telegraph) a signal (also passing by in the close proximity of *P*) from *B* to *A*, but not from *A* to *B*, then the one-sided (asymmetrical) character of time is secured, i.e., there exists no free choice for the direction of the arrow. What is essential in this is the fact that the sending of a signal is, in the sense of thermodynamics, an irreversible process, a process which is connected with the growth of entropy (*whereas, according to our present knowledge*, all elementary processes are reversible).

If, therefore, *B* and *A* are two, sufficiently neighbouring, world-points, which can be connected by a time-like line, then the assertion: ‘*B* is before *A*,’ makes physical sense. But does this assertion still make sense, if the points, which are connectable by the time-like line, are arbitrarily far separated from each other? Certainly not, if there exist point-series connectable by time-like lines in such a way that each point precedes temporally the preceding one, *and if the series is closed in itself*. In that case the distinction ‘earlier-later’ is abandoned for world-points which lie far apart in a cosmological sense, and those paradoxes, regarding the *direction* of the causal connection, arise, of which Mr. Gödel has spoken”.

Seth Lloyd suggests that general relativistic CTCs provide one potential mechanism for time travel, but they need not provide the only one. Quantum mechanics might allow time travel even in the absence of CTCs in the geometry of spacetime. He explores a particular version of CTCs based on combining quantum teleportation (and quantum entanglement) with “postselection”. This combination results in a quantum channel to the past. The entanglement occurs between the forward – and backward going parts of the curve. Post-selection replaces the quantum measurement, allowing time travel to take place: Postselection could ensure that only a certain type of state can be teleported. The states that qualify to be teleported are those that have been postselected to be self-consistent prior to being teleported. Only after it has been identified and approved can the state be teleported, so that, in effect, the state is traveling back in time. Under these conditions, time travel could only occur in a self-consistent, non-paradoxical way. The resulting post-selected closed timelike curves (P-CTCs) provide time-travel (Quantum time machine) that avoids grandfather paradox. Entangled states of P-CTCs, allows time travel even when no space-time CTC exists. Such quantum time travel can be thought of as a kind of quantum tunneling backwards in time, which can take place even in the absence of a classical path from future to past.

But on March 3 1947, Einstein wrote the famous lines to Max Born: “I cannot make a case for my attitude in physics which you would consider at all reasonable. I admit, of course, that there is considerable amount of validity in the statistical approach which you were the first to recognize clearly as necessary given the framework of the existing formalism. I cannot seriously believe in it because the theory cannot be reconciled with the idea that physics should represent a reality in time and space, free from spooky actions at a distance”. (Einstein to Born March 3 1947, letter 84). And at about the same time Professor John Archibald Wheeler recounted the time he was presenting Einstein with a new method of looking at quantum mechanics. The aging Einstein listened patiently for 20 minutes. “Well, I still can’t believe God plays dice”, he replied, adding, “but may be I’ve earned the right to make my mistakes”…

# Einstein and 1915 General Relativity

My new paper shows that a correction of one mistake was crucial for Einstein’s pathway to the first version of the 1915 general theory of relativity, but also might have played a role in obtaining the final version of Einstein’s 1915 field equations. In 1914 Einstein wrote the equations for conservation of energy-momentum for matter, and established a connection between these equations and the components of the gravitational field. He showed that a material point in gravitational fields moves on a geodesic line in space-time, the equation of which is written in terms of the Christoffel symbols. By November 4, 1915, Einstein found it advantageous to use for the components of the gravitational field, not the previous equation, but the Christoffel symbols. He corrected the 1914 equations of conservation of energy-momentum for matter. Einstein had already basically possessed the field equations in 1912 together with his mathematician friend Marcel Grossman, but because he had not recognized the formal importance of the Christoffel symbols as the components of the gravitational field, he could “not obtain a clear overview”. Finally, considering the energy-momentum conservation equations for matter, an important similarity between equations suggests that, this equation could have assisted Einstein in obtaining the final form of the field equations (the November 25, 1915 ones) that were generally covariant.

My new paper on general relativity

# 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

**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 11 ^{th} 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

# Einstein’s Pathway to the Special Theory of Relativity and the General Theory of Relativity

Einstein’s Pathway to the Special Theory of Relativity

*Einstein’s Pathway to the Special Theory* of *Relativity *

Einstein’s discovery of Special Relativity

Einstein Believes in the Ether

Einstein Chases a Light Beam

Einstein recounts the Aarau thought experiment in his Autobiographical Notes, 1949

Magnet and Conductor Thought Experiment, Faraday’s magneto-electric induction

Föppl’s book on Maxwell’s theory

Ether drift and Michelson and Morley’s experiment

The Role of the Michelson-Morley Experiment on the Discovery of Relativity

The Dayton Miller Experiments

Emission theory and ether drift experiments

Paul Ehrenfest and Walter Ritz. Ritz’s Emission Theory

“The Step”

Einstein defined distant simultaneity physically; relativity of simultaneity

The Kyoto lecture notes – Einstein could have visited and consulted his close friend Michele Besso, whom he thanked at the end of his relativity paper. The Patent Office brought them together – their conversations on the way home. Besso was always eager to discuss the subjects of which he knew a great deal – sociology, medicine, mathematics, physics and philosophy – Einstein initiated him into his discovery

Joseph Sauter – Before any other theoretical consideration, Einstein pointed out the necessity of a new definition of synchronization of two identical clocks distant from one another; to fix these ideas, he told him, “suppose one of the clocks is on a tower at Bern and the other on a tower at Muri (the ancient aristocratic annex of Bern)” – synchronization of clocks by light signals.

Did Poincaré have an Effect on Einstein’s Pathway toward the Special Theory of Relativity? Einstein’s reply to Carl Seelig

Poincaré

Einstein’s pathway to the General Theory of Relativity

Entwurf theory – Einstein-Grossmann theory, Hole argument, field equations and the Einstein-Besso manuscript

Gunnar Nordström develops a competing theory of gravitation to Einstein’s 1912-1913 gravitation theory. Einstein begins to study Nordström’s theory and develops his own Einstein-Nordström theory. In a joint 1914 paper with Lorentz’s student Adrian Fokker – a generally covariant formalism is presented from which Nordström’s theory follows if the velocity of light is constant Here

Nordström

The three problems that led to the fall of the entwurf theory –

The gravitational field on a uniformly rotating system does not satisfy the field equations.

Covariance with respect to adapted coordinate system was a flop.

In the Entwurf theory the motion of Mercury’s perihelion came to 18^{0} rather than 45^{0} per century

The General Theory of Relativity – 1915

David Hilbert Enters the Game, the priority dispute – Einstein and Hilbert

In November 18 1915 Einstein calculated rapidly the precession of Mercury’s

Perihelion

Geodesic Equation. Metric tensor. Einstein’s November 4, 11, and 25 field equations.The Riemann-Christoffel Tensor; the Ricci tensor; the Einstein tensor

von Deinem zufriedenen aber ziemlich kaputen

General Theory of Relativity – 1916

Mid December to Mid January 1915: Exchange of letters between Einstein and Ehrenfest

The disk thought experiment; coordinates have no direct physical meaning Euclidean Geometry breaks down; two Globes Thought Experiment; Mach’s Principle; the principle of general relativity; the Equivalence Principle; the principle of general covariance

The Summation Convention

Motion of the Perihelion of the Planetary Orbit; Redshift; Deflection of light in a gravitational field of the sun

Einstein in the Patent Office:

Michele Besso, Joseph Sauter, and Lucian Chavan – Patent Office, Maurice Solovine and Conrad Habicht – the Olympian Academy

*Annus mirabilis* papers

On a Heuristic Viewpoint Concerning the Generation and Transformation of Light – It argues a heuristic manner for the existence of light quanta and derives the photoelectric law

On a New Determination of Molecular Dimensions – doctoral thesis submitted to the mathematical and natural science branch of Zürich University

On the Movement of Particles Suspended in Fluids at Rest, as Postulated by the Molecular Theory of Heat. The Brownian motion paper

On the Electrodynamics of Moving Bodies. The Relativity Paper

Does the Inertia of a Body Depend on its Energy Content? The first derivation of the mass energy equivalence

German Scientists Responded to Einstein’s Relativity Paper – Max Planck wrote Einstein. Max von Laue met Einstein

Einstein teaches his 3 friends from the Patent Office at the University of Bern

Finally Einstein leaves the Patent Office to his first post in the University of Zürich

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