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

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

Grossmann

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 180 rather than 450 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