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Einstein's Uniformly Rotating Disk and the Hole Argument Galina 1 Einstein's Uniformly Rotating Disk and the Hole Argument Galina Weinstein April 22, 2015 Einstein's first mention of the uniformly rotating disk in print was in 1912, in his paper dealing with the static gravitational fields. After the 1912 paper, the rotating disk problem occurred in Einstein's writings only in a 1916 review paper, "The Foundation of the General Theory of Relativity". Einstein did not mention the rotating disk problem in any of his papers on gravitation theory from 1912 until 1916. However, between 1912 and 1914 Einstein invoked the Hole Argument. I discuss the possible connection between the 1912 rotating disk problem and the Hole Argument and the connection between the 1916 rotating disk problem and the Point Coincident Argument. Finally, according to Mach's ideas we see that the possibility of an empty hole is unacceptable. In 1916 Einstein replaced the Hole Argument with the Point Coincidence Argument and later in 1918 with Mach's principle. In September 1909 Paul Ehrenfest explained in his paper, "Gleichförmige Rotation starrer Körper und Relativitätstheorie" (Uniform Rotation of Rigid Bodies and the Theory of Relativity) that rigid bodies cannot exist in the theory of relativity. In 1909 Max Born discussed the problem of rigid bodies in the theory of relativity and showed that, the concept of the rigid body was incompatible with the special theory of relativity because of the Lorentz contraction. Ehrenfest posed a query which dealt with rotating rigid bodies and the Lorentz contraction in the special theory of relativity:1 Consider a relativistic rigid cylinder with radius R. It is gradually set into rotation around its axis until it reaches a state of constant rotation with angular velocity . As measured by an observer at rest, the radius of the rotating cylinder is R'. Then R' has to fulfill the following two contradictory conditions: 1. When the cylinder is moving, an observer at rest measures the circumference of the cylinder to be: due to the Lorentz contraction of lengths relative to its rest length, since each of its elements moves with an instantaneous velocity . 2. If one considers each element along the radius of the cylinder, then the instantaneous velocity of each element is directed perpendicular to the radius. Hence the elements of a radius cannot show any contraction relative to their rest length. This means that: R' = R. 2 Ehrenfest had pointed out that a uniformly rotating rigid disk would be a paradoxical object in special relativity, since on setting it into motion its circumference would undergo a contraction whereas its radius would remain uncontracted. This is "Ehrenfest's paradox".2 On February 28, 1910, Einstein wrote to Vladimir Varićak: "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".3 Later in 1919, Einstein explained why it was impossible for a rigid disk in a state of rest to gradually set into rotation around its axis. Einstein wrote to the philosopher and mathematician Joseph Petzoldt: 4 "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". In 1912, in Prague, Ehrenfest's paradox, however, motivated Einstein to discuss what came to be known as the uniformly (rigid) rotating disk thought experiment, in the first of two papers on static gravitational fields, "Lichtgeschwindigkeit und Statik des Gravitationsfeldes" (The Speed of Light and the Statics of the Gravitational Fields). Einstein considered a system K with coordinates x, y, z in a state of uniform rotation (disk) in the direction of its x-coordinate and referred it to (it is observed from) a non- accelerated system. The origin of K possesses no velocity.5 Hence, Einstein considered a rigid body already in a state of uniform rotation observed from an inertial system. The system K is equivalent to a system at rest (K') in which there exists a certain kind of mass-free static gravitational field. In 1912 Einstein extended the 1911 equivalence principle to uniformly rotating systems. According to Einstein's 1911 equivalence principle: "if we suppose that the systems K and K' are physically exactly equivalent, i.e., if we assume that we may just as well define the system K as being found in space free from gravitational fields, for we must then consider K as uniformly accelerated".6 Consequently, "By assuming this, we obtain a principle which, if it is true, has great heuristic meaning. For we obtained by theoretical consideration of the processes which take place relatively to a uniformly accelerating reference system, information as to the course of processes in a homogeneous gravitational field".7 Einstein explained that, the special measurements are performed by means of measuring rods. When these are compared with one another in a state of rest at the same location in K, they possess the same length. Hence, the laws of Euclidean geometry must hold for the lengths so measured and for the relations between the coordinates and for other lengths; but "they most probably do not hold in a uniformly 3 rotating system in which, owing to the Lorentz contraction, the ratio of the circumference to the diameter would have to be different from if our definition were applied. The measuring rod as well as the coordinate axes are to be conceived as rigid bodies. This is permitted despite the fact that, according to the [special] theory of relativity, the rigid body cannot really exist. For one can imagine the rigid measuring body being replaced by a great number of non-rigid bodies arranged in a row in such a manner that they do not exert any pressure on each other in that each is supported separately".8 Einstein explained the rotating disk problem very succinctly. Even though rigid bodies were not permitted in relativity theory, he still referred to measuring rods as rigid bodies in a coordinate-dependant (static) gravitational theory. "Non-rigid bodies" indicated that Euclidean geometry could not be applied in the rotating disk. 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 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 (are "non-rigid bodies"). 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 . If we adhere to Euclidean geometry, then we arrive at a contradiction because one and the same measuring rod produces what looks like two different lengths due to the Lorentz contraction. From the point of view of a system at rest, the diameter of the uniformly rotating disk would apparently be measured with uncontracted measuring bodies, but the circumference would seem to be longer because the measuring bodies would be contracted. The possibility of Euclidean geometry on a uniformly rotating disk is thus unacceptable. Neither the measurement of the circumference, nor any other feature of the rotating disk could be said to be in accordance with the Euclidean geometry, for the Euclidean geometry could not be applied in the rotating disk. Thus at one blow the rotating disk ostensibly destroyed the basic merits of the laws of Euclidean geometry, which did not hold anymore for the lengths so measured. Thinking about Ehrenfest's paradox and Born's rigidity and taking into consideration the principle of equivalence, Einstein arrived at the conclusion that a system in gravitational field has non-Euclidean geometry. Einstein's first mention of the rotating disk in print, as we have said, was in his paper dealing with the static gravitational fields of 1912. However, at about the same time Einstein started to write a manuscript on the special theory of relativity at the request of the Leipzig physicist, Erich Marx, who hoped to have a contribution on relativity theory from Einstein for his Handbuch der Radiologie. Although the precise circumstances remain obscure, it appears that Einstein worked on this article off and on in Prague and Zurich from 1912 to 1914, producing a 72-page text. 9 While still in Prague Einstein wrote in the manuscript on the special theory of relativity: 10 4 "Physical meaning of spatial determinations. The propositions of Euclidean geometry acquire a physical content through our assumption that there exist objects that possess the properties of the basic structures of Euclidean geometry. We assume that appropriately constructed edges of solid bodies not subjected to external influences have the definitional properties that straight lines have (material straight line), and that the part of a material straight line between two marked material points has the properties of a segment. Then the propositions of geometry turn into propositions concerning the arrangements of material straight lines and segments that are possible when these structures are at relative rest". After the words "and segments that are possible" in the original text, Einstein indicates a note he has appended at the foot of the page: "This term is unclear (rotation)". The reference to rotation is probably an allusion to the problem of the rotating disk discussed in the 1912 paper, "The Speed of Light and the Statics of the Gravitational Fields".11 From 1912 onwards, Einstein adopted the metric tensor as the mathematical representation of gravitation.
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