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Home , Born rigidity, Ehrenfest paradox

Resolution of the in the Dynamic Interpretation of Lorentz Invariance F. Winterberg University of Nevada, Reno, Nevada 89557-0058, USA

Z. Naturforsch. 53a, 751-754 (1998); received July 14, 1998

In the dynamic Lorentz-Poincare interpretation of Lorentz invariance, clocks in absolute motion through a preferred reference system (resp. aether) suffer a true contraction and clocks, as a result of this contraction, go slower by the same amount. With the one-way velocity of light unobservable, there is no way this older pre-Einstein interpretation of can be tested, except in cases involv- ing rotational motion, where in the Lorentz-Poincare interpretation the interaction symmetry with the aether is broken. In this communication it is shown that Ehrenfest's paradox, the Lorentz contraction of a rotating disk, has a simple resolution in the dynamic Lorentz-Poincare interpretation of Lorentz invariance and can perhaps be tested against the prediction of special relativity.

One of the oldest and least understood paradoxes of contraction. With all clocks understood as light clocks special relativity is the Ehrenfest paradox [ 1 ], the Lorentz made up of Lorentz-contraded rods, clocks in absolute contraction of a rotating disk, whereby the ratio of cir- motion go slower by the same amount. For an observer cumference of the disk to its diameter should become less in absolute motion against the preferred reference than n. The paradox gave Einstein the idea that in accel- system, the contraction and observed on an erated frames of reference the metric is non-euclidean object at rest in this reference system is there explained and that by reason of his principle of equivalence the as an illusion caused by the Lorentz contraction and time same should be true in the presence of gravitational fields. dilation of the moving observer, not the observed object The disk to become non-euclidean, though, can only at rest. The reason for this symmetry, giving always the mean that it becomes curved, for example deformed into same impression for an observer in relative motion a dish, as it was conjectured by Ives [2], but this would against an object, is the group structure of the Lorentz not work if the disk is replaced by a rod. The different transformations. It makes it impossible to measure the attempts to resolve the Ehrenfest paradox, none of them one-way velocity of light, and all claims to the contrary satisfactory, have been summarized by Phipps [3]. are false [5]. There are two other experimentally verified effects In Einstein's theory, Lorentz invariance is a purely kin- which seem to contradict the kinematic interpretation of ematic space-time symmetry, explaining the Lorentz Lorentz invariance: The Sagnac effect and unipolar in- contraction as a rotation in space-time. In the Lorentz- duction. In all of these cases rotational motion is in- Poincare theory the contraction is real and caused by volved. Whereas for rectilinear the require- the interaction with an aether. Both interpretations give ment of Born-rigidity in special relativity [4] can be sus- identical results except in the case of rotational motion. tained by a proper acceleration program for different This underlines the importance of experiments involv- parts of a finite size body, this is not possible for rota- ing rotational motion, because they may be capable to tional motion. decide between the Einstein and Lorentz-Poincare theo- It is the purpose of this communication to show that ry [6]. the Ehrenfest paradox can be easily explained with the In the Lorentz-Poincare theory the atoms making up dynamic pre-Einstein by Lorentz and the arms of the Michelson-Morely interferometer, for ex- Poincare. It assumes the existence of a preferred refer- ample, are held together by electromagnetic forces. These ence system (or aether), with all rods in absolute motion forces are derived from potentials which in the aether rest against the preferred reference system to suffer a true frame obey the inhomogeneous wave equation

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For a body in static equilibrium (1) becomes trum. For this reason it can provide the Lorentz invari- ant repulsive force to prevent matter from collapsing. V20 = -4np(r) (2) Returning to the Ehrenfest paradox we first consider with the source p(r) in the body (resp. the arms of the the elastic deformation of a solid rotating cylinder, interferometer). ignoring Lorentz contraction. The solution of this prob- To obtain the corresponding equation after the body lem can be found in the textbook literature [8] and is has been set into absolute motion with the velocity v into given by the x-direction, (1) must be Galilei-transformed by

x' = x - v t, y' - y, z = z, t' = t, (3) where e= e(r) is the radial deformation of the rotating whereby it becomes cylinder of radius R and density p, wis its angular veloc- ity, E the modul of elasticity and p the Poisson number. 1 d2

2 2 2 d 0 | d C> } d

2 dyz dz2 variant form and where the colon stands for the covari- ant derivative. In polar coordinates r, 0, with pa = poo2 r, -4Kp'(x',y,z). (5) (9) is Comparing (5) with (2) demonstrates that (2) can be ob- da; r 2 2 2 2 + (T r -al =-pco r (10) tained from (5) setting dx' = dxA/l - v /c , implying a dr uniform contraction of the body and slower going clocks. The forces transmitted by the electromagnetic potentials Expressing the stress tensor o[ by the stress strain rela- causing an attraction between the atoms of the body must tions one obtains the solution (8). be balanced by a repulsive force to prevent the body from Next we include the effect of Lorentz contraction. Be- cause of Lorentz invariance under linear uniform motion collapsing. In quantum mechanics these repulsive forces we may consider the rotating cylinder to be at rest in the come from the zero point energy fluctuations. In the preferred aether rest frame. The Lorentz contraction act- Lorentz-Poincare theory they come from fluctuations of ing in the ^-direction leads to a relative change in length an electromagnetic background field. According to Ein- of a line element attached to the rotating cylinder, given stein and Hopf [7], a point particle of mass m and electric by charge e, moving with the velocity v through an iso- tropic electromagnetic radiation field with the frequen- Ss _ S-Sq _ cy spectrum/(ea), suffers the friction force 5 •So or (11) co df(co) : f(co)- v. (6) 2c mc 3 d co In the dynamic interpretation of Lorentz invariance it is According to a variational principle by Helmholtz, the caused by a reduction in the attractive force between ad- frictional dissipation of a fluid flow should assume a min- jacent atoms, resulting in a strain in the opposite direc- imum, and it is plausible that this should be also true for tion. In computing the deformation of the rotating rod we the radiation field. It therefore should have the form first assume that the Poisson number is equal to zero, pre- senting the limit of maximum rigidity. In this limit there f(co) = const. OJ\ (7) is no lateral stress, with the additional azimuthal stress given by for which the friction force vanishes. Now, not only is (7) Lorentz invariant, but at the same time it also has the Ss Eco 2 aJ.E^.-Ef. r . (12) form of the quantum mechanical zero point energy spec- F. Winterberg • The Ehrenfest Paradox in the Dynamic Interpretation of Lorentz invariance 753

Adding this stress to g§ on the l.h.s. of (10) one finds One therefore has for the total deformation

r e(r) = _ dO r . _r - „2 „2 I 1 E + Grr = -p(02r2 l- (13) d r 2 pc' (p <ü2/ 8E) (1 - E/(2p c2(1 + p)) [r(3R2 - r2)] (21)

One therefore has to replace in (8) co2 by co2(\ -E/lpc2), with the part due only to Lorentz contraction given by which for p = 0 leads to 2 2 2 2 €L(r) = -(C0 /(l6(l + p)c ))[r(3R -r )], (22)

(14) hence 2 2 3 £L(R) = -(co m\ + p)c ))R (23) while for the centrifugal force alone it would give Accordingly, the radius is now decreased from R to R 2 2 2 0 £=(p(0 /SE)[r(3R -r )]. (15) by

To isolate the effect of the Lorentz contraction, one has 2 2 3 R = R0- (