Rotational Time Dilation and Angle Contraction Ling Jun Wang Department of Physics, Geology and Astronomy University of Tennessee at Chattanooga Chattanooga, TN 37403 U.S.A. Abstract: Rotational time dilation and angular contraction are discussed in comparison with the time deletion and length contraction of special relativity. The related issues of clock paradox and simultaneity are explored in rotational scenario. The non Euclidian nature of a relativistic rotation is analyzed in both θ - t and θ - r subspace. It has been shown that the constancy of the speed of light is intrinsically incompatible with rotational motion. The related issue of simultaneity and rotational space warping are also discussed. Key-words: relativity, rotation, rotating disk, time dilation, angle contraction. 1 Introduction nothing to do with either classical or relativistic The study of rotational motion played an time. His “Galileian system” was nothing short important part in the origin of relativity[1-6]. of an absolute system. The concept of “central The relation between the circumference and the time”[7,14] was criticized as a return to the radius was first pointed out by Ehrenfest [1] as a concept of absolute time. Arzelies’ theory also paradox, and used by Einstein [2] to show that asserted that “the velocity of light is different his new theory was non-Euclidean. The idea of a according to the sense of path”, which was a rotating plane being non Euclidian was objected direct violation of Einstein’s hypothesis of by prominent physicists such as Ehrenfest [1], constant speed of light. Lorentz [3], Eddington [4] and by others [5]. Added to the challenge was Sagnac’s Recently, Wang [15] has derived rigorously experiment [6], which was designed to detect the relativistic rotational transformation the interference of two coherent light beams consistent with Einstein’s field equation and its traveling in the opposite senses around a rotating solutions. Wang’s transformation has yeilded table. The results of Sagnac’s experiment was analytical expressions of rotational time dilation apparently contradicting the fundamental and angle contraction in free space as well as in hypothesis that the speed of light is a constant in gravitational fields. The rotational time dilation any system. The challenge stimulated many to has been used by Einstein himself [16-18] based propose a variety of ad hoc theories of on his time hypothesis, and by Hafele et al. in relativistic rotation [7-14]. These theories their around-the-world-clock experiments [19]. simply applied the special relativity to rotational In this article we will explore the implications motion with some unusual postulations or and significance of these effects. different definition of physical quantities, such as new difinition of uniform rotation[9,10], new 2 The Rotational Time Dilation difinition of rotation vector [11-12] or new In weak gravitational field, the differential definition of time [7,14]. Natuarally, these form of the transformation on the equatorial theories were not generally accepted. For plane is given by [15]: example, Arzelies’ “natural time” [14] has 1 dt = α (dt’+ (r/c)2 ω dφ’) shown explicitly that the time dilation derived dφ = α (ω dt’ + dφ’) (1) rigorously from general relativity agrees locally with the result of special relativity. The α = (1 – r2 ω2/c2)−1/2 (2) acceleration involved in the circular motion does not invalidate local Lorentz transformation and For an object fixed in the reference frame O’, its inverse, and it does not change the essence of dφ’= 0: time dilation and clock paradox. dt = α dt’ = (1- (rw/c)2)-1/2 dt’ (3) Another argument frequently used to resolve the clock paradox was based on the asymmetry where ω is the angular velocity and c is the speed of the twin space travel. One contends that one brother is sitting while the other flying, one is of light. (r,θ,φ) and (r’,θ’,φ’) are the spherical subjected to acceleration while the other is not. coordinates in the two systems. The origins O It was the asymmetry, argued this school of and O’ of the two systems coincide at the center thought, that should be responsible for the of rotation (Figure 1). differential aging[20-27]. Such argument loses its ground if we let both brothers traveling y’ symmetrically in the opposite directions [28]. y The rotational motion provides another scenario of space travel which shows that asymmetry can O,O’ x not be the cause of differential aging. Since relativistic rotational transformation is rigorously valid between any two systems in relative x’ motion, none of them can be assumed to be “at ω rest”. To assume some “rest” reference frame is to assume an “absolute system” which is in direct violation of the principle of relativity. No one can fix an absolute orientation without certain “absolute object”, and if such “absolute Figure 1 object” exists, it can be used as an absolute system. Since the product (rω) is the local linear velocity v, the constant α is the familiar factor γ Darwin[20] used a light-signal-counting of the special relativity. Eq.(1)-(3) show that the technique to resolve the clock paradox. Darwin’s time dilation due to rotational motion is locally technique invalidated the one-way time dilation, identical to that of special relativity. and in essence, it invalidated Lorentz transformation. The rotational transformation Eq.(3) has been used by Einstein [16-18] to now makes it crystal clear that the light signal illustrate the differential aging of twin brothers in counting technique does not work. Since the two a thought experiment of space travel, which observers O and O’ are located at the same point invoked the debate over the clock paradox. To all the time, they can compare their times at any resolve the paradox, one argue that Einstein’s moment without the need of sending light thought experiment involves acceleration and signals. A light signal sent out by any of the two Lorentz transformation does not apply. But such travelers would reach both observers at the same argument violates Einstein’s time hypothesis time because the radius is the same in both which states that the time dilation does not systems. The whole business of light-signal- depend on the acceleration. Our derivation has 2 counting is bankrupted in dealing with the time interpretation is obvious. First, the light signals dilation of rotating systems. One might argue of the simultaneous events will not arrive at an that the light signals sent by one brother have to observer of the rest reference frame unless the be received by the other brother, but not by the two events are located symmetrically with observers O or O’. Namely, the brothers instead respect to the observer. Second, it is possible for of the observers should be the time keepers. But the light signals of some simultaneous events in disqualification of the observers O and O’ as the one reference frame to arrive simultaneously at time keepers leads to fundamental problems. an observer in another reference frame. First, if the time can be measured by brothers, why can it not be measured by the observers? Rotational motion provides another scenario Second, if one asserts that the traveling brother to discuss relativistic simultaneity. Consider and the observer at the center of rotation measure events taking place on a circle with its center at different times, then one is actually saying that the origin. To the observer of one system, ∆t = the system has no common time at all. If no 0, but ∆φ > 0. Translated to another system, we system has a common time, then the concept of will have ∆t’ > 0. But the light signals of these time does not have a meaning in any physical events will arrive at both observers at the origin sense. Remember that there is no such system simultaneously. On the other hand, the with absolutely no motion, translational or simultaneous events on a radius would not arrive rotational. at any observer simultaneously. Apparently, the difference in time of arrival of light signal can Hafele and Keeting [19] had carried out an not be used to explain the relativistic around-the-world-clock experiment in 1971. The simultaneity. experiment was claimed to be an unique macroscopic experiment confirming relativistic 4 Angle Contraction time dilation. Their results were based on the Let us consider certain angular displacement assumption of an absolute system fixed with the dφ’ measured in the O’ system. This angular North star. If we let both clocks fly in such a displacement can be measured by the observer O way that the angular velocities of the two clocks under the condition dt = 0: are equal in magnitude but opposite in sense with respect to the North star, then Hafele’s theory 0 = α (dt’+ (r/c)2 ω dφ’) would lead to a clock paradox [28]. This dφ = α (ω dt’ + dφ’) (4) becomes clear in rotational motion. Since no system can, or need to be considered a non This leads to an angle contraction: absolute reference frame, either one of the two clocks can be considered in rotation or at rest. φ α−1 φ The results based on the assumption of a “rest” d = d ’(5) frame, the North star, are meaningless. Eq.(5) shows that the angular displacement dφ’ measured by O’ would be contracted by a 3 Simultaneity α One of the consequences of time dilation is factor of as measured by the observer O. relativity of simultaneity: Two events occurring Since the transformation goes both ways, an φ simultaneously in one reference frame are not angle d measured by the observer O would be α simultaneous in other reference frames.