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1 Mach's Principle and Higher-Dimensional MACH’S PRINCIPLE AND HIGHER-DIMENSIONAL DYNAMICS B. Mashhoon1 and P.S. Wesson2,3 1Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA 2Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada 3Herzberg Institute of Astrophysics, National Research Council, Victoria, B.C. V9E 2E7, Canada Abstract We briefly discuss the current status of Mach’s principle in general relativity and point out that its last vestige, namely, the gravitomagnetic field associated with rotation, has recently been measured for the earth in the GP-B experiment. Furthermore, in his analysis of the foundations of Newtonian mechanics, Mach provided an operational definition for inertial mass and pointed out that time and space are conceptually distinct from their operational definitions by means of masses. Mach recognized that this circumstance is due to the lack of any a priori connection between the inertial mass of a body and its Newtonian state in space and time. One possible way to improve upon this situation in classical physics is to associate mass with an extra dimension. Indeed, Einstein’s theory of gravitation can be locally embedded in a Ricci-flat 5D manifold such that the 4D energy-momentum tensor appears to originate from the existence of the extra dimension. An outline of such a 5D Machian extension of Einstein’s general relativity is presented. Key words: Mach’s principle, higher-dimensional general relativity PACS numbers: 04.20.Cv, 04.50.Cd, 11.10.Kk Corresponding author: B. Mashhoon (e-mail address: [email protected]). 1 I. Introduction In his treatise on the science of mechanics, Ernst Mach provided a critical analysis of the physical foundations of Newtonian mechanics [1]. The epistemological difficulties associated with the Newtonian notions of absolute space and time, and the ensuing problem of the physical origin of inertial forces, led Mach to suggest that inertia is due to mutual interaction between masses. Mach’s ideas provided the early inspiration for Einstein on his path to general relativity. It is now generally acknowledged, however, that what may loosely be termed Mach’s principle is not properly incorporated into general relativity; in particular, the origin of inertia remains essentially the same as in Newtonian physics. Various attempts at the resolution of difficulties that are encountered in linking Mach’s principle with Einstein’s theory of gravitation have led to many interesting investigations [2-6]; moreover, a perusal of these references illustrates the diversity of opinion regarding Mach’s principle that still lingers. Within the framework of Einstein’s theory of gravitation, the current status of Mach’s principle, especially in connection with the physical origin of inertial forces, is discussed in sections II and III. Briefly, in Einstein’s theory, the global inertial frames of Newtonian physics are replaced by local inertial frames in accordance with Einstein’s local principle of equivalence; nevertheless, so long as only Newtonian gravitational accelerations—due to their universality— can be rendered “relative” in general relativity, non-gravitational accelerated motion retains its absolute character in Einstein’s theory of gravitation. Classical inertial effects thus originate from accelerated motion with respect to the local inertial frames. It is important to recognize that in the course of his critique of Newton’s view of time, space and motion, Mach also identified the essential epistemological weakness of Newtonian mechanics: The internal state of a Newtonian point particle, characterized by its inertial mass, has no immediate connection with the particle’s external state in absolute space and time, characterized by its position and velocity [1]. As described in detail in section IV below, Mach considered how time and space are operationally defined via masses and concluded that time and space, as physical concepts, are in fact different from their operational definitions by means of masses. That is, in Newtonian mechanics, masses are simply “placed” in absolute space and time, which remain external to them. As a consequence of this circumstance, the extrinsic state 2 of a point mass m, namely its position x and velocity v, can be shared by other “comoving” masses. How can this lack of organic connection between m and (x, v) be overcome, apart from Mach’s own solution in terms of relativity of all motion? One way involves the standard extension of classical mechanics to the quantum domain; indeed, this line of thought entails complementarity of absolute and relative motion and has been discussed in references [7-12]. Another way is to associate the internal states of classical particles with an extra dimension. That is, Einstein’s theory of gravitation can be extended to a 5D space-time-matter theory such that the vacuum gravitational field in 5D appears in 4D with an induced energy-momentum tensor as the source of the 4D gravitational field. Indeed, every solution of Einstein’s theory in 4D can be locally embedded in a Ricci-flat 5D manifold in accordance with Campbell’s theorem [13-17]. In such a theory, the free motion of a classical particle in 5D would in general involve the variation of space, time and inertial mass along its 4D world line; therefore, the inertial mass of a test particle varies in space-time. This is in contrast to standard general relativity, where the mass of a test particle is independent of space-time [18, 19]. We briefly describe such a 5D Machian extension of general relativity in section V. II. Relativity of Accelerated Motion? Ernst Mach had a predilection for relativity. In a collection of three essays that he wrote on space and geometry, we find in his discussion of “the relativity of all spatial relations” [20] that: “…All physical determinations are relative. Consequently, likewise all geometrical determinations possess validity only relatively to the measure. The concept of measurement is a concept of relation, which contains nothing not contained in the measure….” It is therefore clear that the absolute space and time of Newtonian physics posed a dilemma for Mach: it was impossible for him to reconcile the great practical success of Newtonian mechanics with the fact that it was based on the notions of absolute space and absolute time, which Mach considered to be unphysical. He imagined that these only formed a temporary framework for 3 physics and would be replaced someday by a future theory based purely on the relativity of all motion [1]. In Newtonian mechanics, absolute motion is defined as movement with respect to the ensemble of inertial frames of reference that are related to each other by Galilean transformations. Moreover, the reality of absolute space and time is revealed via the existence of inertial forces, which are experienced, for instance, by observers at rest in reference systems that accelerate with respect to inertial frames of reference. Mach speculated that the acceleration must instead be relative to the rest of the matter in the universe [1]. That is, inertial forces must be due to an interaction between local and distant masses. Following Mach, Einstein suggested that inertial forces could be of gravitational origin. Such a global Machian equivalence between inertia and gravitation would be consistent with the principle of equivalence of inertial and gravitational masses, since inertial forces acting on a body are all proportional to its inertial mass and hence, via the equivalence principle, its gravitational mass as well. To explain the motivation for his pioneering 1918 work on the gravitational effects of rotating masses in general relativity, Hans Thirring quoted from a 1914 paper of Einstein as follows [21]: “At first it seems that such an extension of the theory of relativity has to be rejected for physical reasons. Namely, let K be a permissible coordinate system in the Galilei-Newtonian sense, K’ a uniformly rotating coordinate system with respect to K. Then centrifugal forces act on masses which are at rest relative to K’, while not on masses at rest relative to K. Already Newton considered this as proof that the rotation of K’ has to be interpreted as “absolute”, and that one cannot thus treat K’ as “at rest” with the same justification as K. This argument is not valid, however, as explained by E. Mach in particular. That is, we need not attribute the existence of those centrifugal forces to the motion of K’; rather, we can attribute them as well to the average rotational motion of the ponderable distant masses in the surrounding relative to K’, whereby we treat K’ as at rest. If the Newtonian laws of mechanics do not admit such a conception, this could well be caused by the defects of this theory….” In short, relativity of accelerated motion necessitates a new theory of gravitational interactions. On the other hand, if such a theory is formulated as a relativistic field theory, then there is an insurmountable conflict with causality as the retardation times between local events and distant 4 masses would be extremely long. Even in our galaxy, it would take thousands of years for local news to reach the stars of our galaxy and similarly for their reaction to get back to earth. Einstein’s theory of gravitation, which is in good agreement with solar system data, is indeed a relativistic field theory modeled after Maxwell’s theory of the electromagnetic field. The modern generally accepted geometric formulation of Einstein’s theory is based upon the local equivalence of an observer in a gravitational field with a certain accelerated observer in Minkowski space-time—in fact, this is the content of Einstein’s heuristic principle of equivalence. Motivated in its infancy by the Machian notion of relativity, the current theory of gravitation is Einstein’s general relativity in which the gravitational field is identified with the Riemannian curvature of space-time [22].
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