Mach's Principle and a Relativistic Theory of Gravitation

INTRODUCTION small mass, its eGect on the metric is minor and can be considered in the weak-field approximation. The observer would, according to general relativity, observe normal behavior of his apparatus in accordance with the usual laws of physics. However, also according to general relativity, the experimenter could set his laboratory rotating by leaning out a window and firing his 22-caliber riQe tangentially. Thereafter the delicate gyroscope in the laboratory would continue to point in a direction nearly fixed relative to the direction of motion of the rapidly receding bullet. The gyroscope would rotate relative to the walls of the laboratory. Thus, from the point of view of Mach, the tiny, almost massless, very distant bullet seems to be more important that the massive, nearby walls of the laboratory in determining inertial coordinate frames and the orientation of the gyroscope. ' It is clear that what is being described here is more nearly an absolute space in the sense of Newton rather than a physical space in the sense of Berkeley and Mach. The above example poses a problem for us. Apparently, we may assume one of at least three things: ' 'T is interesting that only two ideas concerning the ~ nature of space have dominated our thinking since the time of Descartes. According to one of these pictures, space is an absolute physical structure with properties of its own. This picture can be traced from Descartes vortices' through the absolute space of Newton, ' to the ether theories of the 19th century. The contrary view that the geometrical and inertial properties of space are meaningless for an empty space, that the physical properties of space have their origin in the matter contained therein, and that the only meaningful motion of a particle is motion relative to other matter in the universe has never found its complete expression in a physical theory. This picture is also old and can be traced from the writings of Bishop Berkeley' to those of Krnst Mach. 4 These ideas have found a limited expression in general relativity, but it must be admitted that, although in general relativity spatial geometries are aGected by mass distributions, the geometry is not uniquely specified by the distribution. It has not yet been possible to specify boundary conditions on the field equations of general relativity which would bring the theory into accord with Mach's principle. Such boundary conditions would, among other things, eliminate all solutions without mass present. It is necessary to remark that, according to the ideas of Mach, the inertial forces observed locally in an accelerated laboratory may be interpreted as gravitational effects having their origin in distant matter accelerated relative to the laboratory. The imperfect expression of this idea in general relativity can be seen by considering the case of a space empty except for a lone experimenter in his laboratory. Using the traditional, asymptotically Minkowskian coordinate system 6xed relative to the laboratory, and assuming a normal laboratory of 1. that physical space has intrinsic geometrical and inertial properties beyond those derived from the matter contained therein; 2. that the above example may be excluded as nonphysical by some presently unknown boundary condition on the equations of general relativity. 3. that the above physical situation is not correctly described by the equations of general relativity.