Relativistic Gravitation Based on Symmetry

Abstract

We present a Relativistic Newtonian Dynamics ( R N D ) for motion of objects in a gravitational field generated by a moving source. As in General Relativity ( G R ), we assume that objects move by a geodesic with respect to some metric, which is defined by the field. This metric is defined on flat lab spacetime and is derived using only symmetry, the fact that the field propagates with the speed of light, and the Newtonian limit. For a field of a single source, the influenced direction of the field at spacetime point x is defined as the direction from x to the to the position of the source at the retarded time. The metric depends only on this direction and the strength of the field at x. We show that for a static source, the R N D metric is of the same form as the Whitehead metric, and the Schwarzschild metric in Eddington–Finkelstein coordinates. Motion predicted under this model passes all classical tests of G R . Moreover, in this model, the total time for a round trip of light is as predicted by G R , but velocities of light and object and time dilation differ from the G R predictions. For example, light rays propagating toward the massive object do not slow down. The new time dilation prediction could be observed by measuring the relativistic redshift for stars near a black hole and for sungrazing comets. Terrestrial experiments to test speed of light predictions and the relativistic redshift are proposed. The R N D model is similar to Whitehead’s gravitation model for a static field, but its proposed extension to the non-static case is different. This extension uses a complex four-potential description of fields propagating with the speed of light.

Testing Relativistic Time Dilation beyond the Weak-Field Post-Newtonian Approximation

“…This would show which of the coordinate systems is more physical and whether the metric of a spherically symmetric gravitational field is time reversible. As shown in [7], if the additional term is non-zero, this implies a correction to the one-way speed of light in a gravitation field. This correction may be needed for the next level of accuracy in satellite navigation systems and global positioning systems.…”

“…RND does not use Einstein's field equations to define the metric. Rather, the metric in RND is derived [7] entirely from the assumption that the field propagates with the speed of light, the 4D symmetry of the problem and the Newtonian limit. For the gravitational field of a spherically symmetric, non-rotating body of mass M, the line interval of the RND metric in spherical coordinates is…”

“…This metric coincides with Whitehead's metric for such a field. From the derivation [7] of the metric in Equation 1, it follows that this is also the metric of a gravitation field generated by an extended, spherically symmetric body outside this body. The proper time τ of an object (for example, the time of the satellite clock) in a gravitational field is defined by cdτ = ds,…”

“…where J has the meaning of angular momentum per unit mass. The equations for r(τ) are also the same (see [7]):ṙ…”

“…Relativistic Newtonian Dynamics (RND) is an alternative relativistic theory of gravitation, developed recently by Yaakov Friedman and his collaborators (see [7,8]). As in GR, also here motion is by a geodesic with respect to a particular metric.…”