Abstract

In General Relativity, the gravitational field of a spherically symmetric non-rotating body is described by the Schwarzschild metric. This metric is invariant under time reversal, which implies that the power series expansion of the time dilation contains only even powers of v / c . The weak-field post-Newtonian approximation defines the relativistic time dilation of order ϵ (or of order ( v / c ) 2 ) of the small parameter. The next non-zero term of the time dilation is expected to be of order ϵ 2 , which is impossible to measure with current technology. The new model presented here, called Relativistic Newtonian Dynamics, describes the field with respect to the coordinate system of a far-removed observer. The resulting metric preserves the symmetries of the problem and satisfies Einstein’s field equations, but predicts an additional term of order ϵ 3 / 2 for the time dilation. This term will cause an additional periodic time delay for clocks in eccentric orbits. The analysis of the gravitational redshift data from the Galileo satellites in eccentric orbits indicates that, by performing an improved satellite mission, it would be possible to test this additional time delay. This would reveal which of the coordinate systems and which of the above metrics are real. In addition to the increase of accuracy of the time dilation predictions, such an experiment could determine whether the metric of a spherically symmetric body is time reversible and whether the speed of light propagating toward the gravitating body is the same as the speed propagating away from it. More accurate time dilation and one-way speed of light formulas are important for astronomical research and for global positioning systems.

Highlights

  • Einstein’s General Relativity (GR) has succeeded in explaining non-classical behavior in astrophysics

  • The gravitational field defines a metric on this spacetime, and the motion of any object is by a geodesic with respect to this metric

  • Gravitational time dilation is one of the main relativistic effects, which was tested with high accuracy in several experiments

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Summary

Introduction

Einstein’s General Relativity (GR) has succeeded in explaining non-classical behavior in astrophysics. For the gravitational field of a non-rotating, spherically symmetric body, Einstein’s field equations lead to the Schwarzschild metric, usually expressed in Schwarzschild coordinates [2]. In these coordinates, the metric is invariant under time reversal. It shows how to design an improved satellite experiment to determine whether the additional term, predicted by RND, could be observed Such an experiment will test the GR time delay prediction, based on the Schwarzschild coordinates and metric, that terms of order e3/2 must vanish for a spherically symmetric gravitational field. If the experiment produces a non-zero value for this term, this would show that the RND coordinates are more physical and its metric describes the field more precisely

Relativistic Newtonian Dynamics
The Time Delay Factor of Clocks on Satellites in Eccentric Orbits
Time Shift between the Clocks on the Galileo Satellites and EGNSS Clocks
Improved Experiment to Test Relativistic Time Dilation
Discussion
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