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

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.