Gravitational Time Dilation inside the Solid Sphere

Abstract

Gravitational time dilation directly reflects the difference between gravitational potentials at different altitudes in the gravitational field. At the same time this phenomenon is expected to obey the Einstein’s equivalence principle, one of two pillars (apart from general covariance) of general relativity. The experiments aimed at detecting the gravitational time dilation are therefore described as the tests of general relativity or, alternatively, the tests of equivalence principle. When applied to the exterior of a solid sphere, these two interpretations are fully compatible both theoretically and experimentally. However, when applied to the interior of a solid sphere (e.g., to the interior of Earth), they seem to contradict each other. Namely, a strict dependence of the gravitational time dilation on the gravitational potential inside the sphere proves to be at odds with the equivalence principle. This paper reveals this problem and provides solution to it. As a consequence, it is concluded that, contrary to the current belief, the Earth’s center is older, not younger, than the Earth’s surface. Since all the previous experiments have been performed either on or above the Earth’s surface, an experiment performed below the Earth’s surface is proposed.