The Relativistic Gravitational Field of a Spherically Symmetric Extended Body

TL;DR

This paper develops a Lorentz-covariant framework for describing the gravitational field of extended spherically symmetric bodies, revealing weak but measurable corrections to classical and relativistic predictions near compact objects.

Contribution

It introduces a relativistic superposition principle to derive explicit metrics for extended bodies, extending traditional solutions like Schwarzschild to include internal structure effects.

Findings

External gravitational field depends weakly on internal mass distribution

Corrections become significant near compact objects like neutron stars

Measurable differences in light travel times near Earth surface

Abstract

We investigate the gravitational field of an extended spherically symmetric body within the framework of Extended Relativity (ER), a Lorentz-covariant formulation of relativistic gravity on a Minkowski background. Using a relativistic superposition principle for retarded gravitational fields, we derive an explicit metric for an extended body by integrating the contributions of its mass elements. The resulting metric reproduces the standard gravitational time dilation of a point source and agrees with the classical tests of General Relativity in the appropriate limits. However, unlike the exact Newtonian shell theorem and the Schwarzschild exterior solution, the external field depends weakly on the internal mass distribution through higher-order corrections. These corrections decay rapidly with distance but become significant near compact objects. We analyze the corresponding admissible-velocity geometry, derive the motion equations for test particles, and compare the predictions of extended-body and point-source models. For neutron stars, the corrections noticeably modify the local light-velocity structure near the surface. For the Earth, the corrections are small but produce measurable differences in round-trip light travel times to the International Space Station. The formalism provides a transparent relativistic description of extended gravitational sources and offers a framework for studying relativistic corrections due to internal structure in strong-field and precision-measurement regimes.