Relativistic generalization of the inertial and gravitational masses equivalence principle

TL;DR

This paper discusses the need for a relativistic approach to the equivalence principle, especially for electromagnetic sources, emphasizing that non-relativistic approximations are insufficient for their stress-energy tensor.

Contribution

It introduces a relativistic generalization of the equivalence principle for inertial and gravitational masses, considering electromagnetic sources within weak gravitational fields.

Findings

Electromagnetic sources require relativistic treatment in gravitational equations.

Non-relativistic approximations are inadequate for electromagnetic stress-energy tensors.

A relativistic framework aligns better with the properties of electromagnetic sources in gravity.

Abstract

The Newtonian approximation for the gravitational field equation should not necessarily involve admission of non-relativistic properties of the source terms in Einstein's equations: it is sufficient to merely consider the weak-field condition for gravitational field. When a source has electromagnetic nature, one simply {\em cannot} ignore its intrinsically relativistic properties, since there cannot be invented any non-relativistic approximation which would describe electromagnetic stress-energy tensor adequately, even at large distances where the fields become naturally weak. But the test particle on which gravitational field is acting, should be treated as non-relativistic (this premise is required for introduction of the Newtonian potential $\Phi_{\rm N}$ from the geodesic equation).