Derivation of general relativistic gravitational potential energy using principle of equivalence and gravitational time dilation

Abstract

Einstein’s theory of general relativity which has been experimentally proved to be true theory of gravity doesn’t need gravitational potential energy to predict trajectory of particles in space. This is because general relativity is a purely geometric theory. Objects move along the geodesics in the curved space-time. The energy-momentum tensor that warps the space-time as per Einstein’s field equations takes into account only the energy/momentum of matter and radiation. Thus, gravitational potential energy doesn’t come into picture in Einstein’s theory of gravity and its role is taken over by curvature of space-time. However, general relativistically correct expression for gravitational potential energy is required for energy conservation and some energy-based approaches in physics. Conventionally, correct form of gravitational potential energy is derived by using full mathematical formality of general relativity. In this paper, we describe an event by which we derive the same general relativistic expression for gravitational potential energy simply by using the principle of equivalence and gravitational time dilation.