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 a true theory of gravity does not need gravitational potential energy to predict the 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 space–time as per Einstein’s field equations takes into account only the energy–momentum of matter and radiation. Thus, gravitational potential energy does not come into the picture in Einstein’s theory of gravity and its role is taken over by the curvature of space–time. However, a general relativistically correct expression for gravitational potential energy is required for energy conservation and some energy-based approaches in physics. Conventionally, the correct form of gravitational potential energy is derived by using the full mathematical formality of general relativity. In this paper, we derive the same general relativistic expression for gravitational potential energy simply by using the principle of equivalence and gravitational time dilation.