On a strictly local solution to the problem of energy in general relativity

Werner M. Vieira^1*Nadja S. Magalhaes²

• ¹PPG-FIS, Federal University of Sao Paulo, Diadema, SP, Brazil
• ²Physics Dept., Federal University of Sao Paulo, Diadema, SP, Brazil

Abstract. The problem of energy and its conservation is a more than centennial old problem in general relativity (GR), and is considered by many scholars at least not sufficiently solved. The problem's core lays in the abandonnment, in GR, of the non-dynamical Minkowski's spacetime (within the set of inertial coordinates) in favor of dynamical Riemannian space-times (within the set of general coordinates), a choice that, nevertheless, has helped unravel many new phenomena, especially in astrophysics and cosmology. We turn to the subject by briefly revisiting its meaning and describing important approaches to it since the launching of GR: the pseudo-tensor approach and the non-local as well as global approaches. We detail the mathematical difficulties with them to highlight why they are unable to answer the problem at a fundamental level in Riemannian spacetimes, thus hindering the progress on the knowledge not only of cosmology but also quantum gravity. From an analysis of these issues in both mathematical and physical grounds we propose a strictly local approach based on what we named the principle of local balance to provide a consistent response to the energy-momentum problem when the geometry becomes Riemannian. Finally, we show that GR attends only partially to this principle, thus arguing in favor of applying the principle to construct a broader gravitational theory that falls back into GR in some appropriate limit.