Um estudo sobre a energia gravitacional na abordagem do teleparalelismo equivalente à relatividade geral

Abstract

The objective of this thesis is to evaluate the consistency of the gravitational energy obtained when the Teleparallel Equivalent of General Relativity is applied, as well as which teleparallel frame is more suitable for this purpose. In order to do that, we have used this approach to calculate the energy of the gravitational field of the Schwarzschild spacetime and to obtain the gravitational energy-momentum tensor of the Friedmann- Lemaître-Robertson-Walker spacetimes. In the first case, we have used tetrads adapted to Kruskal and Novikov coordinates. We have obtained that the energy of the black/white hole is the famous 𝑀𝑐2 when the wormhole throat is closed. In this result, we have verified that the singularity does not contribute to the energy, which indicates that the energy obtained is purely gravitational. For the second case, we have found the gravitational energy-momentum density tensor of the universe from the tetrad adapted to the Friedmann-Lemaître-Robertson-Walker coordinates.We have decomposed this tensor into its symmetrical and antisymmetrical parts, and we have analysed each one of them. The symmetrical part obeys a radiation-like equation of state, then we have pointed out that, actually, this is a general characteristic of the gravitational energy-momentum density tensor, since it is traceless. The antisymmetric part does not contribute to the energy of the universe. We have observed that the field equations have a well-defined vacuum and that, for spatially flat universes, the total energy density of the universe is zero. In the penultimate chapter, we have also verified the 4-momentum of the Teleparallel Equivalent of General Relativity can be made to coincide with that of the ADM formalism. In this chapter, we discuss the idea that the best frame to analyze the gravitational energy is a frame adapted to free particles and that is free from artificial properties. Based on the results obtained in this thesis, we have concluded the Teleparallel Equivalent of General Relativity is quite suitable for calculating gravitational energy as long as a suitable teleparallel frame is used.