Equivalência Local de Espaços-tempos em (2+1) Dimensões(Local Equivalence of Spacetimes in (2+1) Dimensions)

Abstract

In this work the problem of equivalence of gravitational fields in the framework of theories of gravitation where the space-time is a pseudo-Riemannian manifold with dimension (2+1)D, that is, the question of deciding when two exact solutions apparently different represent the same gravitational field in different coordinate systems, is tackled in its theoretical and practical aspects. The solution of the problem is presented by using a coordinate-invariant description of the gravitational field. Explicit expressions for the dimensions of the group of symmetry of a (2+1)D spacetime and its subgroup of isotropy are given. A minimal set of invariant quantities which describe the local gravitational field is determined, by using the formalism of two-component real spinors. An algorithm for testing the equivalence in practice is developed as an adaptation the algorithm of Karlhede for (3+1)D spactimes, and the Segre classifications of the Ricci and Cotton-York tensors are determined. The algorithm is implemented up to first order covariant derivative of the curvature, by using the computer algebra package GRtensorII of the system Maple, where all calculations are done interactively. Using the equivalence problem techniques for (2+1)D spacetimes, the conditions for space-time homogeneity of (2+1)D spacetimes with Gödel-type metrics are derived and compared with previous works on (3+1)D Gödel- type space-times. The equivalence of (2+1)D Gödel-type space-times is studied and it is shown that they admit a four-dimensional group of isometries and are characterized by two essential parameters.