Three-dimensional space-times
The paper gives a detailed study of three-dimensional space-times following the null tetrad formalism of Newman-Penrose (NP) applied to the three-dimensional case. The theory is developed with a real triad made up of two normalized null vectors and a unit space-like vector with space- time metric signature \((+ + -).\) According to the definitions of the Newman-Penrose Ricci rotations coefficients, the component relations of the Ricci and Bianchi identities are calculated and the algebraic classification (Segré type) of the three-dimensional Ricci tensor is studied showing four distinct types just as in the case of four-dimensional space-times. The field equations in NP-spin components form are given for topologically massive fields following the Deser, Jackiw and Templeton space-time action with topological mass. Using the algebraic classification of the Ricci tensor into these equations, two exact solutions were obtained: one is characterized by a null geodesic eigencongruence of the Ricci tensor (one of the null vector is an eigenvector of the Ricci tensor, Segré type \(\{\) \(3\})\) and the other, a Segré type \(\{\) 111\(\}\), by the fact that all the polynomial invariants constructed from the curvature are constant. The metric forms for both solutions are exhibited explicitly.
scientific article; zbMATH DE number 4024206