Semi-tetrad decomposition of spacetime with conformal symmetry.

Abstract

In this thesis, we study the kinematical and dynamical properties of a general spacetime that admits a conformal Killing vector. A 1+1+2 decomposition of the spacetime is performed using the fluid 4-velocity and a preferred spatial direction in the 3-space. The Lie derivatives of the 4-velocity vector and the preferred spatial direction vector are calculated and analyzed. We compare our results with the 1+3 decomposition of Maartens et al (1986), and find new results in the form of a scalar equation and constraint equation owing to the further decomposition. This provides new insights into the behaviour of the acceleration, expansion, shear and vorticity scalars which are not possible in the 1+3 decomposition. The general energy momentum tensor for an anisotropic fluid is considered and decomposed using the semi-tetrad covariant approach. We take the Lie derivative along the conformal Killing vector and apply to Einstein’s field equations. This makes it possible to generate a set of constraint equations in the new geometrical variables. All the geometrical and thermodynamical quantities are written in terms of the 1+1+2 decomposition variables. This is a new result. We also find a system of equations that must be satisfied by the thermodynamical variables when a conformal symmetry exists applied to the perfect fluid case. We show that the conformal factor satisfies a damped wave equation with a potential.