Singularities in Spacetimes with Torsion

Abstract

We study the singularity theorems of Hawking and Penrose in spacetimes with non-vanishing torsion (Einstein-Cartan theory). Assuming that test particles move on geodesics we manage to generalize those theorems to the case of totally anti-symmetric torsion. In doing so we derive the generalized Raychaudhuri equation for timelike geodesics and arbitrary torsion. For totally anti-symmetric torsion we find that the vorticity of a geodesic is directly related to the torsion. We find that the theorems are not directly extendable to spacetimes that have non totally anti-symmetric torsion; other arguments are needed. However, we formulate a construction that enables one to create a null geodesically incomplete spacetime with vectorial torsion starting from a spacetime that is null geodesically incomplete with respect to the Levi-Civita connection.

The geometric assumptions of the singularity theorems can be translated to assumptions on the matter content of spacetime via the equations of motion of the theory. We studied the two ways of deriving these equations. It turns out that for totally anti-symmetric torsion the metric formalism, in which one takes the metric and torsion as dynamical variables and assumes metric compatibility, is equivalent to the metric-affine formalism, in which one takes the metric and connection as dynamical variables and does not assume metric compatibility. Matter in the Standard model induces totally anti-symmetric torsion. We also generalize the Bianchi identity and conservation of the energy-momentum tensor in general relativity to Einstein-Cartan theory.

We examine the singularity theorems in FLRW spacetimes (assuming vanishing torsion). We argue that one should not use the general definition of a singularity in these spacetimes but should define a singularity as a comoving geodesic that is incomplete. We prove theorems for FLRW spacetimes about the relation between conjugate points and a singularity. In particular we show that for a class of singular spacetimes all points on certain geodesics are conjugate to the point at the singularity. Also when every point on a geodesic is conjugate to a certain point, one must have a singularity. Lastly we show under which condition a geodesic in an FLRW spacetime with flat spacelike three-surfaces has conjugate points.