Abstract
Given a spacetime with nonvanishing torsion, we discuss the equation for the evolution of the separation vector between infinitesimally close curves in a congruence. We show that the presence of a torsion field leads, in general, to tangent and orthogonal effects on the congruence; in particular, the presence of a completely generic torsion field contributes to a relative acceleration between test particles. We derive, for the first time in the literature, the Raychaudhuri equation for a congruence of timelike and null curves in a spacetime with the most generic torsion field.
Highlights
The appearance of singularities in a physical theory ineluctably marks the pillars of Hercules of that model
General Relativity, from this point of view, is not an exception: it did not take a long time for the community of relativists to realize that even the two simplest space-time solutions of Einstein field equations, the Schwarzschild metric and the Friedmann–Lemaître–Robertson–Walker (FLRW) metric, harbour a gravitational singularity, that is, a breakdown of the space-time structure itself! While in the early 50’s Amal Kumar Raychaudhuri was forcedly working on the properties of electronic energy bands in metals, he got interested in the debate around the nature of gravitational singularities and the generic features of Einstein’s theory of General Relativity (GR)
In this paper we derived the equation for the evolution of the separation vector between infinitesimally close curves of a congruence in space-times with non-null generic torsion field, clarifying some of the ambiguities lingering in the literature about the role of the torsion tensor
Summary
The appearance of singularities in a physical theory ineluctably marks the pillars of Hercules of that model. The ECSK theory is characterized by assuming an independent connection (using the so-called Palatini approach to find two sets of independent field equations) and further requiring the anti-symmetric part of the connection to be in general non-vanishing, defining a tensor field dubbed torsion tensor field; note, that the compatibility of the connection with the metric is still imposed, that is, the covariant derivative (defined with the independent connection) of the metric tensor field along any space-time curve is null. In this paper we start filling such a crucial gap in the literature: we will focus on the study of the effects of the most generic torsion field on the kinematics of test particles and derive the Raychaudhuri equation for a congruence of null and time-like curves in the spacetime.
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