Raychaudhuri and optical equations for null geodesic congruences with torsion

Abstract

We study null geodesic congruences (NGCs) in the presence of spacetime torsion, recovering and extending results in the literature. Only the highest spin irreducible component of torsion gives a proper acceleration with respect to metric NGCs, but at the same time obstructs abreastness of the geodesics. This means that it is necessary to follow the evolution of the drift term in the optical equations, and not just shear, twist and expansion. We show how the optical equations depend on the non-Riemannian components of the curvature, and how they reduce to the metric ones when the highest spin component of torsion vanishes.

Spacetime thermodynamics with contorsion

“…This quantity is however not conserved, as we can see from (27b), whose right-hand side does not vanish on-shell. Nevertheless, although τ is not conserved, it is easy to identity an effective energy-momentum tensor which is conserved, thanks to the contorsion decomposition (16). If we take the Levi-Civita 2 Note that the Lie derivatives (24) are not gauge-covariant objects.…”

“…For the reader interested in more details on geodesics with torsion, see e.g. [15,16]. 8 Since in order to recover the Einstein equations we will need to consider arbitrary boost Killing vectors, see discussion below (46), the restriction on torsion (54) should hold for any ξ µ .…”

Static compact objects in Einstein-Cartan theory

“…Now, an LRS space-time is said to be of class I (LRSI) if the congruence of the curves associated with vector field e -defined to have the same direction as the axis of symmetry -is hypersurface orthogonal. 3 If the congruence of curves associated with the vector field u is 2 It should be remarked here that the presence of a generic torsion tensor field affects the definition of the kinematical quantities [35][36][37][38]. See the Appendix A 4 for further details.…”

“…As discussed in Refs. [35][36][37][38], the presence of a generic torsion field will affect the definition of the kinematical quantities that characterize a congruence of curves, such that, θ, σ αβ and ω αβ , Eqs. (A8) -(A10), in general, do not represent the actual geometric -physical -expansion, shear and vorticity of the time-like congruence to which u is tangent.…”

Conserved quantities in the presence of torsion: A generalization of Killing theorem

“…Nevertheless, it may be the case that in other torsionful modified theories of gravity some particles do follow autoparallels, or that one is interested in finding the autoparallels to study some geometrical or topological aspects of spacetime. In fact, recently, people are placing attention to the role of torsion in the Raychaudhuri equation [28,29], which calls for the use of autoparallel congruences, and on the use of Killing horizons to study torsion effects on thermodynamical quantities [30]. Another interesting case where the results of this paper can be applied are the Teleparallel theories of gravity [31] where all the gravitational degrees of freedom are encoded in a torsionful connection.…”