Light propagation and optical scalars in torsion theories of gravity

Abstract

We investigate the propagation of light rays and evolution of optical scalars in gauge theories of gravity where torsion is present. Recently the modified Raychaudhuri equation in the presence of torsion has been derived. We use this result to derive the basic equations of geometric optics for several different interesting solutions of the Poincaré gauge theory of gravity. The results show that the focusing effects for neighboring light rays will be different than general relativity. This in turn has practical consequences in the study of gravitational lensing effects and also determining the angular diameter distance for cosmological objects.

Raychaudhuri and optical equations for null geodesic congruences with torsion

“…With the following convention for the area 2-form on S and the NP tetrad determinant, (2) ǫ := im ∧m, (2) ǫ µν := 2im [µmν] , iǫ µνρσ l µ n ν m ρmσ = 1, (A. 12) where ǫ µνρσ are the components of the spacetime volume form (with conventions ǫ 0123 = √ −g), we have ǫ µνρσ l µ n ν m ρ = −im σ , ǫ µνρσ l µ m νmρ = il σ , ǫ µνρσ n µ m νmρ = −in σ , (A.13)…”

“…We introduce as in the metric case a transverse vector n, and the projector (2), and define the projected tensors B ⊥ , B ′⊥ and their symmetric-traceless, antisymmetric and trace components as in (11). We define the spin coefficients ρ and σ as before in (12), this time using B ⊥ which carries the torsion-full connection, as with the rest of the spin coefficients and curvature scalars. They can be related through (57) to equivalent quantities ρ ′ and σ ′ for B ′⊥ (which shouldn't be thought of as spin coefficients), as well as to their Levi-Civita correspondents for g B ⊥ (namely the spin coefficients determined by the Levi-Civita connection),…”

“…In spite of these limitations, the geodesic deviation and associated Raychaudhuri equations with torsion have been studied in the literature, often motivated by applications to modified theories of gravity, see e.g. [1,8] for early work and more recently [9,10,11,12]. In this brief note we restrict attention to null geodesic congruences (NGCs), rederive results of [9] and extend the analysis to include the null Raychaudhuri and the rest of Sachs' optical equations with torsion.…”