Towards the Raychaudhuri equation beyond general relativity

Abstract

In General Relativity, gravity is universally attractive, a feature embodied by the Raychaudhuri equation which requires that the expansion of a congruence of geodesics is always non-increasing, as long as matter obeys the strong or weak energy conditions. This behavior of geodesics is an important ingredient in general proofs of singularity theorems, which show that many spacetimes are singular in the sense of being geodesically incomplete and suggest that General Relativity is itself incomplete. It is possible that alternative theories of gravity, which reduce to General Relativity in some limit, can resolve these singularities, so it is of interest to consider how the behavior of geodesics is modified in these frameworks. We compute the leading corrections to the Raychaudhuri equation for the expansion due to models in string theory, braneworld gravity, f (R) theories, and Loop Quantum Cosmology, for cosmological and black hole backgrounds, and show that while in most cases geodesic convergence is reinforced, in a few cases terms representing repulsion arise, weakening geodesic convergence and thereby the conclusions of the singularity theorems.

Constraints and horizons for de Sitter with extra dimensions

“…In string theory models, for which corrections of the form (65) can be calculated, we have λ (n) ∼ α n−1 ∼ 2(n−1) s in terms of the string length s . Computing the correction terms in (65) can be challenging for even the simplest backgrounds; for example, [64] computed the corrections arising from Gauss-Bonnet corrections in simple black hole and cosmological backgrounds. We will take the corrections to scale as some power of the curvature H (n) M N N M N N ∼ −[R] n , and we will assume that these corrections maximally violate the NEC in order to arrive at the strongest possible case for the role of these corrections, so that (23) becomes:…”

Raychaudhuri equation with zero point length

“…Similar results exist in the context of space-like singularity formation during collapse of a massive star to a Schwarzschild black hole [36]. Similar consideration of string theory, brane world models and theories beyond general relativity provides mixed results [35]. This is because the nature of additional terms in the Raychaudhuri equation in these contexts depend on the equation of state of the perfect fluid describing matter.…”

“…There are also results from the context of loop quantum cosmology exhibiting avoidance of singularity formation in the cosmological context [31,32]. This is due to the repulsive terms of quantum origin in the Raychaudhuri equation, which takes over when approaching the would-be cosmological singularity [33][34][35]. Similar results exist in the context of space-like singularity formation during collapse of a massive star to a Schwarzschild black hole [36].…”

Geometry of small causal diamonds

“…where the dot represents the derivative with respect to the affine parameter l of the null generators. With vanishing twist on the lightcone, the Raychaudhuri equation and the evolution equation for shear in arbitrary dimension d read [37]:…”