Effective null Raychaudhuri equation

The effects on Raychaudhuri's equation of an intrinsically-discrete or particle nature of spacetime are investigated. This is done through the consideration of null congruences emerging from, or converging to, a generic point of spacetime, i.e. in geometric circumstances somehow prototypical of singularity issues. We do this from an effective point of view, that is through a (continuous) description of spacetime modified to embody the existence of an intrinsic discreteness on the small scale, this adding to previous results for non-null congruences. Various expressions for the effective rate of change of expansion are derived. They in particular provide finite values for the limiting effective expansion and its rate of variation when approaching the focal point. Further, this results in a non-vanishing of the limiting cross-sectional area itself of the congruence.Comment: 7 pages; v2: some comparisons with other approaches adde

Entropy bounds and field equations

For general metric theories of gravity, we compare the approach that describes-derives the field equations of gravity as a thermodynamic identity with the one which looks at them from entropy bounds. The comparison is made through the consideration of the matter entropy flux across (Rindler) horizons, studied by making use of the notion of a limiting thermodynamic scale $l^*$ of matter, previously introduced in the context of entropy bounds. In doing this: i) a bound for the entropy of any lump of matter with a given energy-momentum tensor $T_{ab}$ is considered, in terms of a quantity which is independent of the theory of gravity we use; this quantity is the variation of the Clausius entropy of a suitable horizon when the element of matter crosses it; ii) by making use of the equations of motion of the theory, the same quantity is then expressed as the variation of Wald's entropy of that horizon (and this leads to a generalized form of the generalized covariant entropy bound, applicable to general diffeomorphism-invariant theories of gravity); iii) a notion of $l^*$ for horizons, and an expression for it, is given.Comment: 8 pages, 1 figure; v4: some typos corrected, one reference and some clarifications added, some editin