# Some Result on Cosmological and Astrophysical Horizons and Trapped Surfaces

## TL;DR

This paper investigates the geometric and thermodynamic properties of black hole horizons in various spacetimes, classifies their causal nature, and proves a version of the third law of black hole thermodynamics.

## Contribution

It provides a comprehensive classification of horizons in different cosmological models and establishes stability conditions and thermodynamic laws for these horizons.

## Key findings

- Stability of marginally trapped surfaces requires negative pressure.
- Spacelike MTTs are foliated by stable MTS, preventing shell crossing.
- Explicit proof of the third law of black hole thermodynamics for specific spacetime classes.

## Abstract

We study the evolution of horizons of black holes in the $1+1+2$ covariant setting and investigate various properties intrinsic to the geometry of the foliation surfaces of these horizons. This is done by interpreting formulations of various quantities in terms of the geometric and thermodynamic quantities. We establish a causal classification for horizons in different classes of spacetimes. We have also recovered results by Ben-Dov and Senovilla which put cut-offs on the equation of state parameter $\sigma$, determining the spacelike, timelike and non-expanding horizons in the the Robertson-Walker class of spacetimes. We show that stability of marginally trapped surfaces (MTS) in the Robertson-Walker spacetimes is only achievable under the conditions of negative pressure, and also classify the spacelike future outer trapping horizons (SFOTH) in the Robertson-Walker spacetimes via bounds on the equation of state parameter $\sigma$. For the Lemaitre-Tolman-Bondi (LTB) model, it is shown that a relationship between the energy density and the electric part of the Weyl curvature, $\mathcal{E}$, gives the causal classification of the MTTs. It is further shown that only spacelike MTTs are foliated by stable MTS, and that this stability guarantees no shell crossing. We also provide an explicit proof of the third law of black hole thermodynamics for the LRS II class of spacetimes, and by extension, any spacetime whose outgoing and ingoing null geodesics are normal to the MTS.