Horizon pressure from junction conditions for Schwarzschild and Rindler geometries

TL;DR

This paper investigates the stress tensor and junction conditions at the horizons of Schwarzschild and Rindler geometries, revealing a non-zero surface pressure and vanishing surface energy density, with implications for horizon physics.

Contribution

It demonstrates that a surface pressure exists at horizons under Israel matching conditions, providing new insights into horizon stress-energy characteristics.

Findings

Surface energy density vanishes at horizons.

Surface pressure equals 1/16πl, where l is the proper distance.

Results are consistent for different types of surfaces.

Abstract

We assumed a stress tensor is necessary on the event horizon of a Schwarzschild black hole or on the Rindler horizon for the Israel matching conditions to be satisfied. We found the surface energy density $\rho_{s}$ is vanishing but the surface pressure $p_{s}$ equals $1/16\pi l$ in both cases, where $l$ is the proper distance from the horizon. The junction relations are applied both for $r =$ const. and $T =$ const. surfaces, with the same results for the surface parameters $\rho_{s}$ and $p_{s}$. We emphasize the nonstatic feature of the spacetimes beyond the corresponding horizons.