# Angular Momentum Fluctuations

## TL;DR

This paper investigates angular momentum fluctuations of near-static Schwarzschild black holes in thermal equilibrium, revealing that fluctuations originate from both the black hole and surrounding radiation, with implications for quantum area quantization.

## Contribution

It introduces a semi-classical analysis of black hole angular momentum fluctuations considering small but non-zero angular momentum, linking fluctuations to horizon area and radiation effects.

## Key findings

- Radiation contribution depends on the cutoff distance from the horizon.
- Black hole fluctuations are proportional to the horizon area.
- Results suggest a new quantization rule for the event horizon area.

## Abstract

In this paper, we consider angular momentum fluctuations of a Schwartzschild black hole in thermal equilibrium with radiation which, for the sake of simplicity is here modeled by a scalar field. Important, we do not set the black hole angular momentum $J$ identically to zero at the outset; we allow it to have a small value (in the sense that $J/M<<1$) and then study the conditions for thermodynamical equilibrium; only then take the $J\rightarrow 0$ limit. We calculate the black hole's angular momentum fluctuations which turn out to have two independent contributions: one that comes from the black hole itself, with no respect to the radiation, and another one that arises from the radiation. The result is astonishingly simple: the radiation contribution depends exclusively on the cut-off proper distance from the horizon (or equivalently, the width of the brick wall), while the black hole contribution is proportional to its event horizon area. Accordingly, there are no strictly static black holes in nature, they randomly rotate in all possible directions. Since a black hole is nothing but geometry, we are dealing with geometry fluctuations -- our results are of quantum-gravitational nature (albeit at a semi-classical level). Interestingly enough, if we apply to the black hole fluctuations component the (quantum) rules of angular momentum we obtain an event horizon area quantization rule, albeit with a different spectrum from an equally spaced area spectrum which is widely accepted in the literature.