Mass Dependence of the Entropy Product and Sum

TL;DR

This paper establishes simple criteria for when the entropy product and sum of black holes are independent of mass, based on the metric's Laurent series powers, extending to rotating black holes without needing exact metrics.

Contribution

It introduces a general criterion for mass independence of entropy products and sums using thermodynamics and Vandermonde determinants, applicable to various black hole solutions.

Findings

Entropy product is mass independent if and only if m≥d−2 and n≥4−d.

Entropy sum is mass independent if and only if m≥d−2 and n≥2.

Mass independence of entropy product/sum varies with spacetime dimension, especially for Myers-Perry black holes.

Abstract

For black holes with multiple horizons, the area product of all horizons has been proven to be mass independent in many cases. Counterexamples were also found in some occasions. In this paper, we first prove a theorem derived from the first law of black hole thermodynamics and a mathematical lemma related to the Vandermonde determinant. With these arguments, we develop some general criterion for the mass independence of the entropy product as well as the entropy sum. In particular, if a $d$-dimensional spacetime is spherically symmetric and its radial metric function $f(r)$ is a Laurent series in $r$ with the lowest power $-m$ and the highest power $n$, we find the criteria is extremely simple: The entropy product is mass independent if and only if $m\geq d-2$ and $n\geq4-d$. The entropy sum is mass independent if and only if $m\geq d-2$ and $n\geq 2$. Compared to previous works, our method does not require an exact expression of the metric. Our arguments turn out to be useful even for rotating black holes. By applying our theorem and lemma to a Myers-Perry black hole with spacetime dimension $d$, we show that the entropy product/sum is mass independent for all $d>4$, while it is mass dependent only for $d=4$, i.e., the Kerr solution.