Angular Momentum Independence of the Entropy Sum and Entropy Product for AdS Rotating Black Holes In All Dimensions

TL;DR

This paper demonstrates that the entropy sum of AdS rotating black holes is always independent of angular momentum across all dimensions, while the entropy product's independence depends on specific dimensional and rotational conditions.

Contribution

It introduces a method to analyze entropy relations without requiring explicit metric forms, revealing dimension-dependent angular momentum independence properties.

Findings

Entropy sum is always independent of angular momentum in all dimensions.

Entropy product's independence holds only for dimensions greater than 4 with specific rotation parameters.

Negative cosmological constant does not affect entropy sum and product independence.

Abstract

In this paper, we investigate the angular momentum independence of the entropy sum and product for AdS rotating black holes based on the first law of thermodynamics and a mathematical lemma related to Vandermonde determinant. The advantage of this method is that the explicit forms of the spacetime metric, black hole mass and charge are not needed but the Hawking temperature and entropy formula on the horizons are necessary for static black holes, while our calculations require the expressions of metric and angular velocity formula. We find that the entropy sum is always independent of angular momentum for all dimensions and the angular momentum-independence of entropy product only holds for the dimensions $d>4$ with at least one rotation parameter $a_i=0$, while the mass-free of entropy sum and entropy product for rotating black holes only stand for higher dimensions ($d>4$) and for all dimensions, respectively. On the other hand, we find that the introduction of a negative cosmological constant does not affect the angular momentum-free of entropy sum and product but the criterion for angular momentum-independence of entropy product will be affected.