Angular Momentum-Free of the Entropy Relations for Rotating Kaluza-Klein Black Holes

TL;DR

This paper proves that for certain rotating Kaluza-Klein black holes, the entropy sum and product are independent of angular momentum in higher dimensions, especially when charge is zero, regardless of cosmological constant effects.

Contribution

It demonstrates the angular momentum independence of entropy relations for rotating Kaluza-Klein black holes in higher dimensions, extending previous results to include cosmological constant effects.

Findings

Entropy sum is independent of angular momentum in $d extgreater=4$.

Entropy product is independent when at least one rotation parameter is zero in $d extgreater=5$.

Charge does not affect angular momentum independence in asymptotically flat spacetime.

Abstract

Based on a mathematical lemma related to the Vandermonde determinant and two theorems derived from the first law of black hole thermodynamics, we investigate the angular momentum independence of the entropy sum as well as the entropy product of general rotating Kaluza-Klein black holes in higher dimensions. We show that for both non-charged rotating Kaluza-Klein black holes and non-charged rotating Kaluza-Klein-AdS black holes, the angular momentum of the black holes will not be present in entropy sum relation in dimensions $d\geq4$, while the independence of angular momentum of the entropy product holds provided that the black holes possess at least one zero rotation parameter $a_j$ = 0 in higher dimensions $d\geq5$, which means that the cosmological constant does not affect the angular momentum-free property of entropy sum and entropy product under the circumstances that charge $\delta=0$. For the reason that the entropy relations of charged rotating Kaluza-Klein black holes as well as the non-charged rotating Kaluza-Klein black holes in asymptotically flat spacetime act the same way, it is found that the charge has no effect in the angular momentum-independence of entropy sum and product in asymptotically flat spactime.