The black ring entropy from the Weyl tensor

TL;DR

This paper investigates the relationship between black ring entropy and the Weyl tensor in 5D spacetime, showing that the Weyl scalar invariant's volume integral is proportional to black ring entropy across various configurations.

Contribution

It introduces a novel approach linking Weyl tensor invariants to black ring entropy, extending the analysis to multiple black object configurations and including cosmological constant effects.

Findings

The volume integral of the Weyl scalar invariant is proportional to black ring entropy.

The proportionality holds for black rings with different angular momenta.

The relation remains valid at least at the leading order for extended cases.

Abstract

A black ring is an asymptotically flat vacuum solution of the Einstein equations with an event horizon of topology $S^1\times S^2$. A connection between the black ring entropy and its Weyl tensor $C_{\mu\nu\lambda\rho}$ is explored by interpreting the Weyl scalar invariant $C_{\mu\nu\lambda\rho} C^{\mu\nu\lambda\rho}$ as the entropy density in 5-dimensional space-time. We calculate the proper volume integral of $C_{\mu\nu\lambda\rho} C^{\mu\nu\lambda\rho}$ for a neutral black ring and prove that it is proportional to the entropy of a thin black ring. Similar calculations are extended to more general cases: the black string, the black ring with two angular momenta, and the black ring with a cosmological constant. The proportionality still maintains or is valid at least at the leading order.