Black ring entropy from the Weyl tensor

Abstract

A black ring is an asymptotically flat vacuum solution of the n-dimensional Einstein equations with an event horizon of topology S¹×Sⁿ⁻³. In this study, a connection between the black ring entropy and the Weyl tensor C_μνλρ is explored by interpreting the Weyl scalar invariant C_μνλρC^μνλρ as the entropy density in five-dimensional space-time. It is shown that the proper volume integral of C_μνλρC^μνλρ for a neutral black ring is proportional to the black ring entropy in the thin-ring limit. Similar calculations are extended to more general cases: a black string, a black ring with two angular momenta, and a black ring with a cosmological constant. The proportionality is also found to be valid for these complex black objects at the leading order.

Keywords

black ring / Weyl tensor / entropy / Penrose conjecture

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