Conserved charges and thermodynamics of the spinning Godel black hole

TL;DR

The paper addresses the problem of defining conserved charges for spinning Gödel-type black holes in five-dimensional minimal supergravity, where standard methods fail. It adopts a gauge-theory conserved-charge framework and computes mass, angular momentum, and electric charge by integrating along a solution-space path from the Gödel background to the black hole, obtaining finite expressions: $\mathcal{E}=\frac{3\pi}{4}m-8\pi j^2 m^2-\pi j m l$, $\mathcal{J}^{\phi}=\frac{1}{2}\pi m l-\pi j m l^2-4\pi j^2 m^2 l$, and $\mathcal{Q}=2\sqrt{3}\pi\, j m l$. A generalized Smarr formula is derived and shown to hold together with the first law, though a Gödel-induced anomaly appears when compared to asymptotically flat cases; in the special case $l=0$ a redefinition of the Killing vector can restore a non-anomalous Smarr relation and the first law. Overall, the work provides a consistent thermodynamic framework for Gödel black holes and demonstrates finite-radius charge definitions in theories with Chern–Simons terms.

Abstract

We compute the mass, angular momenta and charge of the Godel-type rotating black hole solution to 5 dimensional minimal supergravity. A generalized Smarr formula is derived and the first law of thermodynamics is verified. The computation rests on a new approach to conserved charges in gauge theories that allows for their computation at finite radius.