Black Hole Thermodynamics without Black Hole Solutions

TL;DR

This work shows that in a string-inspired Einstein-Maxwell-Maxwell-dilaton theory with two gauge fields, the full set of black hole thermodynamic quantities can be derived without solving the field equations. By leveraging a long-range force relation, a weak no-hair conjecture, horizon–asymptotic connections through a non-extremality parameter $\mu$, and thermodynamic homogeneity, the authors express all quantities in terms of three basic variables $Q_1$, $Q_2$, and $\mu$, via $M = \mu f(x,y)$, $\Sigma = \mu g(x,y)$, $S = \mu^2 s(x,y)$, $T = t(x,y)/\mu$, and $\Phi_i = \Phi_i(x,y)$ with $x = Q_1/\mu$, $y = Q_2/\mu$. They derive master equations that determine these functions, validate against known single-charge solutions and special coupling cases, and demonstrate an application by testing Penrose-type thermodynamic bounds; their approach offers a practical route to black hole thermodynamics in regimes where exact solutions are unavailable. The method has potential extensions to other string-inspired theories and p-brane configurations, with caveats related to cosmological constants and rotation.

Abstract

We consider the string-theory inspired Einstein-Maxwell-Maxwell-dilaton theory (EMMD) and show that we can derive the complete set of thermodynamic quantities of charged black holes, without having to solve for the black hole solutions. We argue that the technique can be applied more broadly to string theories, providing an accessible method for determining the thermodynamic properties of large classes of black holes for which exact solutions are typically unavailable.

Figures (1)

• Figure 1: Both panels are associated with $(a_1,a_2)=(2,-1/8)$. We also set $x=1/2$ and $\mu=1$. In the left panel, we scaled the quantity $Z$ by 50 for clarity. We see that both $Y$ and $Z$ are nonnegative and $Z<Y$. In particular, when $y=2$, corresponding to the RN black hole, $Z$ vanished, as expected. In the right panel, we see that the line associated with $U$ is indeed sandwiched by its two bounds.