Caged black hole thermodynamics: Charge, the extremal limit, and finite size effects

TL;DR

This work extends the effective field theory treatment of small Kaluza-Klein black holes to electrically charged cases, deriving thermodynamics to order $λ^2$ via a worldline EFT and constructing an exact extremal solution for validation. It shows that horizon deformations are absent at this order and that key thermodynamic quantities follow from a perturbative analysis matched to the extremal solution. Finite size effects are analyzed through higher-derivative worldline operators, with clear predictions for how the mass, redshift, and tension shift, and extremal matching constrains some Wilson coefficients. The results illuminate how charge influences the KK phase structure and suggest that adding charge delays the transition to a black string, offering a framework for further nonperturbative and numerical exploration.

Abstract

We extend the effective field theory treatment of the thermodynamics of small compactified black holes to the case of charged black holes. The relevant thermodynamic quantities are computed to second order in the parameter λ\sim(r_0/L)^(d-3). We discuss how the addition of charge to a caged black hole may delay the phase transition to a black string. In the extremal limit, we construct an exact black hole solution which serves as a check for our perturbative results. Finite size effects are also included through higher order operators in the worldline action. We calculate how the thermodynamic quantities are modified in the presence of these operators, and show they enter beyond order λ^2 as in the uncharged case. Finally, we use the exact solution to constrain the Wilson coefficients of the finite size operators in the extremal limit.

Figures (7)

• Figure 1: Feynman rules needed for the charged caged BH thermodynamics computations to $\mathcal{O}(\lambda^2)$. The single dashed lines correspond to the KK scalar field $\phi$, which is the propagating scalar mode in the gravitational sector and the double dashed lines denote the electromagnetic scalar $\varphi$. The Feynman rule (a) is given in momentum space, where we suppress the delta function, and (b)-(d) are in coordinate space.
• Figure 2: Diagrams that contribute to the ADM mass to $\mathcal{O}(\lambda^2)$.
• Figure 3: Diagrams that contribute to the asymptotic value of the electric charge.
• Figure 4: Diagrams required to compute the redshift factor $R$ to $\mathcal{O}(\lambda^2)$. Here, $\phi(0)$ denotes that the $\phi$ field propagates back to $\textbf{x}=0$ on the BH worldline but does not have a vertex.
• Figure 5: Schematic of exact solution construction.