Asymptotics of d-dimensional Kaluza-Klein black holes: beyond the Newtonian approximation

TL;DR

This work formulates a worldline effective field theory for small Kaluza-Klein black holes in ${\mathbf R}^{d-1}\times {\mathbf S}^1$, organized by the small parameter $\lambda=(r_s/L)^{d-3}$. Through systematic matching to the full Schwarzschild solution and a careful power-counting analysis, it separates horizon-structure (finite-size) effects from nonlinear gravitational corrections, showing the first horizon-deformation contributions arise at $\lambda^{(d+1)/(d-3)}$. The authors compute ${\cal O}(\lambda)$ and ${\cal O}(\lambda^2)$ corrections to the asymptotic mass $m$ and tension $\tau$ for general $d$, and derive corresponding corrections to the entropy $S$ and temperature $T$ via Smarr relations, providing explicit formulas and consistency with known $d=5,6$ results. Their comparison with Kudoh & Wiseman’s numerical data indicates existing simulations probe point-particle thermodynamics rather than horizon dynamics, and they discuss how horizon effects become more detectable at large $d$. Overall, the EFT framework offers a renormalizable, systematic approach to KK black hole thermodynamics in the $\lambda\ll1$ regime and clarifies when horizon structure can influence macroscopic observables.

Abstract

We study the thermodynamics of small black holes in compactified spacetimes of the form R^(d-1)x S^1. This system is analyzed with the aid of an effective field theory (EFT) formalism in which the structure of the black hole is encoded in the coefficients of operators in an effective worldline Lagrangian. In this effective theory, there is a small parameter $λ$ that characterizes the corrections to the thermodynamics due to both the non-linear nature of the gravitational action as well as effects arising from the finite size of the black hole. Using the power counting of the EFT we show that the series expansion for the thermodynamic variables contains terms that are analytic in $λ$, as well as certain fractional powers that can be attributed to finite size operators. In particular our operator analysis shows that existing analytical results do not probe effects coming from horizon deformation. As an example, we work out the order $λ^2$ corrections to the thermodynamics of small black holes for arbitrary d, generalizing the results in the literature.

Figures (5)

• Figure 1: Leading order contribution of the operators ${\cal O}_{E,B}$ to the effective action tadpoles. The thick square vertex denotes an insertion of ${\cal O}_{E,B}$, and the $\otimes$ corresponds to an insertion of the background graviton field.
• Figure 2: Diagrams contributing to the background field tadpoles at order $\ell^{-1}\lambda^2$. The $\otimes$ denotes an insertion of the background graviton field.
• Figure 3: Diagrams contributing to the background field tadpoles at order $\ell^{-1}\lambda^3$. The $\otimes$ denotes an insertion of the background graviton field.
• Figure 4: Comparison of analytic results with the data of refs. kudoh1kudoh2. (a) compares the $d=5$ results and (b) the $d=6$ results. The dashed lines include only ${\cal O}(\lambda)$ corrections. The solid lines include ${\cal O}(\lambda^2)$ terms.
• Figure 5: Ratio of ${\cal O}(\lambda^2)$ to ${\cal O}(\lambda)$ terms in the perturbative expansion of $TS^{d-2}$ versus $S$ for $d=5$ (dashed line) and $d=6$ (solid line).