Rotating Attractors

TL;DR

The paper extends the entropy-function formalism to rotating extremal black holes in general higher-derivative gravity, showing that the entropy and near-horizon background arise from extremizing an entropy function dependent only on horizon data and charges. A unique extremum yields full attractor behaviour with moduli independence, while flat directions permit background data to depend on asymptotic moduli, yet the entropy remains invariant. The authors illustrate the framework with two-derivative theories, obtaining Kerr and Kerr-Newman results, and then test attractor behaviour in explicit full solutions: rotating Kaluza-Klein black holes and black holes in toroidally compactified heterotic string theory, identifying ergo-free and ergo-branch extremal limits and demonstrating duality-invariant entropy expressions. The work highlights how attractor behaviour generalizes to rotating cases, clarifies the role of ergoregions, and reinforces the universality of the entropy in a broad class of theories and duality frames, with implications for the microscopic counting of states in string theory. It also points to higher-derivative corrections lifting flat directions and restoring full horizon-data determination in broader contexts.

Abstract

We prove that, in a general higher derivative theory of gravity coupled to abelian gauge fields and neutral scalar fields, the entropy and the near horizon background of a rotating extremal black hole is obtained by extremizing an entropy function which depends only on the parameters labeling the near horizon background and the electric and magnetic charges and angular momentum carried by the black hole. If the entropy function has a unique extremum then this extremum must be independent of the asymptotic values of the moduli scalar fields and the solution exhibits attractor behaviour. If the entropy function has flat directions then the near horizon background is not uniquely determined by the extremization equations and could depend on the asymptotic data on the moduli fields, but the value of the entropy is still independent of this asymptotic data. We illustrate these results in the context of two derivative theories of gravity in several examples. These include Kerr black hole, Kerr-Newman black hole, black holes in Kaluza-Klein theory, and black holes in toroidally compactified heterotic string theory.

Figures (4)

• Figure 1: Radial evolution of the scalar field starting with different asymptotic values at three different values of $\theta$. We take $P=Q=4\sqrt{\pi}$, $J={16\pi/ 3}$ for $\Phi_\infty=0$, and then change $\Phi_\infty$ and $P$, $Q$ using the transformation (\ref{['ekk2']}).
• Figure 2: Scalar field profile at the horizon of the Kaluza-Klein black hole. We take $P=Q=4\sqrt{\pi}$, $J={16\pi/ 3}$ for $\Phi_\infty=0$, and then change $\Phi_\infty$ and $P$, $Q$ using the transformation (\ref{['ekk2']}). The figure shows that the scalar field profile at the horizon is independent of $\Phi_\infty$.
• Figure 3: Radial evolution of the scalar field for an ergo-branch black hole starting with different asymptotic values at five different values of $\theta$. We take $P=Q= 2\sqrt\pi$ and $J=4\pi\sqrt{2}$ for $\Phi_\infty=0$, and then change $\Phi_\infty$ and $P$, $Q$ using the transformation (\ref{['ekk2']}).
• Figure 4: Scalar field profile at the horizon for a black hole on the ergo-branch for different asymptotic values of $\Phi$. We take $P=Q= 2\sqrt\pi$ and $J=4\pi\sqrt{2}$ for $\Phi_\infty=0$, and then change $\Phi_\infty$ and $P$, $Q$ using the transformation (\ref{['ekk2']}). Clearly the scalar field profile at the horizon depends on its asymptotic value.