New Near Extremal Black Holes and Love Symmetry

TL;DR

The note constructs a new class of near-extremal black holes in Kleinian signature with an exact integrable structure $SL(2,R) x SL(2,R)$. In the near-horizon throat, described by an Eguchi-Hanson instanton, perturbations are governed by the Love symmetry, enabling an exact treatment of quasinormal modes via matched asymptotics to a far region that resembles self-dual Taub-NUT. Photon-ring physics exhibits a critical scaling $r_c o ext{const} imes T^{2/3}$ and is tied to the emergent $SL(2,R) x SL(2,R)$ symmetry, with QNMs given by $\omega_{n,h} = rac{n}{2M} + rac{1}{2M} T^{2h-1} imes ext{eta}_{n,h}$ and explicit gamma-function coefficients. A Lorentzian tunneling interpretation reproduces the imaginary part of the frequencies, supporting a holographic-like dual description and highlighting the role of integrability in near-extremal gravity.

Abstract

Rotating and charged black holes are known to exhibit remarkable properties close to extremality, including emergent hidden symmetries and holographic duality to 2D theories. In this note, we introduce a new class of near-extremal black holes living in $(2,2)$ signature, strongly resembling the Lorentzian ones but with an exact integrable structure $SL(2,\mathbb{R})\times SL(2,\mathbb{R})$. The exterior of the black hole is a self-dual solution with a photon ring. It develops an infinite near-horizon throat described by the Eguchi-Hanson instanton. An exact version of the so-called Love symmetry controls perturbations on the throat. Furthermore, the normalizable spectrum of this black hole provides quasinormal modes in Lorentzian signature.

Figures (3)

• Figure 1: Topology of the near extremal BH in $r,t$ (each point contains a $(\theta,\phi)$ hyperboloid). The Kleinian solution is regular at the horizon $x=0$. A throat develops at $\mathcal{T}\ll1$ and shrinks to a point for $\mathcal{T}=0$ (this is the hydrogen atom studied in Guevara:2023wlr). In the Lorentzian case the throat has the topology of $AdS_2$ as expected.
• Figure 2: Plot of $\frac{-V(x)}{\Delta(x)}$ for low (dark) to high (light) temperatures. We have continued $\omega \to i\omega$ to reach the Lorentzian slice, resembling the Scharwzschild potential. The unstable maximum corresponds to the location of the photon ring, which approaches the horizon as $\mathcal{T} \rightarrow 0$ (and becomes unbounded at the same time).
• Figure 3: The matching procedure between solutions of the near-horizon and far regions, for near-SD black holes with $\mathcal{T}\ll 1$. Note that in this regime the photon ring is located deep in the near-horizon region. However, as $\mathcal{T}\rightarrow 1$ two things happen: the overlap region shrinks to zero size and the photon ring moves towards the outer boundary of the near-horizon region.