Effective Stringy Description of Schwarzschild Black Holes

TL;DR

The paper connects black-hole wave dynamics to two Riemann-surface constructions: the Z-surface from the tortoise-coordinate map and the W-surface from solving the Fuchsian differential equation via monodromies. For BTZ and Schwarzschild BHs, these surfaces encode the scattering problem and reveal an effective string description in terms of Liouville theory on the W-surface, with quasi-normal modes appearing as poles of Liouville 3-point functions. In solvable limits, such as low-energy near-horizon and high-energy near-singularity regimes, the authors derive explicit QNM spectra, including the famous $\log 3$ real part in the high-damping limit. This framework suggests a universal, stringy effective description of BH dynamics, where the worldsheet Liouville theory and surface monodromies control propagation and emission properties, and potentially constrain the fundamental theory of quantum gravity via CFT-like structures on the worldsheet. The approach is broad, extendable to higher dimensions and rotating BHs, though it remains an effective description with several rules for extracting physical amplitudes to be developed further.

Abstract

We start by pointing out that certain Riemann surfaces appear rather naturally in the context of wave equations in the black hole background. For a given black hole there are two closely related surfaces. One is the Riemann surface of complexified ``tortoise'' coordinate. The other Riemann surface appears when the radial wave equation is interpreted as the Fuchsian differential equation. We study these surfaces in detail for the BTZ and Schwarzschild black holes in four and higher dimensions. Topologically, in all cases both surfaces are a sphere with a set of marked points; for BTZ and 4D Schwarzschild black holes there is 3 marked points. In certain limits the surfaces can be characterized very explicitly. We then show how properties of the wave equation (quasi-normal modes) in such limits are encoded in the geometry of the corresponding surfaces. In particular, for the Schwarzschild black hole in the high damping limit we describe the Riemann surface in question and use this to derive the quasi-normal mode frequencies with the log(3) as the real part. We then argue that the surfaces one finds this way signal an appearance of an effective string. We propose that a description of this effective string propagating in the black hole background can be given in terms of the Liouville theory living on the corresponding Riemann surface. We give such a stringy description for the Schwarzschild black hole in the limit of high damping and show that the quasi-normal modes emerge naturally as the poles in 3-point correlation function in the effective conformal theory.

Figures (9)

• Figure 1: The tortoise map for the BTZ BH. (a) The complex $r$-plane is shown together with the branch cut (bold line); (b) The image of the $r$-plane on the $z$-plane is the strip of width $\pi$. Both horizons are mapped to infinity, zero is mapped to zero, and infinity is mapped to the point $z=-i\pi/2$. Closed contours around the singularity and infinity are shown.
• Figure 2: The surface obtained after all the identifications are performed.
• Figure 3: The W-surface for the BTZ BH
• Figure 4: The tortoise map for the 4d Schwarzschild BH. (a) The complex $r$-plane with a cut placed from the horizon to infinity. The region inside the green curve is mapped into the physical sheet (the strip). The green curve itself becomes the cut on the $z$-plane (b) The $Z$-surface has two sheets. The physical sheet is the strip of width $2\pi$; the unphysical sheet is a copy of the complex plane. The sheets are glued as indicated.
• Figure 5: The topological type of the Schwarzschild BH $Z$-surface is that of a cylinder with one marked point. The marked point corresponds to the singularity.