BTZ Black Holes and Modular CFT Thermodynamics

The BTZ solution turns modular covariance into a quantitative gravity test. A rotating AdS3 black hole with horizons $r_+\ge r_-$ has entropy $2\pi r_+/(4G_3)$, while a two-dimensional CFT with Brown–Henneaux central charge has left- and right-moving Cardy growth. With the vacuum energy and charge conventions matched, the two answers coincide in the controlled high-energy regime.

Required background. AdS3/CFT2 and the Brown–Henneaux Central Charge fixes $c$, and Torus Partition Functions as Bootstrap Data supplies the modular transformation and vacuum-energy shift.

Helpful background. Modular Crossing and Spectral Bounds sharpens the spectral hypotheses, while Imported Wald Entropy in Holographic Higher-Derivative Thermodynamics explains how the entropy formula changes beyond Einstein gravity.

Rotating BTZ thermodynamics

For the Einstein BTZ family, define mass and angular momentum by

$$M=\frac{r_+^2+r_-^2}{8G_3\ell^2}, \qquad J=\frac{r_+r_-}{4G_3\ell}.$$

The horizon generator gives

$$T_H=\frac{r_+^2-r_-^2}{2\pi\ell^2r_+}, \qquad \Omega_H=\frac{r_-}{\ell r_+}, \qquad S_{\rm BH}=\frac{2\pi r_+}{4G_3}.$$

Smoothness of the Euclidean solid torus fixes the thermal identification. Thermal AdS and Euclidean BTZ differ by which primitive boundary cycle is contractible; the modular $S$ transformation exchanges those cycles. This geometric statement is the bulk counterpart of high/low-temperature modular duality.

Cardy growth and the entropy match

On a boundary circle of radius $\ell$, the conserved charges map to shifted conformal weights,

$$h-\frac{c}{24}=\frac{\ell M+J}{2}, \qquad \bar h-\frac{c}{24}=\frac{\ell M-J}{2}.$$

When both shifted weights lie in the asymptotic Cardy regime, vacuum dominance after a modular transformation gives

$$S_{\rm Cardy} =2\pi\sqrt{\frac{c}{6}\left(h-\frac{c}{24}\right)} +2\pi\sqrt{\frac{c}{6}\left(\bar h-\frac{c}{24}\right)}.$$

Substitution of $c=3\ell/(2G_3)$ and the BTZ charges yields $S_{\rm Cardy}=2\pi r_+/(4G_3)$. Rotation matters: the two square roots separately reconstruct $r_++r_-$ and $r_+-r_-$ before their sum removes $r_-$ from the area law.

First application. Match rotating BTZ entropy to the left- and right-moving asymptotic density in its controlled high-energy regime. The calculation must state whether it is microcanonical or canonical, retain the $-c/24$ Casimir shift, and require both chiral excitation energies to be large enough for the chosen Cardy theorem. Near extremality, one sector may require a separate limit rather than the symmetric high-temperature argument.

Universality has hypotheses

Modular invariance alone fixes asymptotic growth at energies parametrically above the vacuum under standard discreteness and vacuum assumptions. Extending a simple Cardy formula down to energies of order $c$ requires additional control, commonly a sufficiently sparse light spectrum. Higher-derivative bulk terms replace the area formula by Wald entropy and modify the relation between anomaly coefficients and couplings. One-loop determinants generate logarithmic and state-dependent corrections rather than changing the leading classical match.

Adversarial control. Insert $h=\bar h=0$ into the high-energy formula or apply it to a dense light spectrum. The square-root expression either leaves its domain or predicts a density not licensed by vacuum dominance. On the gravity side, continuing the same saddle below its thermodynamic dominance similarly confuses a possible Euclidean solution with the canonical equilibrium phase.

Evidence ceiling

The BTZ–Cardy agreement is a robust matching of leading thermodynamic observables in a stated regime. It does not reconstruct individual microstates, prove that a modular saddle sum is one CFT partition function, or determine finite-$c$ degeneracies. Those stronger claims require exact spectral and factorization data.

The rotating black-hole geometry is the BTZ solution Bañados, Teitelboim, and Zanelli 1992, while the asymptotic state-counting step uses Cardy’s modular argument Cardy 1986 and therefore retains its spectral hypotheses.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Bañados, Máximo, Claudio Teitelboim, and Jorge Zanelli. “The Black Hole in Three-Dimensional Spacetime.” Physical Review Letters 69 (1992): 1849–1851. DOI; Open PDF.
• Cardy, John L. “Operator Content of Two-Dimensional Conformally Invariant Theories.” Nuclear Physics B 270 (1986): 186–204. DOI.
• Hartman, Thomas, Christoph A. Keller, and Bogdan Stoica. “Universal Spectrum of 2d Conformal Field Theory in the Large c Limit.” Journal of High Energy Physics 2014, no. 9 (2014): 118. DOI; Open PDF.