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.

What the result establishes

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 compares these assumptions and limitations.

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.