Rotating and Charged AdS Black Holes

Charge and rotation add chemical potentials, extremal limits, and new instabilities to AdS black-hole thermodynamics. The same geometry can be stable at fixed charge and unstable at fixed potential, so a phase statement is incomplete until the ensemble and boundary gauge data are named.

Required background. AdS Black Branes and Holographic Thermodynamics fixes the neutral normalization. Chemical Potentials and Finite-Density Ensembles supplies the thermodynamic variables.

Helpful background. Rotating and Charged Horizons, Superradiance, and Chemical Potentials develops horizon regularity and superradiance.

First application. Analyze a Reissner-Nordstrom-AdS saddle at fixed chemical potential, deriving its temperature, charge, grand potential, and near-extremal limit.

Charged AdS data and normalization

For Einstein–Maxwell theory, a convenient family of spherical solutions can be parameterized as

$$f(r)=1+\frac{r^2}{L^2} -\frac{m}{r^{d-2}} +\frac{q^2}{r^{2d-4}},$$ $$A_t(r)=\mu-c_d\frac{q}{r^{d-2}}, \qquad A_t(r_h)=0.$$

$c_d$ depends on the Maxwell coupling and charge convention; declaring it is essential when comparing different papers. Horizon regularity fixes

$$\mu=c_d\frac{q}{r_h^{d-2}},$$

and the temperature is

$$T=\frac{1}{4\pi} \left[ \frac{d r_h}{L^2} +\frac{d-2}{r_h} -\frac{(d-2)q^2}{r_h^{2d-3}} \right].$$

The physical charge is obtained from the renormalized electric flux and is proportional to $q/g_F^2$ with the sphere volume and normalization restored. Extremality is the double-zero condition $f(r_h)=f'(r_h)=0$, equivalently $T=0$ in this stationary family.

Grand-canonical and fixed-charge actions

At fixed temperature and potential, the saddle computes

$$\Omega(T,\mu)=E-TS-\mu Q, \qquad d\Omega=-S\,dT-Q\,d\mu.$$

At fixed charge,

$$F(T,Q)=E-TS=\Omega+\mu Q, \qquad dF=-S\,dT+\mu\,dQ.$$

The Maxwell boundary term implements this Legendre transform in the Euclidean action. Stability is tested by the Hessian of the potential appropriate to the fixed variables. A negative electric susceptibility can invalidate a grand-canonical saddle even if its fixed-charge heat capacity is positive.

For rotation, replace $\mu Q$ by $\mu Q+\sum_a\Omega_aJ_a$. Smoothness requires a combined Euclidean identification in time and angular coordinates, while the Lorentzian horizon generator is

$$\chi=\partial_t+\sum_a\Omega_a\partial_{\phi_a}.$$

The first law becomes $dE=T\,dS+\mu\,dQ+\sum_a\Omega_a\,dJ_a$. Energy and angular velocity must be measured relative to the chosen nonrotating frame at the AdS boundary.

Near extremality is not automatically a ground-state count

As $T\to0$, an extremal charged black hole can develop a long throat and a finite classical entropy. The throat controls a low-energy sector only after one checks which modes decouple and which boundary conditions connect it to the asymptotic AdS region. The area at extremality is not by itself a microscopic degeneracy: protected counting, quantum corrections, and ensemble definitions are separate questions.

Similarly, the inequality for a bosonic mode to extract horizon energy is schematically

$$0<\omega<m\Omega_H+q_{\mathrm{field}}\Phi_H.$$

AdS reflects radiation, so a superradiant mode can grow rather than escape. Whether it does depends on the full normal-mode spectrum and boundary conditions, not only on the thermodynamic first law.

Adversarial controls

Hold the wrong variable. Differentiating a fixed-potential action as if $Q$ were fixed produces the wrong energy and susceptibility. The boundary variation immediately reveals the mismatch.

Compare unnormalized charges. Rescaling $A$ changes $g_F$, $c_d$, $Q$, and $\mu$ while leaving $\mu Q$ invariant. A charge bound stated only in terms of the metric parameter $q$ is not portable.

Infer stability from extremality. $T\ge0$ constrains the stationary family but does not exclude superradiant, scalar, or localization instabilities. Perturbation spectra and competing saddles must be checked.

Evidence ceiling

A renormalized charged or rotating saddle determines thermodynamic potentials and local stability in its declared ensemble and frame. It can identify extremal and superradiant thresholds. It does not establish global stability across ensembles, the endpoint of an instability, or a microscopic interpretation of the extremal entropy.

The ensemble-dependent charged-AdS phase structure is exhibited explicitly by Chamblin et al. 1999, and the Kerr–AdS first law with consistent asymptotic charges is developed by Gibbons, Perry, and Pope 2005.

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

• Chamblin, Andrew, Roberto Emparan, Clifford V. Johnson, and Robert C. Myers. “Charged AdS Black Holes and Catastrophic Holography.” Physical Review D 60, 064018 (1999). DOI; arXiv:hep-th/9902170.
• Gibbons, G. W., Malcolm J. Perry, and Christopher N. Pope. “The First Law of Thermodynamics for Kerr–Anti-de Sitter Black Holes.” Classical and Quantum Gravity 22, 1503–1526 (2005). DOI; arXiv:hep-th/0408217.
• Hawking, Stephen W., C. J. Hunter, and Marika M. Taylor-Robinson. “Rotation and the AdS/CFT Correspondence.” Physical Review D 59, 064005 (1999). DOI; arXiv:hep-th/9811056.