Rotating and Charged Thermodynamic Potentials

A stationary rotating, charged black hole exchanges not only energy and entropy but also angular momentum and electric charge. The conjugate intensive variables are the horizon angular velocity $\Omega_{\mathrm H}$ and the gauge-invariant potential difference $\Phi_{\mathrm H}$. Both are defined relative to the same normalized horizon generator and boundary gauge that define the mass. This common reference is what makes the work terms in the first law meaningful.

Required background. Gravitational Actions and Boundary Terms supplies the Hamiltonian charges and boundary variations used in the first law.

Helpful background. Quantum Fields in Curved Spacetime supplies thermal weights with chemical potentials; Einstein Equations supplies the Einstein–Maxwell solution; and Black-Hole Geometry and Mechanics supplies stationary horizon generators and mechanical charges.

The corotating horizon generator

Let $t^\mu$ be the stationary Killing field normalized by the boundary clock and let $\phi^\mu$ generate rotations with period $2\pi$. The null generator of a stationary axisymmetric horizon is

$$\chi^\mu=t^\mu+\Omega_{\mathrm H}\phi^\mu.$$

The coefficient $\Omega_{\mathrm H}$ is the angular velocity of the horizon relative to the nonrotating boundary frame. It is not merely a coordinate angular velocity: changing the asymptotic frame by $\phi\mapsto\phi-\Omega_\infty t$ shifts the numerical potential to $\Omega_{\mathrm H}-\Omega_\infty$. The energy entering the first law changes to the Hamiltonian associated with the same new time translation, so the thermodynamic statement remains consistent.

For a four-dimensional asymptotically flat Kerr–Newman black hole, use the Maxwell action $I_{\mathrm{EM}}=-(16\pi)^{-1}\int\sqrt{-g}\,F_{\mu\nu}F^{\mu\nu}d^4x$ and charge $Q=(4\pi)^{-1}\int_{S^2_\infty}{*F}$. With physical mass $M$, $a=J/M$, and $G$ retained, the horizons and angular velocity are

$$\begin{aligned} r_\pm&=GM\pm\sqrt{G^2M^2-a^2-GQ^2},\\ \Omega_{\mathrm H}&=\frac{a}{r_+^2+a^2}. \end{aligned}$$

The domain is $a^2+GQ^2\leq G^2M^2$, with strict inequality for the nonextremal thermodynamic calculation. Here $a$ is a length and the dimensionless spin is $J/(GM^2)$. Declaring the Maxwell action matters: another normalization can move factors of $4\pi$ between charge and potential.

Electric potential as a difference

A gauge transformation can add a constant to $-\chi^\mu A_\mu$, so the thermodynamic electric potential must compare horizon and boundary:

$$\Phi_{\mathrm H} =-\left.\chi^\mu A_\mu\right|_{\mathcal H} +\left.\chi^\mu A_\mu\right|_{\partial\mathcal M}.$$

In the common gauge where the second term vanishes, the Kerr–Newman value in the Maxwell convention above is

$$\Phi_{\mathrm H}=\frac{Qr_+}{r_+^2+a^2}.$$

For the stationary electrovac solution, the Maxwell equations, horizon symmetry, and regularity make $\chi\cdot A$ constant over a connected Killing horizon. A constant gauge shift changes both endpoint values equally and cancels from the difference. This is the electromagnetic analogue of fixing the boundary clock before quoting $\Omega_{\mathrm H}$ or $\kappa$.

First law, Smarr check, and ensembles

The Einstein–Maxwell first law is

$$\delta M =\frac{\kappa}{8\pi G}\,\delta A +\Omega_{\mathrm H}\,\delta J +\Phi_{\mathrm H}\,\delta Q =T_{\mathrm H}\delta S_{\mathrm{BH}} +\Omega_{\mathrm H}\delta J +\Phi_{\mathrm H}\delta Q.$$

The signs follow the convention that $M$ is the energy of the system: increasing $J$ or like-signed $Q$ at fixed entropy requires positive work. A grand-canonical potential instead subtracts those work reservoirs,

$$\mathcal G=M-T_{\mathrm H}S_{\mathrm{BH}} -\Omega_{\mathrm H}J-\Phi_{\mathrm H}Q, \qquad d\mathcal G=-S_{\mathrm{BH}}\,dT_{\mathrm H} -J\,d\Omega_{\mathrm H}-Q\,d\Phi_{\mathrm H}.$$

Thus “holding charge fixed” and “holding electric potential fixed” define different variational problems. The covariant Hamiltonian derivation and its boundary ambiguities are developed by Iyer and Wald 1994, §§3–4, PDF; the four-dimensional black-hole thermodynamic relations are summarized by Wald 2001, §2, PDF.

As an independent normalization check, four-dimensional scaling gives the Smarr relation

$$M=\frac{\kappa A}{4\pi G} +2\Omega_{\mathrm H}J+\Phi_{\mathrm H}Q.$$

Its factors of two differ because area and angular momentum scale quadratically with length whereas mass and charge scale linearly in geometrized units. A formula with the correct differential first law but an inconsistent Smarr relation usually signals a missing normalization, boundary subtraction, or gauge reference.

These equilibrium potentials require a stationary branch. During evaporation, $M$, $J$, and $Q$ evolve and enter mode occupations through $\omega-m\Omega_{\mathrm H}-q\Phi_{\mathrm H}$ only in an adiabatic approximation. That flux problem is separate from defining the stationary thermodynamic variables here.

A final dimensional check is useful. Angular velocity and temperature both have inverse-length dimension, while $\Omega_{\mathrm H}\delta J$ and $T_{\mathrm H}\delta S$ have energy dimension. The electrical term has that same dimension only with the declared Maxwell normalization. Mixing an SI charge with a geometrized potential without restoring the electromagnetic coupling would therefore spoil the comparison even if the symbolic first law looked familiar.

Exercise

Suppose the gauge potential is shifted by a constant one-form along the stationary time, so that $\chi\cdot A$ increases by the same constant $C$ at the horizon and at the boundary. Show what happens to $\Phi_{\mathrm H}$ and the first law.

Solution

The definition uses a difference: $\Phi'_{\mathrm H}=-(\chi\cdot A+C)_{\mathcal H} +(\chi\cdot A+C)_{\partial\mathcal M}=\Phi_{\mathrm H}$. Consequently the coefficient of $\delta Q$ is unchanged, as are all other terms in the first law. Using only the horizon value would spuriously shift the work term and would confuse a gauge choice with a thermodynamic change.

References

• Iyer, V., and R. M. Wald (1994), “Some Properties of Noether Charge and a Proposal for Dynamical Black Hole Entropy,” Physical Review D 50, 846–864. Open PDF.
• Wald, R. M. (2001), “The Thermodynamics of Black Holes,” Living Reviews in Relativity 4, 6. Open PDF.