Black-Hole Temperature and Thermodynamic Potentials

Black-hole thermodynamics becomes precise only after the clock, horizon generator, observer, gauge, and boundary ensemble have been fixed. This chapter builds that data in a controlled order: normalize surface gravity, redshift temperature to local observers, add rotational and electric work, select the ensemble, test its local response, and finally decide whether several horizons can share one equilibrium state. The result is a portable thermodynamic record rather than an unqualified number called “the temperature.”

Helpful background. Quantum Fields in Curved Spacetime provides stationary quantum states; Einstein Equations provides the gravitational solutions; and Black-Hole Geometry and Mechanics provides classical horizons, surface gravity, and first laws.

Enter the thermodynamic problem

The durable question is not simply why a black hole radiates. That fixed-background derivation belongs to Quantum Fields in Curved Spacetime. The question here is how the resulting temperature joins the mechanical charges of a stationary horizon and which thermodynamic experiment those variables describe. Wald’s synthesis makes the key bridge from horizon mechanics to $T_{\mathrm H}=\kappa/(2\pi)$ and entropy Wald 2001, §§2–4, PDF.

Four choices recur:

• a normalized stationary time fixes energy and surface gravity;
• an observer’s lapse converts boundary temperature into a local reading;
• a gauge and rotational frame fix electric and angular potentials; and
• boundary data determine which charges may fluctuate.

Changing one choice while holding the others implicit can create apparent contradictions. A rescaled Killing field changes both energy and temperature. A divergent static-frame temperature near a horizon need not mean divergent freely falling stress. A negative heat capacity is an ensemble-specific local test, not a universal dynamical verdict.

Objects that must not be conflated

Surface gravity is a property of a normalized horizon generator. Local temperature is an observer-dependent Tolman-redshifted quantity. Hawking temperature is the thermal scale relative to the chosen stationary time in a suitable state. These three are related, but they are not interchangeable without their normalization and observer.

Likewise, local and global stability answer different questions. A positive quadratic action on the physical fluctuation space says a saddle is a local minimum against allowed fluctuations. Global dominance compares complete thermodynamic potentials of distinct saddles with identical boundary data. Schwarzschild–AdS illustrates the separation: the large-hole branch becomes locally stable before it overtakes thermal AdS globally.

Multiple horizons add a third distinction. Each horizon may have a local temperature, yet unequal surface gravities prevent one Euclidean period from being smooth at both. A horizon-by-horizon constrained first law can remain well defined even when no global Gibbs state exists.

Reader routes

For the shortest temperature route, read the first two pages and then the ensemble page. This route explains generator normalization, local thermometry, and the boundary meaning of temperature.

For the charged and rotating route, add the potentials page before the ensemble and stability pages. It tracks the common generator and gauge through the work terms and the corresponding Legendre transform.

For the gravitational saddle route, continue through heat capacity and the multiple-horizon page, then proceed to the Euclidean-saddles chapter. The cavity example is the recurring calculation: York fixes a proper wall temperature, finds small and large branches, and tests their action curvature York 1986, §§II–V and VIII.

Chapter map

1. Surface Gravity and Hawking Temperature shows why generator normalization scales energy and temperature together.
2. Tolman Redshift and Local Temperature derives the constant redshifted equilibrium temperature and interprets the horizon divergence of static thermometers.
3. Rotating and Charged Thermodynamic Potentials defines $\Omega_{\mathrm H}$ and $\Phi_{\mathrm H}$ relative to a common frame, gauge, and boundary.
4. Canonical, Grand-Canonical, and Microcanonical Ensembles maps fixed boundary data to $S$, $F$, or $\mathcal G$ and to the allowed fluctuations.
5. Heat Capacity and Local Thermodynamic Stability derives the Schwarzschild sign, cavity stabilization, and the separation between local and global stability.
6. Multiple Horizons and Nonequilibrium Temperatures compares generic unequal-temperature patches with tuned and constrained alternatives.

Completing the first five pages lets a reader formulate a stationary black-hole thermodynamic problem with all intensive and extensive variables defined. The final page adds the diagnostic needed to say when that formulation does—or does not—extend across an entire multi-horizon region.

Review the chapter

Consider a black-hole saddle in a cavity. An author quotes $T=\kappa/(2\pi)$, finds $C\gt0$, and concludes that the black hole is the globally preferred equilibrium. Identify three missing pieces of information and explain what each controls.

Solution

First, the Killing generator and cavity clock must be normalized; otherwise neither $\kappa$ nor the energy conjugate to it is fixed. Second, the ensemble must state which of $J,Q,\Omega,\Phi$ are held fixed, because $C\gt0$ tests only one direction of the relevant response matrix. Third, global preference requires comparing renormalized on-shell actions of every saddle with the same wall geometry and boundary data. Positive heat capacity establishes only local stability, not the smallest free energy.

References

• Wald, R. M. (2001), “The Thermodynamics of Black Holes,” Living Reviews in Relativity 4, 6. Open PDF.
• York, J. W. Jr. (1986), “Black-Hole Thermodynamics and the Euclidean Einstein Action,” Physical Review D 33, 2092–2099. DOI.