Multiple Horizons and Variable Couplings

When a spacetime has several horizons or a theory allows dimensionful couplings to vary, the familiar one-horizon first law does not acquire a unique automatic extension. One must first choose a phase space: which boundaries are included, which generator and energy are used, which quantities may vary independently, and which couplings are fixed. As of 2026-08-12, separate-horizon laws, globally constrained variations, and extended pressure–volume laws are all well-defined in appropriate settings; none is a universal replacement for the others.

Required background. Variational and Symplectic Geometry supplies integrability and boundary flux. Helpful background. Exact Spacetimes supplies de Sitter and AdS families; Causality and Global Structure identifies multiple boundary components; Einstein Equations fixes the couplings; Gravitational Actions and Boundary Terms fixes the variational ensemble; Black-Hole Geometry and Mechanics supplies the single-horizon baseline.

One phase space before one first law

A first law is the equality of Hamiltonian variations on the boundaries of a hypersurface. Its content is therefore conditional on four decisions:

1. the action and its fixed or variable couplings;
2. the permitted solution family and boundary conditions;
3. the evolution vector and its normalization at each boundary;
4. the reference defining energy, potential, and any subtraction terms.

If two horizons bound a static region, the same Killing field can be future directed on one boundary and oppositely oriented on the other. Their signed surface gravities and outward boundary orientations then enter differently. Writing two positive temperatures and adding their area variations without tracking those signs is not a covariant derivation.

Likewise, varying $\Lambda$ changes the theory parameter unless the phase space has explicitly been enlarged. A formula with $V\delta P$ and a formula at fixed $\Lambda$ refer to different tangent spaces. They can both be correct, but they cannot be tested by treating the same $\delta$ as if it meant both variations.

Several horizons: local laws and global constraints

Consider a stationary region bounded by a black-hole horizon $H_b$ and a cosmological horizon $H_c$. Each Killing horizon has its own generator $\chi_i$, area $A_i$, angular velocity $\Omega_i$, potential $\Phi_i$, and surface gravity $\kappa_i$ in a stated normalization. A boundary calculation near one component can yield a local relation of the schematic form

$$\delta E_i =\frac{\kappa_i}{8\pi G}\delta A_i +\Omega_i\delta J_i+\Phi_i\delta Q_i+\cdots.$$

The sign of $\kappa_i$ depends on the chosen generator and orientation; some authors define a positive temperature with $|\kappa_i|$ and move the sign into the energy convention. That is a translation, not a disagreement about the geometry.

A global variation across the region is more restrictive. The horizon radii are roots of one metric function and depend on the same mass, charge, rotation, and cosmological parameters. Consequently $\delta A_b$ and $\delta A_c$ are generally not independent. Varying the shared parameters and differentiating both root equations produces a constrained relation rather than two freely addable first laws.

Schwarzschild–de Sitter makes this explicit. With

$$f(r)=1-\frac{2GM}{r}-\frac{\Lambda r^2}{3},$$

each positive horizon root satisfies

$$0=\delta f(r_i) =-\frac{2G}{r_i}\delta M -\frac{r_i^2}{3}\delta\Lambda +f'(r_i)\delta r_i.$$

The same $\delta M$ and $\delta\Lambda$ drive both $\delta r_b$ and $\delta r_c$. A global ensemble must respect this common origin. It cannot choose both area variations arbitrarily while also holding the shared parameters fixed.

Fixed Λ and extended pressure

At fixed cosmological constant, stationary variations remain within one Einstein theory. The cosmological scale affects the solution and charge renormalization but contributes no $\delta\Lambda$ work term. For rotating AdS black holes, even the asymptotic frame matters: energy and angular velocity must be defined relative to a nonrotating frame at infinity for the standard differential law to hold. Gibbons, Perry, and Pope demonstrate this criterion and the corresponding charges in Gibbons, Perry, and Pope 2005, §§2–4, pp. 1505–1518.

In extended phase space, define

$$P=-\frac{\Lambda}{8\pi G}$$

and permit $P$ to vary. A four-dimensional AdS first law can then take the form

$$\delta M =\frac{\kappa}{8\pi G}\delta A_H +\Omega_H\delta J+\Phi_H\delta Q+V\delta P.$$

Here $M$ is naturally interpreted as enthalpy within that formalism, and $V$ is the derivative of $M$ with respect to $P$ at fixed entropy, angular momentum, charge, and any other declared variables. Kubizňák, Mann, and Teo review the fixed conventions, extended first law, and Smarr scaling in Kubizňák, Mann, and Teo 2017, §2, pp. 5–22.

The competing formulations answer different questions. Fixed-$\Lambda$ mechanics studies a theory with one coupling. Extended mechanics compares solutions across cosmological scales and supports a useful pressure–volume equation of state. The geometric validity of each law is judged by its Hamiltonian integrability and boundary conditions, not by whether the pressure language is adopted universally.

Other variable couplings and defects

Suppose the Lagrangian contains couplings $c_I$ with nonzero length weights. An enlarged phase space may produce

$$\delta M =\frac{\kappa}{2\pi}\delta S_{\mathrm{Wald}} +\Omega_H\delta J+\Phi_H\delta Q +\sum_I\Psi_I\delta c_I,$$

where the entropy is theory dependent and each conjugate $\Psi_I$ follows from the same variational principle. Varying Newton’s constant is especially delicate because it changes both the action normalization and the relation between geometric and physical charges. Holding all geometric fields fixed while changing $G$ is not generally tangent to a solution family.

Conical deficits or string tensions can add length/tension work pairs. Magnetic charge may require gauge patches or dual potentials. Higher-curvature couplings alter both entropy and the scaling relation. These additions should pass three checks: the symplectic charge is integrable for the declared boundary data, Euler scaling includes every dimensionful variable, and a fixed-coupling restriction returns the established law.

Comparing first laws and their assumptions

Through the 2026-08-12 evidence cutoff, the useful comparison is:

Formulation	Boundaries and variables	Strength	Principal limitation
Separate-horizon law	One horizon component with a local generator	Covariant boundary identity under stated conditions	Does not make all horizon areas independent globally
Constrained multi-horizon variation	Shared solution parameters varied across all roots	Exact within a named family or phase space	Energy and sign conventions can be ensemble dependent
Fixed-coupling first law	Couplings such as $\Lambda$ held fixed	Standard Hamiltonian variation within one theory	Omits comparisons across coupling values by design
Extended phase space	$P$, tension, or other couplings varied	Consistent when derived from an enlarged variational problem	Conjugates and interpretation depend on chosen boundary data

No universal verdict follows from this table. A stronger unification would require a covariant phase-space derivation that simultaneously fixes the action, all boundary components, allowed variations, charge references, and integrability for the claimed class. A new derivation of that kind, or a counterexample showing failure under its advertised hypotheses, is the appropriate trigger to revise the comparison.

The unresolved part is therefore not whether the displayed special-case laws can be derived; it is whether a proposed broader formulation covers a claimed class without hiding a boundary condition, dependent horizon variation, or nonintegrable charge. Open questions should be stated relative to that exact formulation rather than as a contest for one universally preferred ensemble.

Test two incompatible variations

For Schwarzschild–de Sitter, suppose one variation holds $M$ and $\Lambda$ fixed but claims independent nonzero $\delta A_b$ and $\delta A_c$. A second variation changes $\Lambda$ and includes $V\delta P$. Diagnose each proposal.

Solution

With $M$ and $\Lambda$ fixed, differentiating $f(r_i)=0$ gives $f'(r_i)\delta r_i=0$. Away from a degenerate root, $f'(r_i)\ne0$, so both $\delta r_i$ and hence both area variations vanish. The claimed independent nonzero changes are not tangent to the solution family. The second proposal can be meaningful only in an extended phase space: it must declare $P=-\Lambda/(8\pi G)$, a compatible mass/energy definition, which other variables are fixed, and the conjugate volume derived from that same Hamiltonian. It must not be combined with a premise that $\Lambda$ is fixed.

The Laws of Black-Hole Mechanics uses this charge framework to state the zeroth, first, area, physical-process, and third laws with their separate hypotheses.

References

• Gibbons, G. W., M. J. Perry, and C. N. Pope. “The First Law of Thermodynamics for Kerr–Anti-de Sitter Black Holes.” Classical and Quantum Gravity 22 (2005): 1503–1526. DOI.
• Kubizňák, David, Robert B. Mann, and Mae Teo. “Black Hole Chemistry: Thermodynamics with Lambda.” Classical and Quantum Gravity 34 (2017): 063001. DOI.