Charges and First Laws in de Sitter

In de Sitter black-hole mechanics, the symbol multiplying $1/r$ in a metric is not automatically a uniquely normalized energy. There is no asymptotically timelike spatial infinity, and a static patch has two horizon boundaries with opposite orientations. A consistent first law must name its evolution vector, boundary, charge prescription, gauge reference, and whether $\Lambda$ varies. Once those choices are fixed, the apparent sign conflicts reduce to compatible variations of the same solution parameters.

Helpful background. Exact Solutions of Einstein’s Equations supplies stationary de Sitter families; Causality and Global Structure supplies spacelike conformal infinity and observer patches; The Einstein Equation supplies the cosmological term; and Black-Hole Geometry and Mechanics supplies horizon charges and first-law identities.

A static derivation with oriented surface gravity

Use Schwarzschild–de Sitter as the cleanest calculation:

$$f(r)=1-\frac{2GM}{r}-\frac{\Lambda r^2}{3}, \qquad \mathcal A_h=4\pi r_h^2.$$

Here $M$ denotes the metric parameter in this declared convention. Solving $f(r_h)=0$ gives

$$M(r_h,\Lambda)=\frac{r_h}{2G} \left(1-\frac{\Lambda r_h^2}{3}\right).$$

For $T=\partial_t$, the signed surface gravity is

$$\kappa_h=\frac12 f'(r_h) =\frac{1-\Lambda r_h^2}{2r_h}.$$

Varying the horizon equation rather than guessing a thermodynamic sign yields

$$\delta M =\frac{\kappa_h}{8\pi G}\,\delta\mathcal A_h -\frac{r_h^3}{6G}\,\delta\Lambda.$$

Define the extended-mechanics variables

$$P=-\frac{\Lambda}{8\pi G}, \qquad V_h=\frac{4\pi r_h^3}{3}.$$

Then the same identity is

$$\delta M =\frac{\kappa_h}{8\pi G}\,\delta\mathcal A_h +V_h\,\delta P.$$

The pressure is negative for positive $\Lambda$ in this convention. Calling $P$ positive while retaining this formula would silently change a sign.

At the black-hole root, $\kappa_b\gt0$. At the cosmological root, $\kappa_c\lt0$. Therefore, at fixed $\Lambda$,

$$\delta M =\frac{\kappa_b}{8\pi G}\delta\mathcal A_b =-\frac{|\kappa_c|}{8\pi G}\delta\mathcal A_c.$$

Increasing $M$ enlarges the black-hole horizon and shrinks the cosmological horizon. The two signs encode one shared variation, not two inconsistent first laws. This direct result is the spherical specialization of the horizon mechanics organized by Gibbons, Lü, Page, and Pope 2005, §§3–4.

Changing the boundary energy convention

Suppose the cosmological boundary is assigned the oppositely oriented energy $E_c=-M$. Its variation becomes

$$\delta E_c =\frac{|\kappa_c|}{8\pi G}\delta\mathcal A_c -V_c\delta P.$$

This form has a positive area coefficient, but the pressure term changes sign as well. Writing only the first term can make two conventions appear to agree while their full variational identities do not.

An overall change of generator has a different effect. If an evolution vector is multiplied by a constant, its Hamiltonian charge, surface gravity, angular velocity coefficients, and electric potential all scale consistently. It is invalid to rescale only $\kappa$ while leaving the energy fixed. The classical first-law derivation and the role of a fixed Killing normalization are given in Bardeen, Carter, and Hawking 1973, pp. 165–169, although their asymptotically flat normalization must not be imported into de Sitter without replacement.

Additive reference choices are independent of normalization. A charge may be shifted so that pure de Sitter has zero energy; such a constant disappears from $\delta E$. A sign change, by contrast, corresponds to reversing a generator or boundary orientation and changes the entire law. Comparisons must state which operation was performed.

Patch, horizon, and conformal-boundary charges

Several legitimate constructions answer different questions.

Construction	Boundary and data	Required choices
Static-patch Hamiltonian	A hypersurface segment between black-hole and cosmological horizons	One evolution field, two boundary orientations, gauge references
Isolated-horizon energy	One null horizon with quasi-local equilibrium data	Horizon evolution vector and integrability condition
Conformal-boundary charge	A two-surface at spacelike de Sitter infinity	Asymptotic class, conformal frame, asymptotic symmetry, flux conditions
Brown–York or finite-boundary charge	A timelike or spacelike regulating boundary	Boundary stress tensor, counterterms or subtraction, orientation

These charges can coincide on a stationary family after conventions are matched, but that agreement is a result to verify, not a definition. In a generic asymptotically de Sitter spacetime, the conformal boundary is spacelike and its unrestricted symmetry group is too large to select a preferred energy automatically. Stronger boundary conditions can reduce the symmetry group but may also remove radiative degrees of freedom. The definitions, charge balance law, and this restriction are analyzed in Ashtekar, Bonga, and Kesavan 2015, §§II, IV, and V.3.

The finite static patch and spacelike conformal infinity also enclose different regions. A flux through one boundary need not be represented by the same integral on the other without an evolution region and matching theorem. Calling both quantities “mass” does not supply that missing relation.

Rotation and electromagnetic potentials

For a rotating horizon, the relevant field is $\chi_h=T+\Omega_h\Phi$, and a schematic horizon law contains

$$\delta E_h =\frac{\kappa_h}{8\pi G}\delta\mathcal A_h +\Omega_h\delta J_h+\Phi_h\delta Q_h+\cdots.$$

Every term inherits the boundary orientation and the chosen basis $(T,\Phi)$. A rigid azimuthal shift changes $\Omega_h$ and the energy associated with the time generator while leaving the geometric vector $\chi_h$ unchanged. For a two-horizon region, potential differences are often more robust than either coefficient separately.

The electromagnetic potential illustrates the same point. With gauge field $A$, define a horizon potential from $-\chi_h\mathbin{\cdot}A$ only after choosing a reference. A constant gauge shift changes individual potentials but not a properly formed difference between boundaries. If charge can cross the region, the Maxwell orientation fixes whether the cosmological term enters with the same written sign as the black-hole term.

The ellipsis may include a cosmological-volume term, matter work terms, or additional rotations. It is not permission to combine quantities from different charge prescriptions. Integrability of the Hamiltonian variation is part of establishing a first law.

Solve the two-area variation

At fixed $\Lambda$, increase the Schwarzschild–de Sitter metric parameter by $\delta M\gt0$. Determine the signs of $\delta\mathcal A_b$ and $\delta\mathcal A_c$. An author instead defines $E_c=-M$ and writes a positive area coefficient at the cosmological horizon. Is that inconsistent?

Solution

The black-hole relation gives

$$\delta\mathcal A_b =\frac{8\pi G}{\kappa_b}\delta M\gt0.$$

The cosmological relation uses $\kappa_c\lt0$:

$$\delta\mathcal A_c =\frac{8\pi G}{\kappa_c}\delta M\lt0.$$

Thus the black-hole area grows while the cosmological area shrinks. Adding the positive magnitudes gives the constraint

$$\kappa_b\delta\mathcal A_b +|\kappa_c|\delta\mathcal A_c=0.$$

Defining $E_c=-M$ is consistent provided the whole identity is transformed: $\delta E_c=|\kappa_c|\delta\mathcal A_c/(8\pi G)$ at fixed $\Lambda$, and the variable-$P$ term is $-V_c\delta P$. The apparent discrepancy is a declared energy and orientation convention, not a physical contradiction.

Dynamical and Quasi-Local de Sitter Horizons next asks which boundary formulations remain useful when stationarity is absent.

References

• Ashtekar, Abhay, Béatrice Bonga, and Aruna Kesavan. “Asymptotics with a Positive Cosmological Constant: I. Basic Framework.” Classical and Quantum Gravity 32 (2015): 025004. DOI. Open manuscript.
• Bardeen, James M., Brandon Carter, and Stephen W. Hawking. “The Four Laws of Black Hole Mechanics.” Communications in Mathematical Physics 31 (1973): 161–170. DOI.
• Gibbons, Gary W., Hong Lü, Don N. Page, and Christopher N. Pope. “The General Kerr–de Sitter Metrics in All Dimensions.” Journal of Geometry and Physics 53 (2005): 49–73. DOI. Open manuscript.