Kerr–de Sitter Horizon Mechanics

Kerr–de Sitter spacetime has several Killing horizons but no asymptotically timelike infinity that canonically fixes a time translation. Horizon area is unambiguous once the azimuthal period is fixed, whereas angular velocity and surface gravity refer to a declared Killing basis and normalization. The mechanics is therefore consistent only when the radial polynomial, horizon component, time generator, and rotating frame are recorded together.

Helpful background. Exact Solutions of Einstein’s Equations supplies the Boyer–Lindquist form; Causality and Global Structure supplies the de Sitter observer patch; The Einstein Equation supplies the vacuum equations with $\Lambda\gt0$; and Black-Hole Geometry and Mechanics supplies Killing generators and surface gravity.

The rotating de Sitter horizon polynomial

Write

$$\ell^2=\frac{3}{\Lambda}, \qquad \Xi=1+\frac{a^2}{\ell^2},$$

and let $m$ be the metric’s length parameter. In four-dimensional Boyer–Lindquist coordinates, the functions relevant to the horizons are

$$\rho^2=r^2+a^2\cos^2\theta, \qquad \Delta_\theta=1+\frac{a^2}{\ell^2}\cos^2\theta,$$ $$\Delta_r=(r^2+a^2)\left(1-\frac{r^2}{\ell^2}\right)-2mr.$$

The inverse radial metric is proportional to $\Delta_r/\rho^2$, so a simple root $r=r_h$ is a candidate Killing horizon. In the regular subextremal range, the positive roots can include an inner Cauchy horizon $r_-$, an outer black-hole horizon $r_b$, and a cosmological horizon $r_c$, ordered $r_-\lt r_b\lt r_c$. The stationary observer region relevant here lies between $r_b$ and $r_c$; one must not call every positive root “the event horizon.”

This specialization follows the Boyer–Lindquist construction and horizon analysis in Gibbons, Lü, Page, and Pope 2005, §§3–4, especially eqs. (3.5), (3.6), and (4.3)–(4.18). Their parameter $\lambda$ is $1/\ell^2=\Lambda/3$ here. The symbol $m$ is not yet called an energy, because de Sitter charge conventions are boundary dependent.

Horizon generators in two azimuthal frames

Let $T=\partial_t$ and $\Phi=\partial_\phi$, with $\phi$ of period $2\pi$. The null generator at any simple root is

$$\chi_h=T+\Omega_h\Phi, \qquad \Omega_h=\frac{a\Xi}{r_h^2+a^2}.$$

The constancy of $\Omega_h$ over a connected horizon is geometric; its numerical value is tied to the chosen Killing basis. To see this explicitly, introduce

$$\widetilde\phi=\phi-\frac{a}{\ell^2}t, \qquad \widetilde T=T+\frac{a}{\ell^2}\Phi.$$

The same vector is then

$$\chi_h=\widetilde T+\widetilde\Omega_h\Phi, \qquad \widetilde\Omega_h =\Omega_h-\frac{a}{\ell^2} =\frac{a(1-r_h^2/\ell^2)}{r_h^2+a^2}.$$

Nothing physical changed: the angular coordinate was shifted by a rigid rotation. In particular,

$$\Omega_b-\Omega_c =\widetilde\Omega_b-\widetilde\Omega_c.$$

Relative horizon angular velocity is insensitive to this common shift. A further rescaling of the time generator rescales angular velocities and surface gravities together. Because conformal de Sitter infinity is spacelike, neither frame is selected by the asymptotically flat prescription $T^2\to-1$. A calculation must therefore state the frame rather than label one value simply “the” angular velocity.

Area and signed surface gravity

For the declared azimuthal period, a horizon cross-section has area

$$\mathcal A_h=\frac{4\pi(r_h^2+a^2)}{\Xi}.$$

Relative to the generator $\chi_h$ above, the signed surface gravity is

$$\kappa_h=\frac{\Delta_r'(r_h)}{2(r_h^2+a^2)}.$$

Using $\Delta_r(r_h)=0$ eliminates $m$ and gives a useful independent form,

$$\kappa_h= \frac{r_h(1-r_h^2/\ell^2)}{r_h^2+a^2} -\frac{1+r_h^2/\ell^2}{2r_h}.$$

For the usual orientation of $T$ in the region $r_b\lt r\lt r_c$, a simple black-hole root has $\kappa_b\gt0$ and the cosmological root has $\kappa_c\lt0$. This follows directly from the signs with which $\Delta_r$ crosses zero. Both magnitudes are positive local acceleration scales, but the signed quantities are the ones that enter oriented boundary identities.

The general Killing-horizon definition $\chi^\nu\nabla_\nu\chi^\mu=\kappa\chi^\mu$ and its dependence on the scaling of $\chi$ are stated in Bardeen, Carter, and Hawking 1973, pp. 164–166. Constancy holds separately on each connected component; it does not require $\kappa_b=|\kappa_c|$.

The two horizon equations also impose a useful shared-parameter check. Every root must yield the same $m$ through

$$m=\frac{(r_h^2+a^2)(1-r_h^2/\ell^2)}{2r_h}.$$

Consequently, $r_b$ and $r_c$ cannot be varied independently while $(a,\ell)$ are held fixed. A proposed pair that gives two different values of $m$ does not belong to one Kerr–de Sitter solution. This is the rotating counterpart of the Vieta check in the static family.

For $a\gt0$ and $r_b\lt r_c$, the frame-invariant difference is

$$\Omega_b-\Omega_c =\frac{a\Xi(r_c^2-r_b^2)} {(r_b^2+a^2)(r_c^2+a^2)}\gt0.$$

Thus the black-hole generator has the larger angular coefficient in either of the two bases used above. This does not mean that a local observer measures that coordinate difference directly: an observer’s angular velocity and orthonormal frame must still be specified. It does show why one rigid azimuthal shift cannot normally make both horizon angular velocities vanish.

Limit checks and degenerate roots

Setting $a\to0$ gives $\Xi\to1$ and $\Delta_r=r^2(1-2m/r-r^2/\ell^2)$. The area becomes $4\pi r_h^2$, both angular velocities vanish, and the surface-gravity formula reduces to $f'(r_h)/2$. This recovers the preceding Schwarzschild–de Sitter conventions.

Taking $\ell\to\infty$ instead gives

$$\Delta_r\to r^2-2mr+a^2, \qquad \Omega_h\to\frac{a}{r_h^2+a^2},$$

together with the Kerr area and surface gravity. Both frame choices coincide because their shift is $a/\ell^2$. These two limits detect the most common sign errors in $\Xi$ and in the cosmological term of $\Delta_r$.

A double root obeys $\Delta_r=\Delta_r'=0$ and hence $\kappa_h=0$ for the unrescaled generator. Which horizons merge matters. Coalescence of $r_-$ and $r_b$ is a rotating black-hole extremal limit; coalescence of $r_b$ and $r_c$ is a rotating Nariai-type limit. The same algebraic condition does not make their limiting geometries or causal roles identical.

The metric parameter $m$, the area, and the generator are enough for local horizon calculations, but not for a complete first law. A Hamiltonian energy requires a boundary, an orientation, an asymptotic symmetry or patch time, and a normalization. Those choices are developed on Charges and First Laws in de Sitter.

Translate a frame-dependent claim

An author quotes $\Omega_h=a\Xi/(r_h^2+a^2)$. Another quotes $a(1-r_h^2/\ell^2)/(r_h^2+a^2)$ for the same horizon and claims a disagreement. Show how to reconcile them. Which comparison between the black-hole and cosmological horizons is unchanged?

Solution

The first value decomposes $\chi_h$ in the basis $(T,\Phi)$. The coordinate change $\widetilde\phi=\phi-at/\ell^2$ replaces the time basis vector by $\widetilde T=T+(a/\ell^2)\Phi$. Therefore the coefficient of $\Phi$ shifts by $-a/\ell^2$:

$$\frac{a\Xi}{r_h^2+a^2}-\frac{a}{\ell^2} =\frac{a(1-r_h^2/\ell^2)}{r_h^2+a^2}.$$

The two formulas describe the same Killing generator. Since the same constant is subtracted at every horizon, $\Omega_b-\Omega_c$ is unchanged. Areas are also unchanged. Surface gravity remains unchanged under this azimuthal shift, although it would rescale if the generator itself were multiplied by a constant.

Two-Horizon Surface Gravities and Nonequilibrium next asks whether the black-hole and cosmological horizon scales can be matched.

References

• 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.