Kerr–AdS Frames, Charges, and Horizons

Kerr–anti-de Sitter spacetime forces a choice that asymptotically flat rotation can hide: the Boyer–Lindquist azimuth rotates relative to a nonrotating frame at conformal infinity. The horizon angular velocity, energy, and angular momentum must therefore be referred to compatible asymptotic generators. In four dimensions, the charges $M=m/(G\Xi^2)$ and $J=ma/(G\Xi^2)$, together with the horizon angular velocity measured relative to nonrotating infinity, satisfy the classical first law. Mixing that mass with the rotating-frame angular velocity does not.

Helpful background. Exact Solutions of Einstein’s Equations supplies the stationary-axisymmetric metric and root analysis; Causality and Global Structure supplies the event-horizon and asymptotic-frame interpretation; The Einstein Equation supplies the vacuum equation with $\Lambda\lt0$; The Gibbons-Hawking-York Boundary Term supplies the boundary structure behind conserved charges; and Black-Hole Geometry and Mechanics supplies Killing generators, surface gravity, and the mechanical first law.

The four-dimensional Kerr–AdS horizon

Write $\Lambda=-3/L^2$ and define

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

The standard global Kerr–AdS family requires $\Xi\gt0$, or $\lvert a\rvert\lt L$. This keeps the angular sector regular at infinity and $\Delta_\theta$ positive. The outer event horizon lies at the largest positive root $r_+$ of $\Delta_r=0$, so its mass parameter may be written

$$m=\frac{(r_+^2+a^2)(1+r_+^2/L^2)}{2r_+}.$$

The horizon area and surface gravity are

$$A=\frac{4\pi(r_+^2+a^2)}{\Xi},$$ $$\kappa= \frac{r_+\left(1+a^2/L^2+3r_+^2/L^2-a^2/r_+^2\right)} {2(r_+^2+a^2)}.$$

These formulas, including the largest-root prescription, are given in Gibbons, Perry, and Pope 2005, §2.1, equations (2.1)–(2.5), Open PDF, pp. 7–8. A root alone is not enough to specify a regular black hole: one must also impose the angular regularity bound, choose the outer root, and check that the intended exterior reaches the chosen conformal boundary.

Two angular velocities, one consistent pairing

In the Boyer–Lindquist coordinates used above, the horizon generator has a coordinate angular velocity

$$\Omega'=\frac{a\Xi}{r_+^2+a^2}.$$

The coordinate frame itself rotates at infinity. Subtracting that asymptotic rotation gives the angular velocity relative to a nonrotating observer at infinity,

$$\Omega=\frac{a(1+r_+^2/L^2)}{r_+^2+a^2}, \qquad \Omega-\Omega'=\frac{a}{L^2}.$$

Neither expression is intrinsically erroneous: each is a difference between the horizon generator and a specified asymptotic time flow. The error is to combine quantities defined by different flows. The frame relation and its physical interpretation are explicit in Gibbons, Perry, and Pope 2005, §2.1, equations (2.6)–(2.8), Open PDF, p. 7.

Operationally, a stationary observer near the conformal boundary first removes the boundary shift and then compares the horizon generator with that nonrotating time direction. This is why $\Omega$, rather than the coordinate rate $\Omega'$, is the work coefficient for the standard mass. A local measurement near the horizon cannot by itself choose between them; the distinction is anchored at infinity.

Using the nonrotating time translation, the conserved charges are

$$M=\frac{m}{G\Xi^2}, \qquad J=\frac{ma}{G\Xi^2}=aM.$$

At fixed $L$, their variations obey the geometric first law

$$\delta M=\frac{\kappa}{8\pi G}\,\delta A+\Omega\,\delta J.$$

This law is more than a consistency slogan. Once $A$, $J$, $\kappa$, and the nonrotating $\Omega$ are fixed, the right-hand side is an exact differential and integrates to $M=m/(G\Xi^2)$, up to the choice that pure AdS has zero mass. By contrast, taking $M'=m/(G\Xi)$ and pairing it with $\Omega'$ fails the first law, and the proposed differential is not exact Gibbons, Perry, and Pope 2005, §§2.1–2.2, equations (2.11)–(2.19), Open PDF, pp. 8–10.

The alternative energy is not unrelated to $M$. Algebraically,

$$M'=M-\frac{a}{L^2}J=\frac{m}{G\Xi}.$$

It is the charge associated with a parameter-dependent rotating time flow. When $a$ varies, that generator varies too, so differentiating $M'$ produces an additional $J\,d(a/L^2)$ contribution. This is why the equality $M'-\Omega'J=M-\Omega J$ for one solution does not imply equality of the corresponding first laws across the family. A charge, a potential, and the vector field being varied must be specified together.

This conclusion concerns classical charges and horizon mechanics. It does not require assigning an entropy to $A$ or a temperature to $\kappa$.

Extremality and horizon shape

A degenerate horizon satisfies $\kappa=0$. The numerator of the surface-gravity formula gives

$$a^2\left(1-\frac{r_+^2}{L^2}\right) =r_+^2\left(1+\frac{3r_+^2}{L^2}\right).$$

To impose the global regularity condition $\Xi\gt0$, set $x=r_+^2/L^2$. The extremality relation becomes

$$\frac{a^2}{L^2}=\frac{x(1+3x)}{1-x}.$$

Positivity first requires $x\lt1$, while $\Xi\gt0$, equivalently $a^2/L^2\lt1$, sharpens the restriction to

$$\frac{x(1+3x)}{1-x}\lt1 \quad\Longleftrightarrow\quad 3x^2+2x-1\lt0 \quad\Longleftrightarrow\quad 0\lt x\lt\frac13.$$

Therefore a regular four-dimensional extremal Kerr–AdS member exists for a given positive horizon radius if and only if $0\lt r_+\lt L/\sqrt{3}$, with $a$ tuned by the relation above. At $r_+=L/\sqrt{3}$ one has $a^2=L^2$ and $\Xi=0$, so the endpoint is excluded. A generic root of $\Delta_r$ is subextremal. In the $L\to\infty$ limit the extremality relation becomes $a^2=r_+^2$, as for extremal Kerr. The same sharp regular-extremal bound is stated in Dias, Figueras, Monteiro, and Santos 2010, §2.1, p. 5, Open PDF.

Rotation also deforms the intrinsic horizon metric even though the cross-section retains $S^2$ topology. Its area depends on $r_+^2+a^2$ and the asymptotic angular normalization $\Xi^{-1}$, not merely on the areal coordinate $r_+$. Consequently $4\pi r_+^2$ is not the Kerr–AdS area, and comparing rotating solutions using $r_+$ alone can reverse an area ordering. The divergence as $\Xi\to0$ is partly the growing range of the normalized azimuth and is another signal that the standard quotient has no regular $a=L$ endpoint.

Checks and limiting regimes

Setting $a=0$ gives $\Xi=1$, $m=GM$, $J=0$, $\Omega=0$, and

$$\kappa=\frac{1+3r_+^2/L^2}{2r_+},$$

which reproduces Schwarzschild–AdS. Taking $L\to\infty$ at fixed $m$, $a$, and $r_+$ gives the asymptotically flat Kerr relations

$$\Xi\to1, \qquad M\to\frac{m}{G}, \qquad J\to\frac{ma}{G}, \qquad \Omega\to\frac{a}{r_+^2+a^2}.$$

These independent limits check both powers of $\Xi$ and the $a/L^2$ frame shift.

The formal limit $a\to L$ is different. At fixed $m$ and with the usual $2\pi$ azimuthal identification, $A$, $M$, and $J$ diverge as powers of $\Xi^{-1}$. It is not a regular endpoint of the standard global family. Finite ultraspinning geometries require a simultaneous coordinate rescaling and a new quotient or, in higher dimensions, a separately controlled limit. They should not be obtained by simply substituting $\Xi=0$ into these formulas.

Worked transfer diagnostic

Let $a=L/2$ and $r_+=L$. Compute both angular velocities and identify the work coefficient compatible with the nonrotating mass. Quantify the error made by using the other coefficient for a small variation $\delta J$ at fixed $L$.

Solution

Here $\Xi=3/4$ and $r_+^2+a^2=5L^2/4$. Therefore

$$\Omega L=\frac{(1/2)(1+1)}{1+1/4}=\frac45, \qquad \Omega' L=\frac{(1/2)(3/4)}{1+1/4}=\frac3{10}.$$

Their difference is $\Omega-\Omega'=1/(2L)=a/L^2$, as required. The mass $M=m/(G\Xi^2)$ is generated by the nonrotating asymptotic time, so its first law uses $\Omega=4/(5L)$. Replacing it by $\Omega'$ undercounts the rotational contribution by

$$(\Omega-\Omega')\delta J=\frac{\delta J}{2L}.$$

Next, AdS Boundary Conditions and the Black-Hole Region separates the causal definition of a black hole from the boundary conditions required for field evolution.

References

• Dias, Oscar J. C., Pau Figueras, Ricardo Monteiro, and Jorge E. Santos. “Ultraspinning Instability of Anti-de Sitter Black Holes.” Journal of High Energy Physics 2010 (2010): 067. DOI. Open PDF.
• Gibbons, G. W., Perry, M. J., and Pope, C. N. “The First Law of Thermodynamics for Kerr–anti-de Sitter Black Holes.” Classical and Quantum Gravity 22 (2005): 1503–1526. doi:10.1088/0264-9381/22/9/002. Open PDF.