Black-Hole Horizon Scales

A black hole has several useful length, time, rotation, thermal, and response scales, but they are not interchangeable. This record fixes one baseline—an uncharged Kerr black hole in four-dimensional asymptotically flat general relativity—and restores SI units. The Equilibrium Black Holes and Horizon Anatomy chapter gives the surrounding geometry; see Kerr horizon and ergoregion anatomy for the spacetime construction and causal surfaces.

Kerr mass and spin package

For mass $M\gt0$ and angular momentum $J$, define

$$\begin{aligned} r_g&=\frac{GM}{c^2}, &t_g&=\frac{GM}{c^3}=\frac{r_g}{c},\\ a&=\frac{J}{Mc}, &\chi&=\frac{a}{r_g}=\frac{Jc}{GM^2}. \end{aligned}$$

Here $r_g$ and $a$ are lengths, $t_g$ is a time, and the signed dimensionless spin obeys $\lvert\chi\rvert\leq 1$ for the Kerr black-hole family. It is convenient to set

$$q\equiv\sqrt{1-\chi^2}.$$

At fixed $\chi$, every Kerr length is $r_g$ times a dimensionless function, and every frequency is $t_g^{-1}$ times a dimensionless function. This mass scaling does not make coefficients belonging to different physical questions interchangeable.

Boyer–Lindquist roots, area, and rotation

The two Boyer–Lindquist algebraic roots are

$$r_\pm=r_g\left(1\pm q\right) =r_g\pm\sqrt{r_g^2-a^2}.$$

These are coordinate values, not circumferential or areal radii. For $0\lt\lvert\chi\rvert\lt1$, both are distinct horizons; at $\chi=0$, the formal lower root $r_-=0$ is the Schwarzschild central curvature singularity, not a horizon. At $\lvert\chi\rvert=1$ the roots merge into one degenerate horizon. The outer event horizon has the invariant intrinsic area

$$\begin{aligned} A_H &=4\pi\left(r_+^2+a^2\right) =8\pi r_g r_+,\\ R_A &\equiv\sqrt{\frac{A_H}{4\pi}} =\sqrt{r_+^2+a^2} =r_g\sqrt{2(1+q)}. \end{aligned}$$

The algebraic roots, angular velocity, and intrinsic area follow from the stationary Kerr geometry; Visser 2008, § 9, printed PDF pp. 24–29, especially Eqs. (104), (112), and (118), PDF gives these quantities while keeping the coordinate and intrinsic two-geometry distinct.

With the asymptotic time translation normalized in the standard way, the outer-horizon angular velocity and its cyclic-frequency counterpart are

$$\Omega_H =\frac{ca}{r_+^2+a^2} =\frac{1}{t_g}\frac{\chi}{2(1+q)}, \qquad f_H=\frac{\Omega_H}{2\pi}.$$

$\Omega_H$ is an angular frequency in radians per second, whereas $f_H$ is in cycles per second. The stationary horizon generator $\xi+\Omega_H\psi$, its normalization, surface gravity, and area enter the black-hole mechanics relations in Bardeen, Carter, and Hawking 1973, pp. 162–166, PDF.

Surface gravity, temperature, and crossing time

Write the outer-horizon surface gravity first as an inverse length,

$$\kappa =\frac{r_+-r_-}{2\left(r_+^2+a^2\right)} =\frac{1}{r_g}\frac{q}{2(1+q)}.$$

Two common physical-unit variants are

$$\kappa_t=c\kappa \quad [\mathrm{s}^{-1}], \qquad g_H=c^2\kappa \quad [\mathrm{m}\,\mathrm{s}^{-2}].$$

Thus a quoted “surface gravity” must state whether it means inverse length, inverse time, or acceleration. Moreover, $\kappa$ depends on the normalization of the horizon-generating Killing field; the expressions here use the stationary asymptotically flat normalization fixed at infinity.

For that same normalization and a stationary nonextremal horizon, Hawking’s thermal conversion gives

$$T_H =\frac{\hbar c\kappa}{2\pi k_B} =\frac{\hbar}{4\pi k_B t_g}\frac{q}{1+q}.$$

The thermal factor and its dependence on normalized surface gravity are derived in Hawking 1975, §§ 2–3, pp. 202–214, PDF. The assumptions and interpretation of this conversion are developed under surface gravity and Hawking temperature.

“Light-crossing time” is not unique until a length is declared. Using the invariant areal radius, this record defines

$$t_{\mathrm{lc}}\equiv\frac{R_A}{c} =t_g\sqrt{2(1+q)}.$$

A diameter-crossing convention would instead give $2t_{\mathrm{lc}}$; $r_+/c$ is a different, coordinate-based shorthand.

Mode-dependent ringdown scaling

For a quasinormal mode labelled by field spin weight $s$, angular indices $(\ell,m)$, and overtone $n$, define

$$\widehat{\omega}_{s\ell mn}(\chi) \equiv \omega_{s\ell mn}t_g.$$

Then

$$\begin{aligned} \omega_{s\ell mn} &=\frac{\widehat{\omega}_{s\ell mn}(\chi)}{t_g},\\ f_{s\ell mn} &=\frac{\operatorname{Re}\widehat{\omega}_{s\ell mn}}{2\pi t_g}, &\tau_{s\ell mn} &=\frac{t_g}{-\operatorname{Im}\widehat{\omega}_{s\ell mn}}. \end{aligned}$$

These signs use the time dependence $e^{-i\omega t}$, so a decaying mode has $\operatorname{Im}\omega\lt0$. The oscillation period follows from the real part, whereas the damping time follows from the imaginary part; neither alone fixes the mode. The coefficient $\widehat{\omega}_{s\ell mn}$ is mode- and spin-dependent; it is not $\Omega_H$, $\kappa_t$, or $T_H$ in another unit system. The horizon boundary condition involves the co-rotating combination $\omega-m\Omega_H$, while the full complex resonance is selected jointly by the horizon and outgoing-infinity conditions. Leaver 1985, Eqs. (12)–(27), pp. 287–291 gives the Kerr continued-fraction construction and boundary behavior. See quasinormal boundary conditions and resonances for the spectral problem.

Limiting checks

For Schwarzschild, $\chi=0$ and $q=1$, so

$$\begin{gathered} r_+=R_A=2r_g, \qquad r_-=0, \qquad A_H=16\pi r_g^2, \qquad \Omega_H=0,\\ \kappa=\frac{1}{4r_g}, \qquad T_H=\frac{\hbar}{8\pi k_Bt_g} =\frac{\hbar c^3}{8\pi GMk_B}, \qquad t_{\mathrm{lc}}=2t_g. \end{gathered}$$

Here the displayed $r_-=0$ is the formal lower algebraic root and the central Schwarzschild curvature singularity; it is not an inner horizon.

In the extremal limit $\lvert\chi\rvert\to1$ and $q\to0$,

$$\begin{aligned} r_+&=r_-=r_g, &A_H&\to8\pi r_g^2, &R_A&\to\sqrt{2}\,r_g,\\ \Omega_H&\to\frac{\chi}{2t_g}, &\kappa&\to0. \end{aligned}$$

Accordingly, the nonextremal stationary formula has $T_H\to0$ and $t_{\mathrm{lc}}\to\sqrt{2}\,t_g$. This is a limiting check; the exact extremal state requires separate care because the nonextremal horizon structure used in the usual thermal derivation degenerates.

Where the scale package stops

• Coordinate versus invariant size. $r_+$ and $r_-$ are Boyer–Lindquist algebraic coordinate values. $A_H$ and the derived $R_A$ are intrinsic outer-horizon quantities; none is a universal Euclidean radius.
• Event, static-limit, and inner surfaces. The outer event horizon is at $r_+$. For nonzero spin, the outer static-limit surface is angle-dependent, $r_{\rm sl}(\theta)=r_g+\sqrt{r_g^2-a^2\cos^2\theta}$, and bounds the ergoregion with $r_+$; it meets the horizon only on the rotation axis. At $\chi=0$, the static limit instead coincides with $r_+$ at every angle and the ergoregion is absent. For $0\lt\lvert\chi\rvert\lt1$, the surface at $r_-$ is the inner Cauchy horizon of exact Kerr; at $\chi=0$, $r_-=0$ is the central curvature singularity.
• Thermal versus ringdown scales. $T_H$ is a stationary semiclassical conversion of normalized $\kappa$. Ringdown frequencies are mode-dependent complex resonances; there is no universal replacement of $\widehat\omega_{s\ell mn}(\chi)$ by a horizon temperature or rotation rate.
• Exact solution versus inference and dynamics. These formulas use the exact stationary Kerr parameters $M$ and $J$. Applying them to empirical mass-and-spin estimates inherits the measurement model and uncertainty, while evolving, distorted, accreting, or merging horizons require dynamical or quasi-local definitions rather than direct substitution into this card.

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. Open PDF.
• Hawking, Stephen W. “Particle Creation by Black Holes.” Communications in Mathematical Physics 43 (1975): 199–220. DOI. Open PDF.
• Leaver, Edward W. “An Analytic Representation for the Quasi-Normal Modes of Kerr Black Holes.” Proceedings of the Royal Society of London A 402 (1985): 285–298. DOI.
• Visser, Matt. “The Kerr Spacetime: A Brief Introduction.” arXiv:0706.0622v3 (2008). Open PDF.