--- title: Rotating and Charged Black Holes module: Black Holes moduleNumber: 8 lessonNumber: 2 order: 802 summary: > A stationary black hole is fixed by three numbers: mass, angular momentum, and charge. The Reissner–Nordström metric adds charge and splits the horizon in two; the Kerr metric adds rotation, drags inertial frames, and wraps the horizon in an ergosphere where nothing can stay still. Inside the ergosphere the Penrose process extracts rotational energy, and the no-hair theorem states that no other detail of the collapsed matter survives. topics: [Black Holes] draft: false sources: - book: Hartle ref: "Gravity, Ch. 15 — Rotating Black Holes" - book: Carroll ref: "Lecture Notes on General Relativity, §7 — The Kerr Solution" --- Real astrophysical bodies rotate and can carry charge, and the collapsed remnant keeps whatever angular momentum and charge the progenitor had. The Schwarzschild solution, with only a mass, is the exception rather than the rule. Two exact solutions extend it: the Reissner–Nordström metric for a static charged hole and the Kerr metric for a rotating one. Rotation is the astrophysically important case, and it introduces structure — frame dragging, an ergosphere, a mechanism for energy extraction — with no Schwarzschild analogue. ## The charged hole: Reissner–Nordström Solving the coupled Einstein–Maxwell equations for a static, spherical body of mass $M$ and charge $Q$ gives, in geometrized units, $$ \d s^2 = -f(r)\,c^2\,\d t^2 + f(r)^{-1}\,\d r^2 + r^2\,\d\Omega^2, \qquad f(r) = 1 - \frac{r_s}{r} + \frac{r_Q^2}{r^2}, $$ with $r_s = 2GM/c^2$ and the charge length $r_Q^2 = G Q^2/(4\pi\varepsilon_0 c^4)$. The horizons are the roots of $f(r) = 0$: $$ r_\pm = \frac{r_s}{2} \pm \sqrt{\frac{r_s^2}{4} - r_Q^2}. $$ Three cases follow from the discriminant. - **$r_Q < r_s/2$ (subextremal):** two distinct horizons, an outer event horizon $r_+$ and an inner Cauchy horizon $r_-$. Charge weakens gravity's grip and pulls the event horizon inward from the Schwarzschild value. - **$r_Q = r_s/2$ (extremal):** the two horizons merge at $r = r_s/2$. Surface gravity vanishes; this is the zero-temperature limit of the next lesson. - **$r_Q > r_s/2$ (overextremal):** $f$ has no real root, and $r = 0$ is a naked singularity with no horizon to hide it. The cosmic censorship conjecture holds that ordinary collapse never reaches this case. Astrophysical holes are effectively neutral: any net charge attracts opposite charges from the surrounding plasma and neutralizes in moments. Reissner–Nordström matters as a solvable model of a two-horizon geometry and as the charged endpoint of the classification, not as a description of a real object. $$ % caption: The Reissner–Nordström metric function f(r). Uncharged, it has a single % zero at rs; charged, it dips to two zeros at the inner and outer horizons; at the % extremal charge the two merge into a single tangent zero. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (-0.2,0) -- (6.0,0) node[right] {r}; \draw[->, black] (0.4,-1.2) -- (0.4,2.4) node[above] {f}; \draw[black, dashed] (0.4,1.5) -- (5.8,1.5) node[right, black] {far limit 1}; % uncharged: dips below then to 1, single crossing \draw[black, thick] (0.7,-1.0) .. controls (1.4,0.6) and (2.2,1.2) .. (3.2,1.38) .. controls (4.2,1.5) and (5.0,1.5) .. (5.8,1.5); \node[black, anchor=west] at (4.3,1.15) {uncharged}; % charged subextremal: two crossings \draw[acc, thick] (0.9,1.6) .. controls (1.3,0.1) and (1.7,-0.55) .. (2.4,-0.55) .. controls (3.2,-0.55) and (4.0,0.9) .. (5.8,1.45); \node[acc, anchor=west] at (2.6,-0.9) {two horizons}; \end{tikzpicture} $$ ## The Kerr metric A rotating hole of mass $M$ and angular momentum $J$ is described by the Kerr solution. Writing the spin parameter $a = J/(Mc)$ (a length) and, in Boyer–Lindquist coordinates, the abbreviations $$ \Sigma = r^2 + a^2\cos^2\theta, \qquad \Delta = r^2 - r_s\,r + a^2, $$ the line element is $$ \d s^2 = -\left(1 - \frac{r_s r}{\Sigma}\right)c^2\,\d t^2 - \frac{2 r_s r\, a \sin^2\theta}{\Sigma}\,c\,\d t\,\d\phi + \frac{\Sigma}{\Delta}\,\d r^2 + \Sigma\,\d\theta^2 + \left(r^2 + a^2 + \frac{r_s r\,a^2\sin^2\theta}{\Sigma}\right)\sin^2\theta\,\d\phi^2. $$ Three features distinguish it from Schwarzschild. - **The cross term $\d t\,\d\phi$** couples time and azimuth. Its presence is frame dragging: the geometry is not invariant under $t \to -t$ alone but only under $t \to -t$ together with $\phi \to -\phi$, so the hole's rotation is imprinted on the metric. - **The horizon is $\Delta = 0$**, at $$ r_\pm = \frac{r_s}{2} \pm \sqrt{\frac{r_s^2}{4} - a^2}, $$ the same algebra as Reissner–Nordström with $a$ in place of $r_Q$. A horizon exists only for $a \le r_s/2$, i.e. $J \le GM^2/c$; the extremal Kerr hole saturates this. Faster spin would expose the singularity, again forbidden by cosmic censorship. - **The singularity is a ring**, not a point. $\Sigma = 0$ requires $r = 0$ and $\theta = \pi/2$ simultaneously, a ring of radius $a$ in the equatorial plane. Setting $a = 0$ recovers Schwarzschild; setting $M \to 0$ with $a$ fixed recovers flat spacetime in spheroidal coordinates. ## The static limit and the ergosphere Frame dragging changes what "standing still" means. Consider an observer trying to remain at fixed $(r, \theta, \phi)$ — stationary with respect to the distant stars. Such a worldline has four-velocity along $\partial_t$, and it is timelike only where $g_{tt} < 0$. The coefficient $$ g_{tt} = -\left(1 - \frac{r_s r}{\Sigma}\right)c^2 $$ vanishes at the **static limit surface** $$ r_{\text{static}}(\theta) = \frac{r_s}{2} + \sqrt{\frac{r_s^2}{4} - a^2\cos^2\theta}. $$ Inside it, $g_{tt} > 0$: no timelike worldline can have $\d\phi = 0$, so no observer can remain at rest relative to infinity. Every observer is forced to co-rotate with the hole. The static limit lies outside the horizon everywhere except at the poles, where the two surfaces touch. The region between them, $$ r_+ < r < r_{\text{static}}(\theta), $$ is the **ergosphere**. > **Definition (Ergosphere).** The ergosphere is the region outside a Kerr > horizon where $g_{tt} > 0$, bounded within by the event horizon $r_+$ and > outside by the static limit surface. Inside it every future-directed timelike > worldline must have $\d\phi/\d t > 0$ — co-rotation with the hole is > compulsory — yet escape to infinity remains possible, because the horizon lies > deeper. Its name marks it as the region from which rotational energy (Greek > _ergon_, work) can be mined. $$ % caption: Meridional cross-section of a Kerr hole. The event horizon is a sphere % of coordinate radius r-plus; the static limit bulges out to rs at the equator and % meets the horizon at the poles; the ergosphere is the lens between them. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,-2.6) -- (0,2.9) node[above] {spin axis}; \draw[->, black] (-2.9,0) -- (2.9,0) node[right] {equator}; % static limit: oblate, meets horizon at poles \draw[acc, very thick] (0,1.5) .. controls (1.4,1.5) and (2.3,0.9) .. (2.3,0) .. controls (2.3,-0.9) and (1.4,-1.5) .. (0,-1.5) .. controls (-1.4,-1.5) and (-2.3,-0.9) .. (-2.3,0) .. controls (-2.3,0.9) and (-1.4,1.5) .. (0,1.5); \node[acc, anchor=west] at (2.0,1.15) {static limit}; % horizon: sphere radius 1.5 (touches static limit at poles) \draw[black, very thick, fill=black!8] (0,0) circle (1.5); \node[black] at (0,0) {horizon}; \node[black, anchor=west] at (1.05,0.55) {ergosphere}; \end{tikzpicture} $$ ## Frame dragging and the ZAMO The compulsory co-rotation has a precise form. A photon or particle with zero angular momentum still acquires an angular velocity as seen from infinity: $$ \omega(r,\theta) = \frac{\d\phi}{\d t} = -\frac{g_{t\phi}}{g_{\phi\phi}} = \frac{r_s r\, a\, c}{(r^2+a^2)\Sigma + r_s r\, a^2\sin^2\theta}. $$ An observer following this $\omega$ is a **zero-angular-momentum observer (ZAMO)**, the local standard of "non-rotating" that the geometry itself defines. Far away $\omega \to r_s r a c / (r^2)^2 = J\,(2G/c^2)/r^3$ falls off as $1/r^3$ — the Lense–Thirring precession that gyroscopes in Earth orbit measure. At the horizon $\omega$ reaches a constant, $$ \Omega_H = \frac{a\,c}{r_+^2 + a^2}, $$ the **angular velocity of the horizon**. A Kerr hole rotates rigidly: its horizon turns at a single rate $\Omega_H$ like a solid body, a fact the next lesson uses in the first law of black-hole mechanics. $$ % caption: Frame dragging. A particle dropped with no angular momentum from far % away spirals in the direction of the hole spin; the induced angular velocity % grows from a 1/r-cubed tail far out to the rigid horizon rate near r-plus. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} \fill[black] (0,0) circle (0.7); \draw[black, very thick] (0,0) circle (0.7); \node[black, font=\scriptsize] at (0,0) {hole}; % spin arrow around hole \draw[acc, thick, ->] (0.95,0.35) arc (25:150:1.0); \node[acc, anchor=south] at (0,1.05) {spin}; % infalling spiral (no ang. momentum, gets dragged) winding into the horizon \draw[black, very thick, ->] (3.4,0.15) .. controls (2.3,0.5) and (1.5,0.95) .. (0.85,0.85) .. controls (0.15,0.72) and (-0.35,0.2) .. (-0.2,-0.45) .. controls (-0.05,-0.72) and (0.3,-0.72) .. (0.5,-0.5); \node[black, anchor=west] at (2.4,0.55) {dragged inward}; \end{tikzpicture} $$ ## The Penrose process Because $\partial_t$ becomes spacelike inside the ergosphere, the conserved energy $E = -c\,p_t$ of a particle can be negative there without violating any local law: locally the particle still has positive energy, but its energy _as measured at infinity_ can be negative. Penrose turned this into an extraction mechanism. Send a particle of energy $E_0 > 0$ into the ergosphere and let it split into two fragments, $$ E_0 = E_1 + E_2. $$ Arrange the split so that fragment 2 has negative conserved energy, $E_2 < 0$, and falls through the horizon; fragment 1 then escapes with $$ E_1 = E_0 - E_2 > E_0. $$ The escaping fragment carries out more energy than the original particle brought in. The deficit is paid by the hole: swallowing a negative-energy, negative-angular- momentum fragment reduces both the hole's mass-energy and its spin. Energy has been mined from the rotation. $$ % caption: The Penrose process. An incoming particle splits inside the ergosphere; % one fragment falls through the horizon carrying negative energy and angular % momentum, so the other escapes with more energy than the incoming particle had. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} \draw[black, very thick, fill=black!10] (0,0) circle (0.9); \node[black, font=\scriptsize] at (0,0) {hole}; \draw[thick] (0,0) ellipse (2.2 and 1.5); \node[anchor=south] at (0,1.55) {ergosphere}; % incoming \draw[black, very thick, ->] (3.6,1.7) -- (1.4,0.6); \node[black, anchor=west] at (2.6,1.6) {incoming E0}; % split point \fill[black!70] (1.4,0.6) circle (1.4pt); % fragment escaping \draw[acc, very thick, ->] (1.4,0.6) -- (3.4,-0.9); \node[acc, anchor=west] at (2.6,-0.9) {escapes with more}; % fragment falling in (negative energy) \draw[black, very thick, ->] (1.4,0.6) -- (0.35,0.0); \node[black, anchor=south, font=\scriptsize] at (0.9,0.35) {E below 0}; \end{tikzpicture} $$ The mining cannot continue without limit. Each Penrose extraction lowers $J$, and when $J = 0$ the ergosphere is gone and the process stops. The extractable energy is the **rotational energy**, the difference between the hole's total mass-energy and its irreducible mass $M_{\text{irr}}$, $$ M_{\text{irr}}^2 = \frac{1}{2}\left(M^2 + \sqrt{M^4 - (Jc/G)^2}\right), $$ and the fraction available is up to $1 - 1/\sqrt{2} \approx 29\%$ of the total mass-energy for an extremal hole. The irreducible mass never decreases under the Penrose process — the first hint of the area theorem and black-hole [thermodynamics](/relativity/black-holes/black-hole-thermodynamics), since $M_{\text{irr}}$ is proportional to the square root of the horizon area. ## The no-hair theorem Every solution above is fixed by a short list of numbers. A collapsing star may begin with an intricate distribution of matter, magnetic fields, and multipole moments, but the final stationary hole retains almost none of it. > **Theorem (No-hair).** A stationary, asymptotically flat, electrovacuum black > hole is completely characterized by three externally observable parameters: > its mass $M$, angular momentum $J$, and electric charge $Q$. All higher > multipole moments are fixed functions of these three; no further information > about the collapsed matter is observable from outside. The general solution is > the Kerr–Newman metric, which reduces to Kerr ($Q=0$), Reissner–Nordström > ($J=0$), and Schwarzschild ($J=Q=0$). The three parameters coincide with the quantities protected by conservation laws with long-range fields to carry them to infinity: mass by the gravitational field, angular momentum by frame dragging, charge by the electromagnetic field. A baryon number, a lepton number, or the detailed shape of the infalling star has no long-range field and leaves no external trace; those quantities are said to fall past the horizon and become inaccessible. The apparent loss of that information is sharpened into a genuine puzzle once the hole is allowed to radiate, the theme the [next lesson](/relativity/black-holes/black-hole-thermodynamics) develops. $$ % caption: The no-hair result. Progenitors with arbitrary shape, composition, and % field structure collapse to a stationary hole labelled by only three numbers: % mass, spin, and charge. \begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0] \definecolor{acc}{HTML}{4A6FA5} % messy inputs \node[black, draw=black, minimum width=1.9cm, minimum height=0.7cm] (a) at (0,1.6) {lumpy star}; \node[black, draw=black, minimum width=1.9cm, minimum height=0.7cm] (b) at (0,0.4) {magnetic lines}; \node[black, draw=black, minimum width=1.9cm, minimum height=0.7cm] (c) at (0,-0.8) {odd multipoles}; % arrows to hole \draw[black, ->] (a.east) -- (2.6,0.9); \draw[black, ->] (b.east) -- (2.6,0.4); \draw[black, ->] (c.east) -- (2.6,-0.1); \node[draw=black, very thick, minimum width=1.7cm, minimum height=1.2cm] (h) at (3.5,0.4) {hole}; % output labels \draw[acc, ->] (h.east) -- (5.4,1.1) node[right, acc] {mass M}; \draw[acc, ->] (h.east) -- (5.4,0.4) node[right, acc] {spin J}; \draw[acc, ->] (h.east) -- (5.4,-0.3) node[right, acc] {charge Q}; \end{tikzpicture} $$ [^hartle-kerr]: **Hartle**, _Gravity_, Ch. 15 — the Kerr metric in Boyer–Lindquist coordinates, frame dragging and the zero-angular-momentum observer, the static limit and ergosphere, the Penrose process, and the irreducible mass. [^carroll-kerr]: **Carroll**, _Lecture Notes on General Relativity_, §7 — the Kerr and Reissner–Nordström solutions, their horizons as roots of $\Delta$, the ring singularity, extremality and cosmic censorship, and the no-hair characterization by mass, spin, and charge.