Rotating and Charged Black Holes

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 Ω^{2}, f (r) = 1 - \frac{r _{s}}{r} + \frac{r _{Q}^{2}}{r ^{2}},

with $r_{s} = 2 GM / c^{2}$ and the charge length $r_{Q}^{2} = G Q^{2} / (4 π ε_{0} c^{4})$ . The horizons are the roots of $f (r) = 0$ :

r_{\pm} = \frac{r _{s}}{2} \pm \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.

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.

The Kerr metric

A rotating hole of mass $M$ and angular momentum $J$ is described by the Kerr solution. Writing the spin parameter $a = J / (M c)$ (a length) and, in Boyer–Lindquist coordinates, the abbreviations

Σ = r^{2} + a^{2} cos^{2} θ, Δ = r^{2} - r_{s} r + a^{2},

the line element is

d s^{2} = - (1 - \frac{r _{s} r}{Σ}) c^{2} d t^{2} - \frac{2 r _{s} r a sin ^{2} θ}{Σ} c d t d ϕ + \frac{Σ}{Δ} d r^{2} + Σ d θ^{2} + (r^{2} + a^{2} + \frac{r _{s} r a ^{2} sin ^{2} θ}{Σ}) sin^{2} θ d ϕ^{2} .

Three features distinguish it from Schwarzschild.

The cross term $d t d ϕ$ 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 $ϕ \to - ϕ$ , so the hole's rotation is imprinted on the metric.
The horizon is $Δ = 0$ , at $r_{\pm} = \frac{r _{s}}{2} \pm \frac{r _{s}^{2}}{4} - a^{2},$ the same algebra as Reissner–Nordström with $a$ in place of $r_{Q}$ . A horizon exists only for $a \leq r_{s} /2$ , i.e. $J \leq G M^{2} / c$ ; 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. $Σ = 0$ requires $r = 0$ and $θ = π /2$ 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 $(r, θ, ϕ)$ — 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} = - (1 - \frac{r _{s} r}{Σ}) c^{2}

vanishes at the static limit surface

r_{static} (θ) = \frac{r _{s}}{2} + \frac{r _{s}^{2}}{4} - a^{2} cos^{2} θ .

Inside it, $g_{tt} > 0$ : no timelike worldline can have $d ϕ = 0$ , 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_{static} (θ),

is the ergosphere.

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.

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:

ω (r, θ) = \frac{d ϕ}{d t} = - \frac{g _{tϕ}}{g _{ϕϕ}} = \frac{r _{s} r a c}{( r ^{2} + a ^{2} ) Σ + r _{s} r a ^{2} sin ^{2} θ} .

An observer following this $ω$ is a zero-angular-momentum observer (ZAMO), the local standard of non-rotating that the geometry itself defines. Far away $ω \to r_{s} r a c / (r^{2})^{2} = J (2 G / c^{2}) / r^{3}$ falls off as $1/ r^{3}$ — the Lense–Thirring precession that gyroscopes in Earth orbit measure. At the horizon $ω$ reaches a constant,

Ω_{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 $Ω_{H}$ like a solid body, a fact the next lesson uses in the first law of black-hole mechanics.

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.

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.

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.

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 $M_{irr}$ ,

M_{irr}^{2} = \frac{1}{2} (M^{2} + M^{4} - (J c / G)^{2}),

and the fraction available is up to $1 - 1/ 2 \approx 29%$ 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, since $M_{irr}$ 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.

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 develops.

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.