Schwarzschild Geometry and Black Holes
The first exact solution of the field equations of The Einstein Field Equations was found within weeks of their publication [Schwarzschild:1916], and it carries a remarkable fraction of the theory's observable content: the orbits behind the classical tests of The Equivalence Principle and Classical Tests, the innermost stable orbit and photon sphere that set the scales of accretion, and — once its coordinate singularity was understood — the event horizon and the black hole. This chapter derives the solution, studies its geodesics, extends it maximally, and follows gravitational collapse to the singularity theorems that make the outcome generic [Penrose:1965].
The second half opens the black-hole family: the Kerr rotating solution [Kerr:1963], its charged relatives, and the uniqueness (no-hair) theorems that reduce every stationary black hole to two observable parameters [Israel:1967] [Carter:1971]. The shadow of the horizon is derived as a prediction here; the observations — stellar orbits about the galactic centre, the Event Horizon Telescope images, ringdown spectroscopy — are deferred with their full apparatus to the black-hole experiment chapter, Experiment: Black-Hole Observations, and to Experiment: Gravitational Waves. Standard treatments are [Misner:1973] [Wald:1984].
The Schwarzschild solution
Derivation and the 1916 paper
A gravitational field is static when spacetime admits a timelike Killing vector (Definition 13.139) whose orthogonal distribution is integrable — there is a time coordinate \(t\) that the geometry neither depends on nor mixes with space — and spherically symmetric when the rotation group acts by isometries with two-dimensional spacelike orbits. On each orbit sphere the metric must be a multiple of the round metric, and the multiple defines the areal radius \(r\): the sphere labelled \(r\) has area \(4\pi r^{2}\) by construction, whatever “distance to the centre” turns out to mean. Wherever \(\dd r\neq0\) these choices reduce the metric to two unknown functions of \(r\) [Wald:1984],
Staticity is assumed here for transparency and proved redundant in Section 45.1.2.
Write \(x^{0}:=ct\), so that primes denote \(\dd/\dd r\) throughout. A mechanical application of Equation (13.299) (torsion zero, per Geometric Formulation of Gravity) gives the nonvanishing Christoffel symbols of Equation (45.1),
and contracting the Riemann tensor Equation (13.306) as in Definition 13.153 yields the Ricci components
with \(R_{\varphi\varphi}=\sin^{2}\theta\,R_{\theta\theta}\) and all off-diagonal components zero.
The unique static, spherically symmetric, asymptotically flat solution of the vacuum field equations \(R_{\mu\nu}=0\) (The Einstein Field Equations) exterior to a body of mass \(M\) is
with the Schwarzschild radius
Rests on Equation (13.306), Theorem 13.150 and Postulate 43.1.
Derivation. Derives Theorem 45.1. Vacuum demands that Equations (45.3) to (45.5) vanish. The combination
kills every second-derivative term at a stroke: \(\alpha+\beta\) is constant, and asymptotic flatness (\(\alpha,\beta\to0\) as \(r\to\infty\)) sets the constant to zero, so \(\beta=-\alpha\). Inserting this into \(R_{\theta\theta}=0\) gives
which integrates to \(r\ee^{2\alpha}=r-C\) with \(C\) an integration constant:
Direct substitution confirms that \(R_{00}=0\) then holds identically — as it must, since the contracted Bianchi identity (Equation (13.308)) makes the field equations degenerate.
The constant \(C\) is fixed by the Newtonian limit. A slow test particle on a geodesic (Postulate 43.1) has \(\dd t/\dd\tau\simeq1\) and, by Equation (45.2),
at large \(r\). Newtonian gravity requires \(\ddot{r}=-GM/r^{2}\) for the same source, so \(C=2GM/c^{2} =r_{\mathrm{s}}\), which is Equations (45.6) and (45.7). Uniqueness within the ansatz is manifest — every step was forced — and the ansatz itself is generic by the reduction above.
∎The scale Equation (45.7) is tiny for ordinary bodies: \(r_{\mathrm{s}}=2.95\,\mathrm{km}\) for the Sun (\(M_{\odot}=1.989\times 10^{30}\,\mathrm{kg}\)) and \(8.87\,\mathrm{mm}\) for the Earth, which is why every solar-system test of Experiment: The Classical Tests of General Relativity probes the far field \(r\gg r_{\mathrm{s}}\), where Equation (45.6) deviates from flatness at the level \(r_{\mathrm{s}}/r\sim10^{-6}\) or less.
Schwarzschild found the solution in the winter of 1915–16, on the Russian front, using a radial coordinate for which his “\(r\)” was not the areal radius [Schwarzschild:1916]; the form Equation (45.6), in the coordinates now universally called Schwarzschild coordinates, is due independently to Droste [Droste:1917]. Neither author regarded the locus \(r=r_{\mathrm{s}}\) as physical: for every known star it lies far inside the matter, where the vacuum solution does not apply. The interior problem — matching Equation (45.6) to a matter-filled region and the resulting structure equations — is developed with the compact stars that need it, in Compact Stars and Relativistic Astrophysics.
Two static observers at radii \(r_{1}<r_{2}\) exchange light. Staticity means successive wave crests follow congruent paths, so the coordinate time between crests is the same at both ends, while each observer's proper time is \(\dd\tau=\sqrt{1-r_{\mathrm{s}}/r}\;\dd t\) by Equation (45.6). The received frequency is therefore shifted by
whose weak-field form is the tower redshift measured by Pound, Rebka and Snider and the clock-rate budget of satellite navigation (The Equivalence Principle and Classical Tests and Experiment: The Classical Tests of General Relativity). As \(r_{1}\to r_{\mathrm{s}}\) the shift diverges: the horizon is an infinite-redshift surface for static sources, a fact that controls the appearance of collapse in Section 45.4.1.
Birkhoff's theorem
Staticity was an input above. It is in fact a consequence of spherical symmetry alone — a theorem proved by Jebsen two years before the customary source [Jebsen:1921] [Birkhoff:1923].
Every spherically symmetric solution of the vacuum field equations is locally isometric to the Schwarzschild solution Equation (45.6). In particular the field outside any spherically symmetric body is static, however violently the body itself pulsates, collapses or explodes, so long as it does so spherically. Rests on Theorem 45.1 and Equation (13.306).
Derives Theorem 45.4. Spherical symmetry alone reduces the metric, wherever the areal radius has spacelike gradient, to the form Equation (45.1) with \(t\)-dependent functions \(\alpha(t,r)\), \(\beta(t,r)\): the orbit spheres fix the angular block, the areal radius supplies \(r\), and the residual freedom in relabelling \(t\) removes the single cross term \(g_{tr}\) [Misner:1973] [Wald:1984]. The time dependence adds exactly one new nonvanishing Ricci component,
with \(\pp_{0}=\pp/\pp x^{0}\), and adds to \(R_{00}\) and \(R_{rr}\) terms each of which carries at least one factor of \(\pp_{0}\beta\) or \(\pp_{0}^{2}\beta\). Vacuum makes \(R_{0r}\) vanish, so \(\beta=\beta(r)\); all the new terms in \(R_{00}\) and \(R_{rr}\) then vanish identically, and the static computation of Theorem 45.1 applies verbatim at each instant. It gives \(\pp_{r}(\alpha+\beta)=0\), i.e. \(\alpha=-\beta+f(t)\); the redefinition \(\dd\tilde{t}=\ee^{f(t)}\dd t\) absorbs \(f\), and \(R_{\theta\theta}=0\) integrates as before to \(\ee^{-2\beta}=1-C(t)/r\) — with \(C\) in fact constant because \(\beta\) is \(t\)-independent. The result is Equation (45.6).
∎A radially pulsating star cannot radiate gravitationally: its exterior field is exactly the static Equation (45.6) at all times, so no disturbance whatever propagates outward. Gravitational radiation begins at the quadrupole (Gravitational-Wave Theory), and Theorem 45.4 is the exact, nonlinear statement of which the absence of monopole terms in the quadrupole formula is the weak-field shadow. Rests on Theorem 45.4.
Theorem 45.4 is also the tool that makes collapse computable: whatever happens inside a collapsing spherical star, the exterior is Equation (45.6) with a constant \(M\). That is the key to Section 45.4.1.
Geodesics and orbits
The effective potential
The metric Equation (45.6) has two obvious Killing vectors, \(\vect{\xi}=\pp_{t}\) and \(\vect{\eta}=\pp_{\varphi}\), and each gives a conserved quantity along geodesics (Proposition 13.141). For a timelike geodesic with four-velocity \(u^{\mu}=\dd x^{\mu}/\dd\tau\), normalized as always to \(g_{\mu\nu}u^{\mu}u^{\nu}=-c^{2}\) (Proposition 40.3, in this part's signature), define the conserved specific energy and specific angular momentum
Spherical symmetry confines each orbit to a plane: with initial conditions \(\theta=\pi/2\), \(\dd\theta/\dd\tau=0\), the \(\theta\) geodesic equation gives \(\dd^{2}\theta/\dd\tau^{2}=0\), so the motion stays equatorial, and we take \(\theta=\pi/2\), \(L=r^{2} \dd\varphi/\dd\tau\) throughout.
Along an equatorial timelike geodesic,
with the effective potential
Rests on Theorem 45.1, Proposition 13.141 and Proposition 40.3.
Derivation. Derives Proposition 45.6. Write out the normalization \(g_{\mu\nu}u^{\mu}u^{\nu}=-c^{2}\) in the equatorial plane,
and eliminate the cyclic velocities with Equation (45.9): \(\dd t/\dd\tau=\mathcal{E}/\bigl(c^{2}(1-r_{\mathrm{s}}/r)\bigr)\) and \(\dd\varphi/\dd\tau=L/r^{2}\). Multiplying through by \(\tfrac{1}{2}(1-r_{\mathrm{s}}/r)\),
Expanding the product and using \(r_{\mathrm{s}}c^{2}=2GM\) gives Equations (45.10) and (45.11) exactly — no weak-field approximation has been made.
∎The first two terms of Equation (45.11) are precisely the Newtonian effective potential of the Kepler problem (Equation (27.32) with \(U=-GM/r\)), with \(\mathcal{E}_{\mathrm{N}}\) reducing at \(r\gg r_{\mathrm{s}}\), where \(\mathcal{E}\simeq c^{2}+E_{\mathrm{Newt}}\), to the Newtonian specific energy. The whole of general relativity's modification of bound motion is the single attractive term \(-GML^{2}/(c^{2}r^{3})\), and it changes the phase portrait qualitatively: in Newtonian mechanics any \(L\neq0\) erects an infinite centrifugal barrier (Section 27.3.2), whereas here \(V_{\mathrm{eff}}\to-\infty\) as \(r\to0\) for every \(L\). The barrier survives only for \(L>L_{\mathrm{crit}}=2\sqrt{3}\,GM/c\): below that angular momentum a particle is captured from any energy — there are orbits that spiral into the centre, which Newtonian gravity forbids.
In terms of the inverse radius \(u:=1/r\), an equatorial timelike geodesic with \(L\neq0\) obeys
Dropping the \(u^{2}\) term gives the Binet equation of the Kepler problem (Equation (27.28) with the inverse-square force), whose solutions are the closed conics of Section 27.4. Rests on Proposition 45.6 and Equation (27.28).
Derivation. Derives Proposition 45.8. Since \(\dd\varphi/\dd\tau=Lu^{2}\),
so the first integral of Proposition 45.6 reads
Differentiating with respect to \(\varphi\) and dividing by \(2L^{2}\,\dd u/\dd\varphi\) (legitimate except at turning points, which are isolated),
which is Equation (45.12) on using \(r_{\mathrm{s}}c^{2}=2GM\).
∎A bound orbit in the field of a spherical mass \(M\) does not close: the periastron advances in the direction of the motion by
per revolution, to lowest order in \(GM/(c^{2}a)\), for an orbit of semi-major axis \(a\) and eccentricity \(e\). Two measurements span nine orders of magnitude in central mass and confirm the same formula. For Mercury the advance is the 43 seconds of arc per century that Le Verrier isolated as a residual half a century before there was a theory to explain it [LeVerrier:1859] [Einstein:1915a]; for the star S2 orbiting the compact mass at the Galactic centre it is about twelve minutes of arc per sixteen-year orbit, detected by near-infrared interferometry [Abuter:2020]. Rests on Equation (45.12) and Proposition 45.8.
Derivation. Derives Phenomenon 45.9. The orbit equation Equation (45.12) is exact; the advance is its lowest-order consequence. The relativistic term is smaller than the linear terms by \(3L^{2}u^{2}/c^{2}\sim v^{2}/c^{2}\), so one solves perturbatively about the Keplerian ellipse: substituting the conic \(u_{0}=(GM/L^{2})(1+e\cos\varphi)\) into the \(u^{2}\) term produces, among bounded forced terms, one resonant term proportional to \(\cos\varphi\), and its secular growth rotates the apsidal line by Equation (45.13) per revolution. That computation is carried out in full — including the numerical budget that turns it into Mercury's 43 seconds of arc per century — in the derivation of Phenomenon 53.1 in Experiment: The Classical Tests of General Relativity, which takes Equation (45.12) from this chapter as its input; it is not repeated here. For S2 the same formula, with the mass and orbital elements measured in Experiment: Black-Hole Observations, gives the twelve-minute advance per orbit reported by the GRAVITY interferometer [Abuter:2020].
∎Circular orbits, the ISCO and the photon sphere
Circular orbits sit at extrema of Equation (45.11), and the systematic treatment of what follows — including the rotating generalization — is [Bardeen:1972].
Circular timelike geodesics of radius \(r\) exist iff \(r>3GM/c^{2}\), with
and carry specific energy
They are stable iff \(r\geq r_{\mathrm{ISCO}}=6GM/c^{2}\); at the innermost stable circular orbit, \(\mathcal{E}=\sqrt{8/9}\,c^{2}\) and \(L=2\sqrt{3}\,GM/c\). Rests on Proposition 45.6.
Derivation. Derives Proposition 45.10. Write \(m:=GM/c^{2}\). Setting \(V_{\mathrm{eff}}'(r)=0\) in Equation (45.11) and multiplying by \(r^{4}\),
which is Equation (45.14); a positive solution requires \(r>3m\), and \(L^{2}\to\infty\) as \(r\to3m\). For the energy, evaluate \(\mathcal{E}^{2}=c^{4}+2c^{2}V_{\mathrm{eff}}(r)\) on Equation (45.14): putting the terms of Equation (45.11) over the denominator \(2r(r-3m)\),
whose square root is Equation (45.15). For stability, differentiate Equation (45.11) twice and insert Equation (45.14):
positive for \(r>6m\), negative for \(3m<r<6m\): the inner branch of circular orbits is unstable, and the marginal case is \(r_{\mathrm{ISCO}}=6m=6GM/c^{2}\). There Equations (45.14) and (45.15) give \(L^{2}=GM\cdot36m^{2}/(3m)=12G^{2}M^{2}/c^{2}\) and \(\mathcal{E}/c^{2}=(1-\tfrac13)/\sqrt{1-\tfrac12} =\tfrac{2}{3}\sqrt{2}=\sqrt{8/9}\).
∎As \(L\to\infty\) the unstable branch radius tends to \(3GM/c^{2}\): the faster the particle, the closer its unstable circular orbit sits to the limiting value, which is the circular orbit of light itself. The same radius reappears in Section 45.7 as the top of the null effective potential, Equation (45.59), where its instability makes it the rim of the shadow rather than a place where light accumulates. Nothing in the Kepler problem corresponds to either radius: both are creatures of the \(r^{-3}\) term of Equation (45.11).
Matter accreting onto a compact object radiates a fraction of order ten per cent of its rest energy, an order of magnitude above the \(0.7\%\) that hydrogen burning can release. This is what allows a quasar to outshine its entire host galaxy from a region light-days across [Schmidt:1963], and it is the argument that identified quasars with accreting massive black holes in the nuclei of ordinary galaxies [LyndenBell:1969]. The scale is set by the innermost stable circular orbit: circular geodesics of the Schwarzschild geometry exist inward to \(r=3GM/c^{2}\) but are stable only outside
where a unit mass has specific energy \(\sqrt{8/9}\,c^{2}\). Matter spiralling slowly inward through a thin disc therefore radiates at most
of its rest energy before it crosses the horizon [Bardeen:1972] [Misner:1973]. For a rapidly rotating hole the innermost stable orbit moves inward and the figure rises toward about \(0.4\), which is why the observed efficiencies are informative about spin. Rests on Proposition 45.10.
Derivation. Derives Phenomenon 45.12. In a geometrically thin, optically thick disc (Experiment: Black-Hole Observations for the observational side) the gas drifts inward through a sequence of nearly circular orbits, each lasting many orbital periods, so its specific energy at radius \(r\) is Equation (45.15); the difference from \(c^{2}\) has been carried away by radiation. The drift ends at \(r_{\mathrm{ISCO}}\): inside, no stable circular orbit exists (Equation (45.16)), and the remaining infall to the horizon is dynamical, too fast to radiate efficiently, so the energy still aboard is advected in. The radiated fraction is therefore the binding energy at the innermost stable orbit,
which is Equation (45.18). For a rotating hole the co-rotating ISCO moves inward — to \(GM/c^{2}\) in the extremal limit — and the same computation on the Kerr geometry gives binding energies up to \(1-1/\sqrt{3}=0.42\); those orbits are catalogued in [Bardeen:1972], and the spin dependence is taken up in Section 45.5.1.
∎Null geodesics: deflection and delay
Light follows null geodesics, for which proper time is unavailable; let \(\lambda\) be an affine parameter with dimensions of length and define the conserved quantities
for an equatorial ray (\(E\) dimensionless, \(L\) a length; both are the constants of Proposition 13.141 in convenient normalizations). The null condition \(g_{\mu\nu}\,\dd x^{\mu}\dd x^{\nu}=0\), treated exactly as in the derivation of Proposition 45.6, gives the first integral
Passing to \(u=1/r\) as in Proposition 45.8,
the orbit equation Equation (45.12) with the mass term absent — it came from the rest mass. As \(r\to\infty\) the first form of Equation (45.21) degenerates to \((\dd u/\dd\varphi)^{2}+u^{2}=E^{2}/L^{2}\), whose straight-line solution passes the centre at perpendicular distance
the impact parameter: the ratio of the two conserved quantities is the one geometric datum a distant observer can assign to a ray.
Solving Equation (45.21) perturbatively about the straight line \(u_{0}=\sin\varphi/b\) shifts the two asymptotes of the ray and bends it, for \(b\gg r_{\mathrm{s}}\), through
which is Equation (53.3). The computation — and the observational history from the 1919 eclipse through modern radio interferometry, where the formula is confirmed to a part in \(10^{4}\) — is carried out in full in Experiment: The Classical Tests of General Relativity (derivation of Phenomenon 53.2), taking Equation (45.21) from this chapter as its input [Dyson:1920]. Half the value comes from the equivalence principle alone; the other half is the spatial curvature visible in Equation (45.23) below.
The substitution \(r=\rho\left(1+r_{\mathrm{s}}/(4\rho)\right)^{2}\) brings Equation (45.6) to the isotropic form
whose spatial part is conformally flat. To first order in \(r_{\mathrm{s}}/\rho\) it reads \(\dd s^{2}\simeq-\left(1-r_{\mathrm{s}}/\rho\right)c^{2}\dd t^{2} +\left(1+r_{\mathrm{s}}/\rho\right) \left(\dd\rho^{2}+\rho^{2}\dd\Omega^{2}\right)\). Rests on Theorem 45.1.
Derivation. Derives Proposition 45.14. Write \(x:=r_{\mathrm{s}}/(4\rho)\). Differentiating the substitution, \(\dd r=\left(1+x\right)\left(1-x\right)\dd\rho\), and a two-line computation gives
Hence \(\left(1-r_{\mathrm{s}}/r\right)^{-1}\dd r^{2} =\left(1+x\right)^{4}\dd\rho^{2}\) and \(r^{2}\dd\Omega^{2}=\left(1+x\right)^{4}\rho^{2}\dd\Omega^{2}\): every spatial term acquires the same conformal factor, which is Equation (45.23). Expanding \(\left((1-x)/(1+x)\right)^{2}\simeq1-4x\) and \(\left(1+x\right)^{4}\simeq1+4x\) gives the first-order form.
∎In the isotropic form the coordinate speed of light is isotropic and equals \(c\left(1-r_{\mathrm{s}}/\rho\right)\) to first order — one factor from time dilation, one from spatial curvature. Integrating the deficit along a straight path between radii \(r_{1}\) and \(r_{2}\) passing the mass at distance \(b\) gives the round-trip excess delay
which is Equation (53.5), proposed by Shapiro as the fourth classical test [Shapiro:1964]. The integration, and the radar and spacecraft measurements that verify it — now at the \(2\times10^{-5}\) level via the Cassini link — are in Experiment: The Classical Tests of General Relativity (derivation of Phenomenon 53.4), which starts from Equation (45.23).
The horizon
Coordinate and curvature singularities
At \(r=r_{\mathrm{s}}\) the metric component \(g_{rr}\) of Equation (45.6) diverges and \(g_{tt}\) vanishes. Whether anything physical diverges there is a question the coordinates cannot answer; curvature invariants can, because they are scalars.
In the orthonormal frame of a static observer the nonvanishing components of the Riemann tensor of Equation (45.6) are
(up to index symmetries), so the Kretschmann scalar is
It is finite at \(r=r_{\mathrm{s}}\) and diverges only at \(r=0\): the horizon is a coordinate singularity, the centre a curvature singularity. Rests on Theorem 45.1 and Equation (13.306).
Derivation. Derives Proposition 45.16. Mechanical evaluation of Equation (13.306) on Equation (45.2) with \(\ee^{2\alpha}=\ee^{-2\beta}=1-r_{\mathrm{s}}/r\), followed by projection onto the orthonormal frame \(\hat{e}_{0}=(1-r_{\mathrm{s}}/r)^{-1/2}\,c^{-1}\pp_{t}\), \(\hat{e}_{r}=(1-r_{\mathrm{s}}/r)^{1/2}\pp_{r}\), \(\hat{e}_{\theta}=r^{-1}\pp_{\theta}\), \(\hat{e}_{\varphi}=(r\sin\theta)^{-1}\pp_{\varphi}\), gives Equation (45.24). In the contraction each of the six independent pair components enters with multiplicity four and, in an orthonormal frame, with a positive square, which is the bracket in Equation (45.25).
∎By geodesic deviation (Equation (43.3)) the frame components Equation (45.24) are tidal accelerations per unit separation: radial stretching \(2GM/r^{3}\), transverse compression \(GM/r^{3}\), in exact correspondence with the Newtonian tidal tensor \(\pp_{i}\pp_{j}\Phi\) — and finite at the horizon, where they equal \(2GM/r_{\mathrm{s}}^{3}=c^{6}/(4G^{2}M^{2})\) radially. The scale falls with mass: for \(M=M_{\odot}\) it is \(10^{10}\,/\mathrm{s}^{2}\) — an infalling body is shredded long before reaching the horizon — while for the \(4.3\times10^{6}\,M_{\odot}\) object at the Galactic centre it is \(5.6\times 10^{-4}\,/\mathrm{s}^{2}\), milder than the tide in a fast elevator. Locally the horizon of a large black hole is unremarkable; what distinguishes it is global, as the next construction shows.
The diagnosis of \(r=r_{\mathrm{s}}\) as an artefact is completed by exhibiting coordinates regular there. Define the tortoise coordinate and the advanced time
so that \(\dd r_{*}=\left(1-r_{\mathrm{s}}/r\right)^{-1}\dd r\) and ingoing radial light rays are \(v=\text{const}\). Substituting \(c\,\dd t=\dd v-\dd r_{*}\) into Equation (45.6),
the ingoing Eddington–Finkelstein form: every component is regular for all \(r>0\), and the determinant \(g=-r^{4}\sin^{2}\theta\) never vanishes there. The coordinate \(v\) was introduced by Eddington [Eddington:1924a], its meaning recognized by Lemaître [Lemaitre:1933] and sharpened by Finkelstein [Finkelstein:1958].
In the geometry Equation (45.27), every future-directed causal curve at \(r\leq r_{\mathrm{s}}\) has \(\dd r\leq0\), with equality only for radial null curves on \(r=r_{\mathrm{s}}\) itself. The surface \(r=r_{\mathrm{s}}\) is a null hypersurface that can be crossed inward but not outward: the event horizon. Rests on Theorem 45.1.
Derivation. Derives Proposition 45.18. Radial null directions of Equation (45.27) satisfy \(\dd v\left[\left(1-r_{\mathrm{s}}/r\right)\dd v-2\,\dd r\right]=0\): either \(\dd v=0\), the ingoing family, along which \(r\) strictly decreases toward the future; or \(\dd r/\dd v=\tfrac{1}{2}\left(1-r_{\mathrm{s}}/r\right)\), the would-be outgoing family — positive outside the horizon, zero on it, negative inside. A causal curve lies inside or on the local light cone spanned by the two, so inside the horizon both bounding directions, and hence every causal direction, have \(\dd r<0\). On \(r=r_{\mathrm{s}}\) the normal \(n_{\mu}=\pp_{\mu}\left(r-r_{\mathrm{s}}\right)\) has \(n^{\mu}n_{\mu}=g^{rr}=1-r_{\mathrm{s}}/r=0\): the surface is null, generated by the marginally outgoing rays that run along it forever.
∎Nothing locally distinguishes the horizon — the curvature there is finite and can be arbitrarily small (Remark 45.17) — and no signal sent from inside ever reaches a distant observer. The horizon is a globally defined surface: the boundary of the region from which future null infinity cannot be reached [Hawking:1973]. Inside it the roles of \(r\) and \(t\) interchange — \(g_{rr}<0\), \(g_{tt}>0\) in Equation (45.6) — so decreasing \(r\) is there a direction in time: reaching \(r=0\) is as unavoidable as reaching tomorrow.
The Kruskal–Szekeres extension
Eddington–Finkelstein coordinates extend the exterior across the future horizon. The maximal analytic extension, found independently by Kruskal and Szekeres [Kruskal:1960] [Szekeres:1960], is reached by one further exponential substitution.
On the exterior \(r>r_{\mathrm{s}}\) define null coordinates \(u:=ct-r_{*}\), \(v:=ct+r_{*}\) and
Then
with \(r(T,X)>0\) determined implicitly by
The metric Equation (45.29) is regular on the whole domain \(T^{2}-X^{2}<1\), which is the maximal analytic extension of the Schwarzschild geometry. Rests on Theorem 45.1.
Derivation. Derives Proposition 45.19. In null coordinates the \((t,r)\) block of Equation (45.6) is \(-\left(1-r_{\mathrm{s}}/r\right)\dd u\,\dd v\). From Equation (45.28),
and by Equation (45.26), \(\ee^{r_{*}/r_{\mathrm{s}}} =\left(r/r_{\mathrm{s}}-1\right)\ee^{r/r_{\mathrm{s}}}\) outside the horizon. Hence
since \(\left(1-r_{\mathrm{s}}/r\right)/\left(r/r_{\mathrm{s}}-1\right) =r_{\mathrm{s}}/r\). Rewriting \(\dd U\,\dd V\) in \((T,X)\) gives Equation (45.29); the identities \(UV=T^{2}-X^{2}\) and \(UV=-\ee^{r_{*}/r_{\mathrm{s}}}\) give the first of Equation (45.30), and \(-V/U=\ee^{ct/r_{\mathrm{s}}}\) the second. The right-hand side of Equation (45.30) is a monotonic function of \(r\) mapping \((0,\infty)\) onto \((-1,\infty)\), so \(r\) is well defined and positive exactly where \(X^{2}-T^{2}>-1\), and the conformal factor in Equation (45.29) is analytic and nonzero there. The extension is maximal because every geodesic either runs to infinite affine parameter or ends on the curvature singularity \(T^{2}-X^{2}=1\), i.e. \(r=0\) (Proposition 45.16).
∎The two lines \(T=\pm X\) divide the domain into four wedges. Region I (\(X>\abs{T}\)) is the original exterior, with static observers on hyperbolae of constant \(X^{2}-T^{2}\) and constant-\(t\) lines through the origin. Region II (\(T>\abs{X}\)) is the black-hole interior: every future-directed causal curve entering it ends on the singularity \(T=\sqrt{1+X^{2}}\), the interior statement of Proposition 45.18. Region III (\(T<-\abs{X}\)) is the time-reverse — a white hole, out of which everything must pass — and region IV (\(X<-\abs{T}\)) is a second asymptotically flat exterior, causally disconnected from region I: the two are joined only through the spacelike “throat”, and no causal curve crosses from one to the other. The conformal compactification of this diagram is the prototype Penrose diagram [Misner:1973]. Regions III and IV are artefacts of exact vacuum eternity: in a black hole formed by collapse the matter's interior occupies the part of the diagram where they would lie, and only regions I and II survive (Section 45.4.1). The white hole and the second universe are not predictions about Nature; the black hole is.
Gravitational collapse
Oppenheimer–Snyder collapse
The simplest collapse that can be followed exactly is a uniform ball of pressureless dust [Oppenheimer:1939b]. Outside, spherical symmetry forces the geometry to be Schwarzschild (Theorem 45.4); the surface of the ball falls along a radial timelike geodesic, so the first computation is radial free fall.
A test particle released from rest at \(r=R_{0}\) falls along the cycloid
reaching \(r=0\) at the finite proper time
identical to the Newtonian free-fall time from the same radius. The horizon is crossed strictly earlier, also at finite proper time. Rests on Proposition 45.6.
Derivation. Derives Proposition 45.21. Set \(L=0\) in Proposition 45.6; release from rest at \(R_{0}\) fixes \(\mathcal{E}=c^{2}\sqrt{1-r_{\mathrm{s}}/R_{0}}\), so
This is exactly the Newtonian energy equation — the relativistic corrections have been absorbed into the meaning of \(r\) and \(\tau\) — and the substitution Equation (45.31) solves it:
using \(\sin^{2}\eta=(1-\cos\eta)(1+\cos\eta)\) and \(R_{0}/r=2/(1+\cos\eta)\). The collapse time is \(\eta=\pi\) in Equation (45.31); since \(r=r_{\mathrm{s}}\) is reached at some \(\eta<\pi\), the horizon crossing is earlier and finite. For a ball of one solar mass released from the Sun's present radius, Equation (45.32) gives \(\tau_{\mathrm{collapse}}=1.8\times 10^{3}\,\mathrm{s}\) — half an hour of proper time from star to singularity.
∎To a distant static observer the same infall never ends: the areal radius of the falling surface asymptotes to \(r_{\mathrm{s}}\), and light received from it at late observer time \(t\) is redshifted as
an e-folding time \(4GM/c^{3}=2.0\times 10^{-5}\,\mathrm{s}\) per solar mass. The luminosity dies on the same timescale: within milliseconds a stellar collapse is, observationally, black. Rests on Proposition 45.21 and Remark 45.3.
Derivation. Derives Proposition 45.22. Near the horizon write \(r=r_{\mathrm{s}}(1+\epsilon)\), \(\epsilon\ll1\). Dividing the radial equation of Proposition 45.21 by \(\dd t/\dd\tau=\mathcal{E}/\bigl(c^{2}(1-r_{\mathrm{s}}/r)\bigr)\) gives \(\dd r/\dd t\simeq-c\left(1-r_{\mathrm{s}}/r\right) \simeq-c\,\epsilon\) as \(\epsilon\to0\) (the remaining factors tend to constants), so \(r_{\mathrm{s}}\,\dd\epsilon/\dd t\simeq-c\,\epsilon\) and \(\epsilon\propto\ee^{-ct/r_{\mathrm{s}}}\): the surface approaches the horizon exponentially in coordinate time and never crosses it at any finite \(t\).
For the redshift, compound two factors at emission radius \(r\). Relative to the static observer there, the faller has Lorentz factor \(\gamma=-g_{\mu\nu}u_{\mathrm{fall}}^{\mu} u_{\mathrm{stat}}^{\nu}/c^{2} =\mathcal{E}/\bigl(c^{2}\sqrt{1-r_{\mathrm{s}}/r}\bigr) \propto\epsilon^{-1/2}\), so light it emits radially outward is Doppler-shifted by \(\sqrt{(1-v/c)/(1+v/c)}\simeq1/(2\gamma)\propto\sqrt{\epsilon}\); climbing out then costs the gravitational factor \(\sqrt{1-r_{\mathrm{s}}/r}\simeq\sqrt{\epsilon}\) (Equation (45.8)). The received frequency is suppressed by the product, of order \(\epsilon\). Finally, the outgoing photon spends a coordinate time \(\simeq(r_{\mathrm{s}}/c)\ln(1/\epsilon)\) escaping the near zone (integrating \(\dd t=\dd r_{*}/c\)), which at \(\epsilon\propto\ee^{-ct_{\mathrm{em}}/r_{\mathrm{s}}}\) doubles the elapsed time: \(t\simeq2t_{\mathrm{em}}\). Hence \(1+z\propto\epsilon^{-1} \propto\ee^{ct_{\mathrm{em}}/r_{\mathrm{s}}} =\ee^{ct/(2r_{\mathrm{s}})}\), which is Equation (45.33); with \(M=M_{\odot}\), \(2r_{\mathrm{s}}/c=1.97\times 10^{-5}\,\mathrm{s}\). The luminosity, proportional to the received photon rate times the energy per photon, falls at least as fast.
∎The interior completes the picture, and its computation is the matching of two exact solutions along the falling surface.
Let a uniform, pressureless, momentarily static ball of dust of mass \(M\) and initial areal radius \(R_{0}>r_{\mathrm{s}}\) collapse under its own gravity. Then: (i) the interior geometry is a portion \(0\leq\chi\leq\chi_{0}\) of a closed matter-dominated Friedmann–Lemaître–Robertson–Walker spacetime (Evidence-Based Cosmology),
whose scale factor traverses the collapsing half of the dust cycloid; (ii) the exterior is the Schwarzschild geometry Equation (45.6), and the two match smoothly across the surface — induced metric and extrinsic curvature both continuous — iff
with \(a_{\mathrm{m}}\) the initial (maximal) scale factor; equivalently \(M=\tfrac{4}{3}\pi\rho_{0}R_{0}^{3}\) with \(\rho_{0}\) the initial density — the Schwarzschild mass is exactly the Newtonian mass of the ball; (iii) every dust particle reaches the central singularity at the same finite proper time \(\tau=\pi a_{\mathrm{m}}/(2c)\), the surface crossing the horizon strictly earlier, while a distant observer sees the exponential freezing of Proposition 45.22. Rests on Theorem 45.4 and Proposition 45.21.
Full derivation in Appendix A.
Derives Theorem 45.23.
Dark objects are observed with masses far above the maximum that any cold equation of state consistent with causality can hold up. X-ray binaries yield dynamical masses between about \(5\,M_{\odot}\) and \(20\,M_{\odot}\) for the accreting component [Remillard:2006], and the gravitational-wave merger catalogue reaches tens of solar masses [Abbott:2016] [Abbott:2019], against a neutron-star maximum bounded near \(3\,M_{\odot}\) (Compact Stars and Relativistic Astrophysics). No equilibrium configuration is available to such objects, and the idealized collapse of a pressureless homogeneous ball ends at a horizon after finite proper time on the infalling matter, while a distant observer sees the surface asymptote to \(r_{\mathrm{s}}\) with exponentially growing redshift and never quite arrive [Oppenheimer:1939b]. The observational record of these objects is Experiment: Black-Hole Observations. Rests on Theorem 45.23, Proposition 45.22 and Theorem 45.27.
Derivation. Derives Phenomenon 45.24. The absence of equilibria above the maximum mass is the structure theory of Compact Stars and Relativistic Astrophysics. Given that, the collapse statements are Proposition 45.21 (finite proper time to horizon and singularity), Proposition 45.22 (exponential fading and freezing outside), and Theorem 45.23 (the complete interior–exterior solution, proved in Appendix A.2). The idealization — exact sphericity, zero pressure — is dispensable: once the surface is inside \(r_{\mathrm{s}}\), trapped surfaces exist (Remark 45.26), and Theorem 45.27 makes the singular outcome generic, independent of symmetry. What no theorem supplies is the horizon itself in the general case; that is the censorship conjecture of Section 45.4.3.
∎The singularity theorems
Is the singularity an artefact of exact spherical symmetry — all the matter aimed at one point? Penrose's 1965 theorem [Penrose:1965], for which the 2020 Nobel prize in physics was awarded, says no: what forces it is not symmetry but a trapped surface, and trapped surfaces are stable under perturbations.
Let \(S\) be a closed spacelike 2-surface. Through each point of \(S\) run two future-directed families of null geodesics orthogonal to \(S\) — outgoing and ingoing — and let \(\theta_{\pm}\) be their expansions: the fractional rate of change of the area element dragged along each family. \(S\) is trapped when both \(\theta_{+}<0\) and \(\theta_{-}<0\): even the outgoing light front loses area. In flat spacetime every round sphere has \(\theta_{+}>0\); trapping is the statement that gravity has bent the outgoing light cone back onto itself.
In the Eddington–Finkelstein geometry Equation (45.27), consider the sphere \((v,r)=\text{const}\) with \(r<r_{\mathrm{s}}\). The two orthogonal null families are those of Proposition 45.18: the ingoing rays have \(\dd r<0\), and the “outgoing” ones have \(\dd r/\dd v=\tfrac{1}{2}(1-r_{\mathrm{s}}/r)<0\) as well. Since the area is \(4\pi r^{2}\), both expansions are negative: every sphere inside the horizon is trapped. In the Oppenheimer–Snyder spacetime the exterior is exactly this geometry once the surface has passed \(r_{\mathrm{s}}\) (Theorem 45.23), so trapped surfaces form in finite proper time — and they persist under any sufficiently small perturbation of the initial data, because \(\theta_{\pm}<0\) are open conditions.
Let \((M,g)\) satisfy the null convergence condition \(R_{\mu\nu}k^{\mu}k^{\nu}\geq0\) for every null vector \(k\) — which the field equations translate into the null energy condition on matter (Section 44.8) — and contain a noncompact Cauchy hypersurface (a spacelike hypersurface met exactly once by every inextendible causal curve). If \(M\) contains a trapped surface, then \(M\) is future null geodesically incomplete: some outgoing light ray ends at finite affine parameter [Penrose:1965]. Rests on Definition 45.25.
Let \((M,g)\) satisfy the timelike and null convergence conditions (strong energy condition via the field equations, Section 44.8), the generic condition (every causal geodesic feels some curvature), and contain no closed timelike curves. If in addition it contains a trapped surface, or a compact achronal hypersurface, or a point whose past light cone starts reconverging, then \(M\) contains at least one incomplete timelike or null geodesic [Hawking:1970]. Rests on Definition 45.25.
Both proofs are global variational arguments — Raychaudhuri focusing plus a topological obstruction — and belong to mathematics; the standard complete accounts are [Penrose:1965] [Hawking:1970] and the monograph [Hawking:1973], and this treatise imports them. What the theorems deliver is deliberately weak: geodesic incompleteness, a freely falling history that cannot be continued, not a place of infinite curvature — though in every explicitly known collapse solution the incompleteness is accompanied by divergent curvature, as in Proposition 45.16. Their hypotheses matter: quantum fields violate the pointwise energy conditions (Axiomatic Quantum Field Theory), which is one reason the theorems say nothing about what replaces the singularity, a question quarantined to Quantum Gravity: The Honest Status. The cosmological instance of Theorem 45.28 — the universe's own past incompleteness — belongs to Evidence-Based Cosmology.
Cosmic censorship
The singularity theorems guarantee incompleteness; they do not guarantee a horizon around it. The weak cosmic censorship conjecture [Penrose:1969] asserts that, for generic asymptotically flat initial data with physically reasonable matter, the singularities of gravitational collapse are hidden behind event horizons — no “naked” singularity is visible from infinity, and the exterior remains predictable. Despite half a century of effort it remains a conjecture: no proof exists, and the fine-tuned counterexamples known (critical collapse, certain non-generic dust configurations) are exactly the non-generic cases the conjecture excludes. This treatise therefore treats censorship as a working hypothesis, not a result: it underwrites the astrophysical use of the Kerr geometry below, it is what the area theorem (Theorem 45.47) assumes, and its only observational support to date is indirect — every measured compact-object spin respects the Kerr bound (Phenomenon 45.35). The honest accounting of its status sits with the other open problems in What We Observe but Do Not Understand.
Rotating and charged black holes
The Kerr solution
Astrophysical bodies rotate, and their collapse remnants must carry angular momentum. The stationary axisymmetric vacuum solution that describes them was found by Kerr [Kerr:1963]; no elementary route to it exists, and its derivation is imported — Kerr's own, via algebraically special metrics, and the systematic construction in [Chandrasekhar:1983].
For each mass \(M\) and angular momentum \(J\), define
The metric, in the coordinates of Boyer and Lindquist [Boyer:1967],
is a stationary, axisymmetric, asymptotically flat solution of the vacuum field equations [Kerr:1963] [Boyer:1967]. At \(a=0\) it reduces to Equation (45.6); at \(M=0\) it is flat spacetime in spheroidal coordinates; at large \(r\) its \(g_{t\varphi}\) identifies \(J\) as the source's angular momentum. Rests on Theorem 45.1.
That Equation (45.37) solves \(R_{\mu\nu}=0\) is a finite mechanical verification; the discovery was finding it. Its properties, by contrast, are derivable in a few lines each, and they are the physics.
The surfaces \(\Delta=0\), at
are null hypersurfaces; \(r_{+}\) is the event horizon. They exist iff \(\abs{a}\leq GM/c^{2}\), i.e. iff the dimensionless spin \(\chi=cJ/(GM^{2})=ac^{2}/(GM)\) satisfies \(\abs{\chi}\leq1\); at \(\abs{\chi}=1\) (the extremal hole) the two coincide, and for \(\abs{\chi}>1\) the metric has no horizon at all — its ring singularity at \(\Sigma=0\) would be naked. Rests on Theorem 45.30.
Derivation. Derives Proposition 45.31. \(\Delta=r^{2}-r_{\mathrm{s}}r+a^{2}\) is a quadratic in \(r\) with roots Equation (45.38), real iff \(\left(r_{\mathrm{s}}/2\right)^{2}\geq a^{2}\), which is \(\abs{a}\leq GM/c^{2}\). A surface of constant \(r\) has normal \(n_{\mu}=\pp_{\mu}r\) with norm \(g^{\mu\nu}n_{\mu}n_{\nu}=g^{rr}=\Delta/\Sigma\), which vanishes exactly on \(\Delta=0\): those surfaces are null, hence one-way membranes for causal curves exactly as in Proposition 45.18. Curvature invariants diverge only on \(\Sigma=0\) — the equatorial ring \(r=0\), \(\theta=\pi/2\) — so \(r_{\pm}\) are coordinate singularities removable by Eddington–Finkelstein-type coordinates. With \(a=\chi GM/c^{2}\), the reality condition is \(\abs{\chi}\leq1\).
∎(i) Static observers (fixed \(r,\theta,\varphi\)) exist only outside the static limit
which lies outside \(r_{+}\) everywhere except at the poles. In the ergosphere \(r_{+}<r<r_{E}\) every observer is dragged around the hole. (ii) An observer with zero angular momentum orbits at
falling off at large \(r\) as \(\Omega\simeq2GJ/(c^{2}r^{3})\); on the horizon the dragging rate becomes uniform,
the angular velocity of the black hole. Rests on Theorem 45.30.
Derivation. Derives Proposition 45.32. (i) A static worldline has tangent \(\propto\pp_{t}\), timelike iff \(g_{tt}<0\), i.e. \(\Sigma>r_{\mathrm{s}}r\); the boundary \(r^{2}-r_{\mathrm{s}}r+a^{2}\cos^{2}\theta=0\) gives Equation (45.39), which exceeds \(r_{+}\) of Equation (45.38) wherever \(\sin\theta\neq0\). Inside it \(\pp_{t}\) is spacelike: remaining at rest with respect to infinity would require moving faster than light against the dragging. (ii) Zero angular momentum means \(u_{\varphi}=g_{\varphi t}\,u^{t}+g_{\varphi\varphi}u^{\varphi}=0\), so \(\Omega=u^{\varphi}/u^{t}=-g_{t\varphi}/g_{\varphi\varphi}\). Reading the components off Equation (45.37) and using the identity \(\Sigma\left(r^{2}+a^{2}\right)+r_{\mathrm{s}}r\,a^{2}\sin^{2}\theta =\left(r^{2}+a^{2}\right)^{2}-a^{2}\Delta\sin^{2}\theta\) gives Equation (45.40); the far-field fall-off follows from \(r_{\mathrm{s}}a\,c=2GJ/c^{2}\). On \(\Delta=0\) the denominator is \(\left(r_{+}^{2}+a^{2}\right)^{2}\) and \(r_{\mathrm{s}}r_{+}=r_{+}^{2}+a^{2}\) (from \(\Delta(r_{+})=0\)), whence Equation (45.41), independent of \(\theta\): the horizon rotates rigidly. The weak-field limit of Equation (45.40) is the Lense–Thirring precession rate whose measurement by Gravity Probe B and lunar laser ranging is reported in Experiment: The Classical Tests of General Relativity.
∎Inside the ergosphere the Killing energy \(\mathcal{E}=-\xi_{\mu}u^{\mu}\) of a particle can be negative, because \(\vect{\xi}=\pp_{t}\) is there spacelike. A body that splits in the ergosphere can therefore send one fragment into the hole with negative energy and eject the other to infinity with more energy than went in: rotational energy is extracted [Penrose:1969]. The process cannot be repeated indefinitely — each extraction spins the hole down, and the bookkeeping of what is extractable is exactly the irreducible-mass inequality Equation (45.50), quantified in Section 45.6.1: at most \(1-1/\sqrt{2}\approx29\%\) of the mass–energy of an extremal hole. Its astrophysical descendants (magnetically mediated extraction from accreting holes) power the relativistic jets observed in Experiment: Black-Hole Observations.
The Kerr metric has only the two Killing vectors \(\pp_{t}\) and \(\pp_{\varphi}\) — two constants of motion where the Schwarzschild case, with its full rotation group, had three. Geodesic motion is nevertheless integrable: Carter found a fourth invariant by separating the Hamilton–Jacobi equation (Hamilton–Jacobi Theory and the Optical–Mechanical Analogy, whose separability machinery, Section 23.2, is exactly what is applied) in Boyer–Lindquist coordinates [Carter:1968]. The invariant is quadratic in the momenta and derives from a rank-two Killing tensor (Definition 13.142) that is not a product of Killing vectors — the physically important instance of that definition, and a hidden symmetry with no isometry behind it. All Kerr geodesics used below (accretion orbits, photon rings, the spin-dependent ISCO of [Bardeen:1972]) rest on this integrability.
The compact objects in accreting binaries and the remnants of gravitational-wave mergers carry angular momentum. Writing the dimensionless spin as
measured values are non-zero in many systems and in no case exceed \(\abs{\chi}=1\), the value at which the two horizons of the Kerr solution merge and beyond which the solution has no horizon at all. The broad, gravitationally redshifted iron K\(\alpha\) line emitted by the inner accretion disc is one measurement channel [Tanaka:1995]; the remnant of the first detected binary merger, for which \(\chi\approx0.67\), is another [Abbott:2016]. That the bound is respected wherever it has been tested is the only observational evidence bearing on cosmic censorship (Section 45.4.3). Rests on Proposition 45.31.
Derivation. Derives Phenomenon 45.35. The bound itself is Proposition 45.31: horizons exist iff \(\abs{\chi}\leq1\), the extremal case being the merger of \(r_{\pm}\) at \(r=GM/c^{2}\), so a measured \(\abs{\chi}>1\) would mean either a naked singularity or a non-Kerr object — either way a violation of the framework whose consistency is being tested. The two channels lean on different parts of this chapter: the iron-line profile is set by the spin-dependent inner edge of the disc (Remark 45.36: the co-rotating ISCO sweeps from \(6GM/c^{2}\) at \(\chi=0\) to \(GM/c^{2}\) at \(\chi=1\), and the gravitational redshift Equation (45.8) from that edge is what broadens the line [Tanaka:1995]); the merger-remnant spin is inferred from the ringdown frequencies, whose dependence on \((M,J)\) is Equation (45.45) below. Both are assembled, with their apparatus, in Experiment: Black-Hole Observations and Experiment: Gravitational Waves.
∎Repeating the circular-orbit analysis of Section 45.2.2 on Equation (45.37) — the computation, using Carter's integrability, is carried out in [Bardeen:1972] and its results are imported here — gives an ISCO that depends on spin and on orbital sense: for co-rotating equatorial orbits it descends from \(6GM/c^{2}\) at \(\chi=0\) to \(GM/c^{2}\) at \(\chi=1\), while counter-rotating orbits retreat to \(9GM/c^{2}\). The bounding energy rises correspondingly: the extremal co-rotating ISCO has \(\mathcal{E}=c^{2}/\sqrt{3}\), so thin-disc accretion can radiate up to \(1-1/\sqrt{3}=0.42\) of the rest energy — the origin of the “toward \(0.4\)” in Phenomenon 45.12, and the lever by which disc spectroscopy measures spin.
Charge: Reissner–Nordström and Kerr–Newman
Charge is the one other parameter a stationary black hole can carry (Section 45.5.3). The static charged solution was found within two years of Schwarzschild's [Reissner:1916] [Nordstrom:1918], and unlike Kerr's it is an exercise: the Schwarzschild derivation reruns with the Coulomb field's stress–energy as source.
The unique spherically symmetric, asymptotically flat solution of the Einstein–Maxwell equations with charge \(Q\) and mass \(M\) is
with the radial Coulomb field \(E=Q/(4\pi\epsilon_{0}r^{2})\) and the charge length
Horizons sit at \(r_{\pm}=r_{\mathrm{s}}/2\pm\sqrt{r_{\mathrm{s}}^{2}/4-r_{Q}^{2}}\), existing iff \(r_{Q}\leq r_{\mathrm{s}}/2\). Rests on Theorem 45.1.
Derivation. Derives Proposition 45.37. Spherical symmetry admits only a radial electrostatic field. The curved-space source-free Maxwell equations reduce, in the ansatz Equation (45.1) with \(\alpha+\beta=0\), to \(\pp_{r}\left(r^{2}E\right)=0\) — the flux integral — so \(E=Q/(4\pi\epsilon_{0}r^{2})\) exactly as in flat space. Its stress–energy is that of an electromagnetic field with energy density \(u_{\mathrm{em}}=\epsilon_{0}E^{2}/2\) (Section 60.4.2), radial tension equal to the energy density and isotropic transverse pressure: \(T^{0}{}_{0}=T^{r}{}_{r}=-u_{\mathrm{em}}\), \(T^{\theta}{}_{\theta}=T^{\varphi}{}_{\varphi}=+u_{\mathrm{em}}\). Because \(T^{0}{}_{0}=T^{r}{}_{r}\), the combination that gave \(\alpha+\beta=\text{const}\) in the proof of Theorem 45.1 does so again, and asymptotic flatness sets \(\beta=-\alpha\). The tensor is traceless, so \(R=0\) and the field equations read \(R_{\mu\nu}=(8\pi G/c^{4})T_{\mu\nu}\); the \(\theta\theta\) component becomes
which integrates to \(r\ee^{2\alpha}=r-r_{\mathrm{s}}+r_{Q}^{2}/r\), i.e.\ Equation (45.43). The horizon condition \(f=0\) is a quadratic with the stated roots.
∎The rotating charged solution — charge \(Q\) grafted onto Equation (45.37) by \(\Delta\to r^{2}-r_{\mathrm{s}}r+a^{2}+r_{Q}^{2}\), with the accompanying electromagnetic field — was found by Newman and collaborators [Newman:1965] and is imported here; it completes the stationary family of Section 45.5.3. Astrophysically, charge is negligible by two crushing margins. For charge to matter geometrically requires \(r_{Q}\sim r_{\mathrm{s}}/2\), i.e.\ \(Q\sim\sqrt{4\pi\epsilon_{0}G}\,M \approx1.7\times 10^{20}\,\mathrm{C}\) per solar mass. But a hole charged to just \(Q_{\mathrm{eq}}=4\pi\epsilon_{0}GMm_{\mathrm{p}}/e \approx1.5\times 10^{2}\,\mathrm{C}\) per solar mass already attracts protons electrostatically as strongly as it does gravitationally: surrounding plasma neutralizes any astrophysical hole to about one part in \(10^{18}\) of the geometrically relevant charge. Charge is therefore a logical parameter of the classification, not an observable of astronomy, and \(\left(M,J\right)\) is the observable list.
The no-hair theorems
Collapse is messy; its endpoint is not. The stationary states available to it are classified by uniqueness theorems of remarkable strength, and they are pure mathematics: this treatise imports them with hypotheses stated.
A static, asymptotically flat vacuum spacetime whose event horizon is a connected, regular, non-degenerate closed 2-surface is isometric to the Schwarzschild solution Equation (45.6); the horizon is then a round sphere. The electrovac counterpart, with Maxwell field, is Reissner–Nordström [Israel:1967] [Israel:1968]. Rests on Theorem 45.1 and Proposition 45.37.
A stationary, axisymmetric, asymptotically flat vacuum spacetime, regular on and outside a connected non-degenerate event horizon, is a member of the two-parameter Kerr family Equation (45.37), labelled by \((M,J)\) with \(\abs{\chi}<1\): Carter reduced the problem to two-parameter families [Carter:1971], Robinson closed it [Robinson:1975]. With electromagnetism the endpoint is Kerr–Newman, labelled by \((M,J,Q)\) [Mazur:1982]. Rests on Theorem 45.30.
Axisymmetry is not an extra assumption in the end: Hawking's rigidity theorem shows a stationary rotating black hole must be axisymmetric [Hawking:1973] — though its original proof assumes analyticity of the metric, a technical gap still not closed in full generality. “Non-degenerate” excludes the extremal case \(\abs{\chi}=1\), which needs and has separate treatment. And “vacuum or electrovac” is essential: with other matter — Yang–Mills fields, complex scalars with harmonic time dependence — stationary “hairy” solutions exist mathematically. None is known to be both stable and producible by collapse, and no observation requires one; within the matter content actually observed (Epistemology and the Scientific Method's scope rule), the theorems say a stationary black hole is characterized completely by \((M,J,Q)\), and astrophysically (Remark 45.38) by \((M,J)\). Wheeler's slogan for this — “a black hole has no hair” — entered the literature through [Ruffini:1971]: every other memory of the progenitor (baryon number, magnetic moments, mountains, history) is radiated away in the ringdown.
No stationary black hole has yet been observed to carry any property beyond a mass and an angular momentum. The sharpest test available is the ringdown of a merger remnant: for a Kerr hole the frequencies and damping times of the quasi-normal oscillations are fixed functions of \(M\) and \(J\) alone, so measuring two of them over-determines the remnant and any third measurement is a consistency check. The remnant of the first observed binary black-hole merger rings at the frequency and damping time predicted for a Kerr hole of the mass and spin inferred independently from the inspiral [Abbott:2016], and the growing merger catalogue has produced no counterexample [Abbott:2019]. Rests on Theorems 45.39 and 45.40.
Derivation. Derives Phenomenon 45.42. The two-parameter character of the spectrum follows from the uniqueness theorems by dimensional analysis, with no wave computation at all. A perturbed remnant settles to Kerr (Theorem 45.40); its perturbations obey linear wave equations on the Kerr background with outgoing radiation at infinity and ingoing at the horizon, and those two boundary conditions make the frequency spectrum discrete and complex — damped resonances, the quasi-normal modes [Regge:1959] [Vishveshwara:1970]. The background carries exactly one dimensional scale, \(GM/c^{3}\) (a time), and one dimensionless parameter, \(\chi\); therefore every mode must take the form
with \(F_{n\ell m}\) universal dimensionless functions — computed once and for all, never fit per source. Their values are imported: for Schwarzschild the fundamental quadrupole mode is \(\omega=\left(0.374-0.089\,\ii\right)c^{3}/(GM)\) [Vishveshwara:1970] [Chandrasekhar:1983]. The test is then over-determination: one complex frequency (one \(f\), one damping time) inverts Equation (45.45) for \((M,\chi)\); any second mode, or the \((M,\chi)\) inferred independently from the inspiral, must agree — and for GW150914, ringing at about \(250\,\mathrm{Hz}\) with the mass and spin the inspiral gave, it does [Abbott:2016]. A single confirmed extra parameter in the spectrum would falsify the uniqueness theorems' applicability to Nature.
∎Black-hole mechanics
The four laws
Black holes obey mechanical laws with the exact structure of thermodynamics [Bardeen:1973]. The dictionary needs two quantities: the horizon area, and the surface gravity.
A static observer at radius \(r\) in Equation (45.6) requires proper acceleration
which diverges at the horizon. The redshifted acceleration — the force per unit mass an operator at infinity must supply, holding the observer on an ideal massless tether — stays finite, and its horizon limit defines the surface gravity:
For Kerr the same construction on the horizon-generating Killing field \(\pp_{t}+\Omega_{\mathrm{H}}\pp_{\varphi}\) gives
uniform over the horizon (the zeroth law); the Kerr evaluation is imported [Bardeen:1973] [Wald:1984]. Rests on Theorems 45.1 and 45.30.
Derivation. Derives Proposition 45.43. The static four-velocity has \(u^{0}=c\left(1-r_{\mathrm{s}}/r\right)^{-1/2}\) (normalization) and acceleration \(a^{\mu}=u^{\nu}\nabla_{\nu}u^{\mu}\) with single nonvanishing component \(a^{r}=\Gamma^{r}{}_{00}\left(u^{0}\right)^{2}\). From Equation (45.2) with \(\ee^{2\alpha}=1-r_{\mathrm{s}}/r\), \(\Gamma^{r}{}_{00} =\left(r_{\mathrm{s}}/2r^{2}\right) \left(1-r_{\mathrm{s}}/r\right)\), so \(a^{r}=GM/r^{2}\) and the invariant magnitude is \(\mathrm{a}=\sqrt{g_{rr}}\,a^{r}\), which is Equation (45.46). The tether argument for the redshift factor is the energy bookkeeping of Equation (45.8): work done at infinity is redshifted by \(\sqrt{-g_{tt}}/c\) relative to work delivered locally. The limit is elementary. For the Schwarzschild check of Equation (45.48): \(r_{+}=r_{\mathrm{s}}\), \(r_{-}=0\), \(a=0\) gives \(c^{2}/(2r_{\mathrm{s}})=c^{4}/(4GM)\), agreeing with Equation (45.47). Numerically \(\kappa=1.5\times 10^{13}\,\mathrm{m}/\mathrm{s}^{2}\) for \(M=M_{\odot}\), scaling as \(1/M\) — enormous, yet the temperature it will buy in Section 45.6.2 is tens of nanokelvin.
∎The area of the Kerr horizon is
Rests on Proposition 45.31.
Derivation. Derives Lemma 45.44. On \(r=r_{+}\), \(t=\text{const}\) the induced metric from Equation (45.37) is \(\Sigma_{+}\dd\theta^{2} +\bigl(\left(r_{+}^{2}+a^{2}\right)^{2}\sin^{2}\theta/ \Sigma_{+}\bigr)\dd\varphi^{2}\), writing \(\Sigma_{+}:=\Sigma(r_{+},\theta)\) and using the identity from the proof of Proposition 45.32 at \(\Delta=0\). Its area element is \(\left(r_{+}^{2}+a^{2}\right)\sin\theta\,\dd\theta\,\dd\varphi\) — the \(\Sigma_{+}\) factors cancel — so \(A=4\pi\left(r_{+}^{2}+a^{2}\right)\), and \(r_{+}^{2}+a^{2}=r_{\mathrm{s}}r_{+}\) gives the second form.
∎Define the irreducible mass by \(A=:16\pi G^{2}M_{\mathrm{irr}}^{2}/c^{4}\). Then
so \(M\geq M_{\mathrm{irr}}\), with equality iff \(J=0\) [Christodoulou:1970]. In particular, since the area theorem below forbids \(M_{\mathrm{irr}}\) ever to decrease, at most
of a hole's mass–energy is extractable by any process — the bound saturated only from the extremal state, where \(M_{\mathrm{irr}}=M/\sqrt{2}\). Rests on Lemma 45.44 and Proposition 45.31.
Derivation. Derives Proposition 45.45. By Lemma 45.44, \(16\pi G^{2}M_{\mathrm{irr}}^{2}/c^{4} =8\pi\left(GM/c^{2}\right)r_{+}\), i.e.\ \(r_{+}=2GM_{\mathrm{irr}}^{2}/\left(Mc^{2}\right)\). Substitute this into \(r_{\mathrm{s}}r_{+}=r_{+}^{2}+a^{2}\) with \(a=J/(Mc)\):
and multiplication by \(M^{2}c^{4}/\left(4G^{2}\right)\) gives \(M^{2}M_{\mathrm{irr}}^{2} =M_{\mathrm{irr}}^{4}+c^{2}J^{2}/\left(4G^{2}\right)\), which is Equation (45.50). At extremality \(J=GM^{2}/c\), and Equation (45.50) becomes \(M^{2}=M_{\mathrm{irr}}^{2}+M^{4}/\left(4M_{\mathrm{irr}}^{2}\right)\), whose solution is \(M_{\mathrm{irr}}^{2}=M^{2}/2\); extracting all the angular momentum at fixed \(M_{\mathrm{irr}}\) leaves a Schwarzschild hole of mass \(M/\sqrt{2}\), which is Equation (45.51).
∎Along the Kerr family, with \(E=Mc^{2}\),
with \(\kappa\) and \(\Omega_{\mathrm{H}}\) as in Equations (45.41) and (45.48). Moreover the homogeneity of the mass formula yields Smarr's integral form [Smarr:1973],
Derivation. Derives Proposition 45.46. Differentiate Equation (45.50) at fixed functional form, using \(\dd M_{\mathrm{irr}}^{2} =c^{4}\dd A/\left(16\pi G^{2}\right)\):
Two substitutions, both via \(M_{\mathrm{irr}}^{2}=Mc^{2}r_{+}/(2G)\) (from Lemma 45.44), turn the coefficients geometric. The full \(\dd J\) coefficient is
by Equation (45.41), using \(J=Mca\) and \(r_{\mathrm{s}}=2GM/c^{2}\). In the \(\dd A\) coefficient, \(c^{2}J^{2}/\left(4G^{2}M_{\mathrm{irr}}^{4}\right) =a^{2}/r_{+}^{2}\) by the same substitution, and \(r_{+}r_{-}=a^{2}\) (product of the roots of \(\Delta\)) turns the bracket into \(1-r_{-}/r_{+}=\left(r_{+}-r_{-}\right)/r_{+}\); assembling,
the middle equality by \(r_{+}^{2}+a^{2}=r_{\mathrm{s}}r_{+}=2GMr_{+}/c^{2}\), matching Equation (45.48). This is Equation (45.52). For Equation (45.53): under \((A,J)\to(\lambda^{2}A,\lambda^{2}J)\) the mass formula gives \(E\to\lambda E\); differentiating \(E(\lambda^{2}A,\lambda^{2}J)=\lambda E(A,J)\) at \(\lambda=1\) gives \(2A\,\pp E/\pp A+2J\,\pp E/\pp J=E\), which with the coefficients just computed is Equation (45.53). The Schwarzschild case checks in one line: with Equations (45.47) and (45.49), \(\dd A=32\pi G^{2}M\,\dd M/c^{4}\) and \(\left(\kappa c^{2}/8\pi G\right)\dd A=c^{2}\,\dd M\) exactly.
∎In an asymptotically predictable spacetime (cosmic censorship, Section 45.4.3, holds) whose matter satisfies the null energy condition (Section 44.8), the total area of event horizons is non-decreasing toward the future [Hawking:1971]. Rests on Proposition 45.18.
The parallel with thermodynamics (Classical Thermodynamics) is now structural, law by law [Bardeen:1973]:
| Law | Black-hole mechanics | Thermodynamics |
|---|---|---|
| Zeroth | $\kappa$ uniform over the horizon of a stationary hole | $T$ uniform at equilibrium |
| First | $\dd E=\dfrac{\kappa c^{2}}{8\pi G}\,\dd A +\Omega_{\mathrm{H}}\,\dd J$ | $\dd U=T\,\dd S-p\,\dd V$ |
| Second | $\dd A\geq0$ (Theorem 45.47) | $\dd S\geq0$ |
| Third | $\kappa=0$ (extremality) unattainable in any finite process | $T=0$ unattainable |
The correspondence pairs \(E\) with internal energy, \(\kappa\) with temperature and \(A\) with entropy, up to constants; Bekenstein argued on information-theoretic grounds that the pairing is physical and the entropy real [Bekenstein:1973]. Classically that seems absurd — a classical black hole absorbs everything and emits nothing, so its temperature ought to be exactly zero; the area theorem's own hypotheses (null energy condition) are what quantum fields refuse to satisfy, and quantum theory is exactly what turns the analogy literal (Section 45.6.2). One consequence of Theorem 45.47 is already observable bookkeeping: two merging Schwarzschild holes of mass \(M\) each have final mass at least \(\sqrt{2}M\) (areas add, \(A\propto M^{2}\)), so at most \(1-1/\sqrt{2}\approx29\%\) of the total can be radiated — GW150914's measured \(3.0\,M_{\odot}\) out of \(65\,M_{\odot}\), under \(5\%\), sits comfortably inside the bound [Abbott:2016].
Hawking radiation: prediction without observation
Quantum fields on the spacetime of a body collapsing to a black hole of surface gravity \(\kappa\) settle, at late times, into a state in which the hole emits thermal radiation to infinity at the temperature
and the four laws of Section 45.6.1 become literal thermodynamics with entropy
[Hawking:1975]. Rests on Propositions 45.43 and 45.46.
The derivation is imported [Hawking:1975] [Wald:1994]: it is a computation in quantum field theory on the fixed collapse background, comparing the field's in-vacuum with the late-time out-modes. Its structural core, however, is already in this treatise: Proposition 109.74 establishes — as a theorem of axiomatic field theory, via the KMS condition — that the vacuum restricted to a wedge is thermal with respect to the boost flow, at the temperature \(T=\hbar a/(2\pi ck_{\mathrm{B}})\) for the observer of proper acceleration \(a\) [Unruh:1976]. Near the horizon a static observer is precisely such an accelerated observer, and Equation (45.54) is the wedge temperature evaluated at the surface gravity and redshifted to infinity (Equation (45.47)); the Euclidean route — regularity of the imaginary-time geometry forcing periodicity \(\hbar/(k_{\mathrm{B}}T_{\mathrm{H}})\) — gives the same number [Gibbons:1977]. Consistency of three independent derivations is why the prediction is regarded as robust given its premise: quantum field theory remains valid on a classical black-hole background. That premise is untested, and everything that hangs on the answer — the information problem, the microscopic meaning of Equation (45.55) — is quarantined with the quantum-gravity programme in Quantum Gravity: The Honest Status.
A hole radiating thermally at Equation (45.54) from area Equation (45.49) loses mass as \(\dd M/\dd t\propto-1/M^{2}\) and evaporates in
where the numerical prefactor is the photon-only, geometric-optics Stefan–Boltzmann estimate; emitted-species and greybody corrections change it by factors of order unity [Hawking:1975]. Rests on Theorem 45.49.
Derivation. Derives Proposition 45.51. Treat the hole as a black body of area \(A=16\pi G^{2}M^{2}/c^{4}\) at temperature Equation (45.54) and apply the Stefan–Boltzmann law (Classical Thermodynamics), \(P=\sigma AT_{\mathrm{H}}^{4}\) with \(\sigma=\pi^{2}k_{\mathrm{B}}^{4}/(60\hbar^{3}c^{2})\):
by integrating \(M^{2}\dd M\). Inserting \(M_{\odot}\) gives \(6.6\times 10^{74}\,\mathrm{s}\), i.e. about \(2\times 10^{67}\,\mathrm{yr}\) — \(10^{57}\) Hubble times. The estimate ignores the frequency-dependent horizon absorptivity and the species count, which is why only its scaling and order of magnitude are asserted.
∎The honest evidence status must be stated plainly, in the terms of Epistemology and the Scientific Method: Hawking radiation has never been observed, and for every black hole known to exist it never can be by any foreseeable instrument. The arithmetic is unforgiving: Equation (45.54) gives \(T_{\mathrm{H}}=6.2\times 10^{-8}\,\mathrm{K}\) for one solar mass — colder than the \(2.725\,\mathrm{K}\) microwave background by nearly eight orders of magnitude, so every astrophysical hole absorbs background radiation faster than it emits, and will until cosmic expansion has cooled the background below its temperature. A hole hotter than the background must be lighter than \(4.5\times 10^{22}\,\mathrm{kg}\) (half a lunar mass, horizon smaller than a tenth of a millimetre); one completing its evaporation today — the potentially detectable terminal \(\gamma\)-ray burst — must have been born primordial at about \(10^{11}\)–\(10^{12}\,\mathrm{kg}\) [Hawking:1975], and searches for such bursts have found nothing attributable. The status is therefore exactly that recorded for the acceleration temperature in Remark 109.75, and for the same structural reason: the KMS mathematics is theorem-solid, the physical consequence unobserved. What would count as evidence: the terminal burst of a primordial hole with the predicted spectrum, or — more remotely — thermal spectra from any horizon at the predicted temperature. Analogue systems (sonic horizons in condensates) confirm the mode-conversion mathematics, not the gravitational claim [Unruh:1981]. No phenomenon environment appears in this section: the book reserves that environment for observed facts, and this is a prediction — the sharpest untested prediction general relativity and quantum theory jointly make.
The shadow
Seen from outside, a black hole's mark is what it does to light. Rays with impact parameter below a critical value \(b_{\mathrm{c}}\) are captured (Section 45.2.3); an observer looking at a hole backlit or surrounded by emission therefore sees a dark disc of radius \(b_{\mathrm{c}}\) — the capture cross-section, not the horizon — bounded by a bright ring where rays that nearly orbited the photon sphere pile up. The escape-cone geometry was computed by Synge [Synge:1966], and the first realistic image of a hole with a thin accretion disc — ring, asymmetry and all — by Luminet [Luminet:1979]. The same construction is carried out in Figure 45.1 for a rapidly spinning hole viewed almost edge-on — the configuration rendered for the film Interstellar, whose images leave out the Doppler and gravitational frequency shifts deliberately, so that the disc appears left–right symmetric [James:2015]; the second panel restores them. The angular size is set by \(b_{\mathrm{c}}\approx2.6\,r_{\mathrm{s}}\): about ten gravitational radii across, only weakly dependent on spin and inclination [Falcke:2000], which is what makes the measurement a test of the geometry rather than of astrophysical modelling. The imaging observations themselves — and the stellar-orbit mass that calibrates them — are deferred, with the full apparatus, to Experiment: Black-Hole Observations.
![[figure: bh-lensed-disc.pdf]](/images/treatise/en/figure/bh-lensed-disc.svg)
A thin accretion disc around a rapidly spinning black hole, computed by integrating the Kerr null geodesic equations backwards from the observer. The hole has \(a=0.999\,GM/c^{2}\) and is seen \(5^\circ\) out of the equatorial plane; the disc is opaque, geometrically thin, and reaches from the innermost stable circular orbit at \(1.18\,GM/c^{2}\) out to \(20\,GM/c^{2}\), radiating with the time-averaged flux of a steady thin disc [Page:1974]. Lensing does all the work. Light from the far side is bent over and under the hole, so the far side of the disc appears as an arc above the dark region and its underside as a second arc below, while the near side crosses in front as a thin band: a flat disc is wrapped into the shape one sees. The dark region is the capture cross-section, flattened on the left — the side on which co-rotating photons appear — by the dragging of inertial frames, and the thin ring is the analytic boundary of that cross-section, which the disc's own images partly cover and on which the higher-order images pile up too closely to be drawn. Panel (a) omits the Doppler and gravitational frequency shifts, as the film Interstellar did [James:2015]; panel (b) restores them, multiplying the surface brightness by \(g^{4}\), where \(g\) is the redshift factor of a prograde circular orbit. The approaching side is on the left, and outshines the receding one by orders of magnitude. Brightness is logarithmic over four decades and both panels share one normalisation, so they differ by exactly the factor \(g^{4}\). The first image of this kind was computed by Luminet [Luminet:1979].
A black hole surrounded by an optically thin emitting region appears as a bright ring enclosing a central brightness depression. The depression is bounded by the photon-capture impact parameter, so a Schwarzschild hole of mass \(M\) seen from distance \(D\) subtends an angular diameter
that is \(\theta_{\mathrm{sh}}D=3\sqrt{3}\,r_{\mathrm{s}} \simeq5.20\,r_{\mathrm{s}}\), some ten gravitational radii and only weakly dependent on spin and on viewing angle [Synge:1966] [Falcke:2000]. Rings of about this size have been imaged at millimetre wavelengths around the nucleus of M87 and around Sgr A* [Akiyama:2019] [Akiyama:2022]; the observations, with their apparatus, are assembled in Experiment: Black-Hole Observations. Rests on Equation (45.20), Theorem 45.1 and Proposition 13.141.
Derivation. Derives Phenomenon 45.52. Take the first integral Equation (45.20) of the null geodesic equation, established in Section 45.2.3, which for a ray confined to the equatorial plane reads
with \(E\) and \(L\) the constants conjugate to the two Killing vectors and \(b:=L/E\) the impact parameter measured at infinity (Equation (45.22)) [Misner:1973]. Radial turning points are the zeros of the right-hand side, so a ray of impact parameter \(b\) turns where
An incoming ray escapes if it meets a turning point outside the horizon and is captured if it does not, so the critical impact parameter separating the two behaviours is fixed by the maximum of \(W\). Differentiating,
which vanishes at
the photon sphere, and \(W''<0\) there, so this is the maximum. Its value is
so that \(b_{\mathrm{c}}^{2}=27r_{\mathrm{s}}^{2}/4\) and \(b_{\mathrm{c}}=\tfrac{3}{2}\sqrt{3}\,r_{\mathrm{s}} =3\sqrt{3}\,GM/c^{2}\), which is Equation (45.57). The apparent diameter of the capture cross-section is twice this, \(2b_{\mathrm{c}}=3\sqrt{3}\,r_{\mathrm{s}}=5.196\,r_{\mathrm{s}}\).
Two remarks. The shadow is larger than the horizon by the factor \(3\sqrt{3}/2\approx2.60\), because the impact parameter is measured at infinity while the horizon radius is a coordinate radius: what is imaged is the capture cross-section, not the horizon. And rays with \(b\) slightly above \(b_{\mathrm{c}}\) orbit many times near \(r_{\mathrm{ph}}\) before escaping, which piles emission up at the rim and produces the bright ring rather than a mere edge.
∎