The geometry of black holes

Think of radial motion as a ball rolling in a one-dimensional potential well $V_{\text{eff}}(r)$: circular orbits sit where the well is flat ($dV_{\text{eff}}/dr=0$), and they are stable only where the well curves upward ($d^2V_{\text{eff}}/dr^2>0$) rather than downward. Because the relativistic potential has an extra $1/r^3$ term absent in Newtonian gravity, the well develops a maximum close to the black hole — inside that radius no stable circular orbit exists at all, no matter how fast the particle spins around; matter simply plunges in.

Step 1 (conserved quantities and the radial equation). Along any geodesic (timelike $\kappa=-c^2$, or null $\kappa=0$) confined to the equatorial plane $\theta=\pi/2$, the metric's independence of $t$ and $\phi$ gives two conserved quantities $\tilde E = (1-r_s/r)c^2\dot t$ and $\tilde L = r^2\dot\phi$ (dot $=d/d\tau$). Substituting these into the normalization $g_{\mu\nu}\dot x^\mu\dot x^\nu=\kappa$ and simplifying yields, for timelike motion, $$\left(\frac{dr}{d\tau}\right)^2 = \tilde E^2 - \left(1-\frac{r_s}{r}\right)\left(c^2+\frac{\tilde L^2}{r^2}\right) \equiv \tilde E^2 - V^2_{\text{eff}}(r)$$

Step 2 (circular-orbit condition). Expanding gives $$V^2_{\text{eff}}(r) = c^2 - \frac{c^2 r_s}{r} + \frac{\tilde L^2}{r^2} - \frac{\tilde L^2 r_s}{r^3}$$ A circular orbit has constant $r$, i.e. $dr/d\tau=0$ at all times, which requires both $\tilde E^2=V^2_{\text{eff}}(r)$ and (so that $r$ does not drift away) $dV^2_{\text{eff}}/dr=0$: $$\frac{d}{dr}V^2_{\text{eff}} = \frac{c^2 r_s}{r^2} - \frac{2\tilde L^2}{r^3} + \frac{3\tilde L^2 r_s}{r^4} = 0$$

Step 3 (solve for the angular momentum of a circular orbit). Multiplying through by $r^4$ and solving for $\tilde L^2$ gives the angular momentum needed to sustain a circular orbit at radius $r$: $$\tilde L^2(r) = \frac{c^2 r_s r^2}{2r-3r_s}$$ This already carries information about the photon sphere: as $r\to (3/2)r_s$ from above, the denominator $2r-3r_s\to0^+$ and $\tilde L^2\to+\infty$ — no finite angular momentum sustains a circular orbit that close, which is exactly the massless (photon) limit reached below.

Step 4 (marginal stability — the ISCO). As $r$ decreases from infinity, $\tilde L^2(r)$ first decreases, reaches a minimum, then blows up at $r=3r_s/2$; a circular orbit is stable exactly where increasing $\tilde L^2$ is needed to shrink $r$ further (precisely, stability turns on the sign of $d\tilde L^2/dr$). Differentiating Step 3's result, $$\frac{d\tilde L^2}{dr} = \frac{2c^2 r_s\,r(r-3r_s)}{(2r-3r_s)^2}$$ which vanishes at $r=0$ (excluded), $r=3r_s$, and nowhere else for $r>3r_s/2$. This single interior root is the marginally stable radius: $$r_{\text{ISCO}} = 3r_s = \dfrac{6GM}{c^2}$$ For $r>3r_s$ the orbit is stable ($d\tilde L^2/dr>0$, larger $r$ needs more angular momentum, as in Kepler); for $3r_s/2<r<3r_s$ it is unstable.

Step 5 (the photon sphere from the null geodesic). For null geodesics the same substitution with $\kappa=0$ gives instead $$\left(\frac{dr}{d\lambda}\right)^2 = E^2 - \frac{L^2}{r^2}\left(1-\frac{r_s}{r}\right) \equiv E^2 - V^2_{\gamma}(r)$$ with $$V^2_\gamma(r) = L^2\left(\frac{1}{r^2}-\frac{r_s}{r^3}\right)$$ Setting the derivative of the bracket to zero, $$\frac{d}{dr}\left(\frac{1}{r^2}-\frac{r_s}{r^3}\right) = -\frac{2}{r^3}+\frac{3r_s}{r^4}=0 \;\Longrightarrow\; r=\frac{3r_s}{2}$$ directly locates the unique radius of circular photon orbits, $$r_{\text{ph}} = \dfrac{3}{2}r_s = \dfrac{3GM}{c^2}$$ matching the limit found in Step 3, and it is unstable because the bracket $1/r^2-r_s/r^3$ has a local maximum, not minimum, there — any inward nudge sends the photon spiraling into the horizon, any outward nudge sends it escaping to infinity.

Newtonian gravity, treated as a force on a fast-moving particle, predicts some bending, but it only accounts for the curvature of time (clocks run slow near mass). General relativity adds an equal contribution from the curvature of space (rulers shrink radially near mass), and the two effects add, doubling the deflection — the numerical factor that let the 1919 eclipse expedition distinguish Einstein's theory from Newton's.

Step 1 (the orbit equation for light). For a null geodesic confined to the equatorial plane, the same conserved quantities $E=(1-r_s/r)c^2\dot t$ and $L=r^2\dot\phi$ used for the photon sphere, substituted into $g_{\mu\nu}\dot x^\mu\dot x^\nu=0$ and converted from $\tau$-derivatives to a $\phi$-derivative of $u\equiv1/r$ (using $\dot r = -L\,du/d\phi$), give after one more differentiation the standard light-bending orbit equation $$\frac{d^2u}{d\phi^2}+u = \frac{3r_s}{2}u^2, \qquad u\equiv\frac1r$$

Step 2 (zeroth order: the straight line). Dropping the right-hand side (setting $r_s=0$, flat spacetime) gives $u_0''+u_0=0$, whose solution passing at closest distance $b$ at $\phi=\pi/2$ is the straight line $$u_0(\phi) = \frac{\sin\phi}{b}$$ — the undeflected path in polar form.

Step 3 (first-order perturbation). Write $u=u_0+r_s u_1$ and substitute into Step 1's equation, keeping only terms linear in $r_s$; the $u_0''+u_0=0$ part cancels and what remains is a driven linear oscillator for $u_1$: $$u_1''+u_1 = \frac{3}{2b^2}\sin^2\phi$$ A particular solution matching the symmetric bending expected on both sides of closest approach is $$u_1(\phi) = \frac{1}{2b^2}\left(1+\cos^2\phi\right)$$ so the full first-order trajectory is $$u(\phi) = \frac{\sin\phi}{b} + \frac{r_s}{2b^2}\left(1+\cos^2\phi\right)$$

Step 4 (extract the total bending angle). Flat space has $u\to0$ exactly at $\phi=0$ and $\phi=\pi$; with the $r_s$ correction, $u\to0$ at $\phi=-\delta_1$ and $\phi=\pi+\delta_2$ for small $\delta_1,\delta_2$. Expanding $u(-\delta_1)=0$ to first order in $\delta_1$ and $r_s$ gives $-\delta_1/b+r_s/b^2=0$, so $\delta_1=r_s/b$; the same computation at the other end gives $\delta_2=r_s/b$. The total angle by which the outgoing asymptote misses being exactly antiparallel to the incoming one is $$\Delta\phi = \delta_1+\delta_2 = \frac{2r_s}{b} = \frac{4GM}{c^2 b}$$ which, restoring $r_s=2GM/c^2$, is the claimed formula.

Gravity, in general relativity, always focuses light rather than defocusing it (that is what the energy condition encodes); a bundle of light rays that is already converging on both sides at some closed surface therefore keeps converging, and geometrically convergent light rays must cross — but two light rays crossing (a conjugate point) inside a region that a well-behaved, globally hyperbolic spacetime cannot make sense of without producing a boundary to the light ray's existence. The theorem turns this focusing intuition into a hard inequality (the Raychaudhuri equation) and a global topological obstruction.

Step 1 (trapped surface, in symbols). Let $S$ be a closed spacelike 2-surface, and let $\theta$ denote the expansion of a bundle of null geodesics leaving $S$ orthogonally — the fractional rate at which the bundle's cross-sectional area grows. Far from any mass, the outgoing bundle expands ($\theta_{\text{out}}>0$) and the ingoing one contracts ($\theta_{\text{in}}<0$). $S$ is trapped if instead $\theta_{\text{out}}<0, \qquad \theta_{\text{in}}<0$ on all of $S$: even the outgoing light is being dragged inward.

Step 2 (the Raychaudhuri equation). Differentiating the definition $\theta=\nabla_a k^a$ of the expansion along the null congruence with tangent $k^a$, and using the definition of the Riemann tensor to commute derivatives, gives the exact kinematic identity $$\frac{d\theta}{d\lambda} = -\frac{\theta^2}{2} - \sigma_{ab}\sigma^{ab} - R_{ab}k^ak^b$$ where $\sigma_{ab}$ is the shear tensor of the congruence. This equation is purely geometric — a statement about how any family of light rays must bend, following from the definition of curvature, before any physics is assumed.

Step 3 (energy condition forces further focusing). Physically reasonable matter satisfies the null energy condition $$R_{ab}k^ak^b \ge 0$$ and the shear term $\sigma_{ab}\sigma^{ab}\ge0$ always (it is a sum of squares); dropping both non-negative terms from the right of Step 2's identity gives the inequality $$\frac{d\theta}{d\lambda} \le -\frac{\theta^2}{2}$$

Step 4 (the Riccati inequality forces a finite-time blow-up). On a trapped surface $\theta_0<0$ at $\lambda=0$. As long as $\theta$ stays negative, dividing the inequality by $\theta^2>0$ and rearranging gives $$\frac{d}{d\lambda}\left(\frac{1}{\theta}\right) \ge \frac12 \;\Longrightarrow\; \frac{1}{\theta(\lambda)} \ge \frac{1}{\theta_0}+\frac{\lambda}{2}$$ Since the right side grows without bound as $\lambda$ increases while $1/\theta$ (negative) must satisfy this lower bound, $1/\theta$ is forced up toward $0^-$, i.e. $\theta\to-\infty$, at some finite affine parameter $\lambda_*\le 2/|\theta_0|$: the null congruence orthogonal to $S$ must develop a conjugate point (a caustic, where neighboring light rays cross) within finite affine time.

Step 5 (from local focusing to global incompleteness). A standard result of causal structure theory (used but not re-derived here) shows that a null geodesic in a globally hyperbolic spacetime cannot remain achronal — cannot stay on the boundary of the future of a set — past a conjugate point. Combined with the non-compact Cauchy surface, this rules out the geodesic looping back or terminating in an ordinary way, and forces at least one null geodesic orthogonal to $S$ to be future-incomplete: it cannot be extended to arbitrarily large affine parameter. By definition, this incompleteness is what Penrose calls a singularity — the theorem shows it must occur without any assumption of symmetry, whenever a trapped surface forms.