Null Geodesics and the Photon Sphere

Light rays are null geodesics of the Schwarzschild metric, and the same conserved-quantity machinery that governed massive-particle orbits governs them, with one change: the four-momentum of a photon is null rather than timelike, so the normalization is $g_{μν} u^{μ} u^{ν} = 0$ instead of $- c^{2}$ . Dropping that one term reshapes the effective potential into a form with a single maximum. That maximum is the photon sphere, an orbit of light; the impact parameter relative to it decides whether a ray escapes or is swallowed; and the weak deflection of a grazing ray is the effect Eddington measured in 1919.

Null geodesics and the impact parameter

A light ray in the equatorial plane carries the same two conserved quantities as a massive orbit, but parametrized by an affine parameter $λ$ rather than proper time (a photon has no proper time). The energy and angular momentum constants are

E = (1 - \frac{r _{s}}{r}) c^{2} \frac{d t}{d λ}, L = r^{2} \frac{d ϕ}{d λ} .

Only their ratio has invariant meaning for a photon, since rescaling $λ$ rescales both. That ratio is the impact parameter

b = \frac{c L}{E},

the perpendicular distance between the mass and the straight line the ray would follow if the mass were absent, measured far away where spacetime is flat. It is the one physical label of a light ray.

Imposing the null condition $g_{μν} u^{μ} u^{ν} = 0$ in the equatorial plane and substituting the constants gives the radial equation

(\frac{d r}{d λ})^{2} = \frac{E ^{2}}{c ^{2}} - \frac{L ^{2}}{r ^{2}} (1 - \frac{r _{s}}{r}) .

Dividing by $L^{2} / c^{2}$ and writing everything in terms of $b$ isolates a purely geometric potential,

\frac{1}{b ^{2}} = \frac{1}{r ^{2}} (1 - \frac{r _{s}}{r}) + \frac{1}{c ^{2} L ^{2}} (\frac{d r}{d λ})^{2} \geq W (r), W (r) = \frac{1}{r ^{2}} (1 - \frac{r _{s}}{r}) .

The radial velocity vanishes at a turning point, where $1/ b^{2} = W (r)$ . Whether a ray has a turning point — and so escapes — is decided by comparing $1/ b^{2}$ against the maximum of $W$ .

The photon sphere

The photon potential $W (r)$ has one extremum, from $W^{'} (r) = 0$ :

- \frac{2}{r ^{3}} + \frac{3 r _{s}}{r ^{4}} = 0 ⟹ r_{ph} = \frac{3}{2} r_{s} = \frac{3 GM}{c ^{2}} .

Unlike the massive-particle potential, which has both a minimum and a maximum, $W (r)$ has only this single maximum. It is a circular light orbit — the photon sphere — and because it sits at a maximum of the potential it is unstable: a photon nudged inward spirals to the horizon, one nudged outward escapes to infinity. No stable circular orbit of light exists in the Schwarzschild geometry.

The maximum value of the potential fixes a critical impact parameter through $1/ b_{c}^{2} = W (r_{ph})$ . Evaluating,

W (r_{ph}) = \frac{1}{( 3 GM / c ^{2} ) ^{2}} (1 - \frac{2}{3}) = \frac{c ^{4}}{27 G ^{2} M ^{2}}, b_{c} = \frac{1}{W ( r _{ph} )} = 33 \frac{GM}{c ^{2}} \approx 2.60 r_{s} .

The photon effective potential W(r) has a single unstable maximum at the photon sphere r = 3 GM/c-squared. A ray with 1/b-squared below the peak (dashed line) has a turning point and escapes; one above the peak has none and is captured.

Capture, escape, and the critical ray

The maximum of $W$ divides light rays into two classes by their impact parameter.

$b > b_{c}$ — then $1/ b^{2} < W (r_{ph})$ , the line $1/ b^{2}$ meets the rising branch of $W$ at a turning point outside the photon sphere, and the ray reaches a minimum radius and escapes, deflected by an angle set below.
$b < b_{c}$ — then $1/ b^{2} > W (r_{ph})$ , no turning point exists, and the ray falls monotonically through the photon sphere and across the horizon. The photon is captured.
$b = b_{c}$ — the marginal ray, whose turning point lands right on the photon sphere. It spirals in and asymptotically wraps the photon sphere infinitely many times without crossing it, the boundary case between capture and escape.

The capture cross-section for light is therefore a disk of radius $b_{c}$ : a black hole of Schwarzschild radius $r_{s}$ absorbs every ray aimed within $b_{c} = 2.60 r_{s}$ of its center and deflects the rest. The geometric cross-section $π b_{c}^{2} = 27 π (GM / c^{2})^{2}$ exceeds the horizon's own area cross-section $π r_{s}^{2}$ by a factor $27/4$ .

Rays sorted by impact parameter. Wide rays (b greater than b-c) reach a turning point and escape with a bend; the critical ray (b = b-c) spirals onto the photon sphere; narrow rays (b less than b-c) cross the horizon and are captured.

The deflection of a grazing ray

For a ray that stays far from the mass, $b ≫ r_{s}$ , the deflection is small and computable in closed form. The total change in $ϕ$ as the ray comes from infinity, reaches its turning point $r_{0}$ , and returns to infinity is

Δ ϕ_{total} = 2 r_{0} \int \infty \frac{d r}{r ^{2} \frac{1}{b ^{2}} - \frac{1}{r ^{2}} ( 1 - \frac{r _{s}}{r} )} .

A straight line accumulates exactly $π$ , so the deflection angle is $δ = Δ ϕ_{total} - π$ . Expanding the integrand to first order in the small quantity $r_{s} / r$ and carrying out the integral gives

δ = \frac{2 r _{s}}{b} = \frac{4 GM}{c ^{2} b} .

The result comes out to twice what a naive calculation gives by treating light as a Newtonian particle moving at speed $c$ past the mass, which yields $2 GM / (c^{2} b)$ . The extra factor of two comes from the curvature of space — the $g_{r r}$ part of the metric — which a purely Newtonian argument, involving only the time dilation $g_{tt}$ , omits. Half the bending is the gravitational pull on the light's energy; the other half is the spatial geometry the ray traverses.

A light ray with impact parameter b passing a mass M is deflected by delta = 4 GM / (c-squared b). The incoming and outgoing straight asymptotes (dashed) meet at the deflection angle; the actual ray (solid) bends toward the mass.

The shadow of a black hole

An observer far from a black hole, looking at it against a bright background, sees a dark disk. Every line of sight whose impact parameter is below $b_{c}$ terminates on the horizon, so no background light arrives along it; lines of sight with $b > b_{c}$ collect light that swings past and escapes. The boundary is the critical impact parameter, and the dark disk — the shadow — has apparent angular radius

θ_{sh} = \frac{b _{c}}{D} = \frac{3 3 GM}{c ^{2} D}

for a hole at distance $D$ . The shadow is larger than the horizon: its radius $b_{c} = 2.60 r_{s}$ exceeds $r_{s}$ because the photon sphere bends grazing rays around the hole, so the silhouette is set by $r_{ph}$ and the critical ray, not by the horizon itself. The shadow is the observable that the Event Horizon Telescope images resolve, and its size measures $GM / c^{2}$ directly.

The black-hole shadow. Lines of sight with impact parameter below b-c end on the horizon and return no light, forming a dark disk of apparent radius b-c = 3 sqrt-3 GM/c-squared, larger than the horizon because the photon sphere bends grazing rays around the hole.

The three lengths of the Schwarzschild geometry now stand in a fixed ratio: the horizon at $r_{s}$ , the photon sphere at $\frac{3}{2} r_{s}$ , and the innermost stable orbit of matter at $3 r_{s}$ , with the shadow edge projected from $b_{c} = 2.60 r_{s}$ . The tests module takes the weak-field limits of these results — precession, deflection, redshift, and time delay — into the numbers the solar system supplies, and the black-holes module follows the captured rays across the horizon.