---
title: Null Geodesics and the Photon Sphere
module: The Schwarzschild Solution
moduleNumber: 6
lessonNumber: 3
order: 603
summary: >
  Light follows null geodesics, governed by a photon effective potential with a
  single unstable maximum at $3GM/c^2$, the photon sphere. The impact parameter
  sorts rays into those that escape with a deflection and those captured, with the
  critical value $b_c = 3\sqrt{3}\,GM/c^2$ dividing them. A grazing ray bends by
  $4GM/(c^2 b)$, twice the naive Newtonian value, and the critical impact
  parameter sets the edge of a black hole's shadow.
topics: [The Schwarzschild Solution]
draft: false
sources:
  - book: Hartle
    ref: "Gravity, Ch. 9–10 — Light Rays in the Schwarzschild Geometry"
  - book: Carroll
    ref: "Lecture Notes on General Relativity, §7 — Null Geodesics and Deflection of Light"
---

Light rays are null geodesics of the
[Schwarzschild metric](/relativity/the-schwarzschild-solution/the-schwarzschild-metric),
and the same conserved-quantity machinery that governed
[massive-particle orbits](/relativity/the-schwarzschild-solution/geodesics-and-orbits-in-schwarzschild)
governs them, with one change: the four-momentum of a photon is null rather than
timelike, so the normalization is $g_{\mu\nu}u^\mu u^\nu = 0$ 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 $\lambda$ rather than
proper time (a photon has no proper time). The energy and angular momentum
constants are

$$
\mathcal E = \left(1 - \frac{r_s}{r}\right)c^2\,\frac{\d t}{\d\lambda},
\qquad
\mathcal L = r^2\,\frac{\d\phi}{\d\lambda}.
$$

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

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

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 $g_{\mu\nu}u^\mu u^\nu = 0$ in the equatorial plane
and substituting the constants gives the radial equation

$$
\left(\frac{\d r}{\d\lambda}\right)^2 = \frac{\mathcal E^2}{c^2} - \frac{\mathcal L^2}{r^2}\left(1 - \frac{r_s}{r}\right).
$$

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

$$
\frac{1}{b^2} = \frac{1}{r^2}\left(1 - \frac{r_s}{r}\right) + \frac{1}{c^2\mathcal L^2}\left(\frac{\d r}{\d\lambda}\right)^2
\;\ge\; W(r),
\qquad
W(r) = \frac{1}{r^2}\left(1 - \frac{r_s}{r}\right).
$$

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 $W'(r) = 0$:

$$
-\frac{2}{r^3} + \frac{3 r_s}{r^4} = 0
\quad\Longrightarrow\quad
r_{\rm ph} = \frac{3}{2}\,r_s = \frac{3GM}{c^2}.
$$

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.

> **Definition (Photon sphere).** The sphere $r = 3GM/c^2 = \tfrac{3}{2}r_s$ on
> which light can orbit the mass on unstable circular null geodesics. It lies
> outside the horizon $r_s$ and inside the massive-particle ISCO $6GM/c^2$, and it
> governs both the capture of light and the apparent size of a black hole.

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

$$
W(r_{\rm ph}) = \frac{1}{(3GM/c^2)^2}\left(1 - \frac{2}{3}\right) = \frac{c^4}{27\,G^2M^2},
\qquad
b_c = \frac{1}{\sqrt{W(r_{\rm ph})}} = 3\sqrt{3}\,\frac{GM}{c^2} \approx 2.60\,r_s .
$$

$$
% caption: The photon effective potential W(r) has a single unstable maximum at
% the photon sphere r = 3 GM/c-squared. A ray with 1/b-squared below the peak
% (dashed line) has a turning point and escapes; one above the peak has none and is
% captured.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (-0.2,0) -- (6.0,0) node[right] {r};
  \draw[->, black] (0,-0.2) -- (0,2.3) node[above] {W(r)};
  % W curve, x=0.55(r-2), y=40 W
  \draw[acc, very thick] plot coordinates
    {(0,0)(0.275,1.28)(0.55,1.48)(0.825,1.40)(1.10,1.25)(1.65,0.96)(2.20,0.74)(3.30,0.47)(4.40,0.32)(5.50,0.23)};
  % peak marker photon sphere
  \fill[acc] (0.55,1.48) circle (1.7pt);
  \node[acc, anchor=south west] at (0.5,1.54) {photon sphere};
  % escape level (turning point on rising branch)
  \draw[black, dashed] (0,0.96) -- (5.7,0.96);
  \fill[black] (1.65,0.96) circle (1.5pt);
  \node[black, anchor=south west] at (1.7,0.98) {turning point (escape)};
  % capture level above peak
  \draw[black, densely dotted] (0,1.95) -- (5.7,1.95);
  \node[black, anchor=south] at (2.9,1.97) {no turning point: capture};
\end{tikzpicture}
$$

## Capture, escape, and the critical ray

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

- **$b > b_c$** — then $1/b^2 < W(r_{\rm ph})$, 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 $1/b^2 > W(r_{\rm ph})$, 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
$\pi b_c^2 = 27\pi (GM/c^2)^2$ exceeds the horizon's own area cross-section
$\pi r_s^2$ by a factor $27/4$.

$$
% caption: Rays sorted by impact parameter. Wide rays (b greater than b-c) reach a
% turning point and escape with a bend; the critical ray (b = b-c) spirals onto the
% photon sphere; narrow rays (b less than b-c) cross the horizon and are captured.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % horizon and photon sphere
  \fill[black!78] (0,0) circle (0.5);
  \draw[dashed] (0,0) circle (0.75);
  % critical radius circle
  \draw[black, densely dotted] (0,0) circle (1.30);
  \node[anchor=west] at (1.5,-1.3) {photon sphere};
  \draw (1.45,-1.25) -- (0.55,-0.52);
  % escaping ray (b large): comes from left high, bends around top
  \draw[black, very thick] plot coordinates
    {(-4.2,1.75)(-2.2,1.65)(-0.8,1.35)(0.4,1.5)(1.8,2.1)(3.4,2.9)};
  \node[black, anchor=south] at (-2.6,1.9) {escape};
  % critical ray: grazes and wraps the photon sphere
  \draw[acc, very thick] plot coordinates
    {(-4.2,1.30)(-2.2,1.22)(-1.2,1.02)(-0.5,0.78)(0.05,0.78)(0.55,0.55)(0.74,0.05)(0.5,-0.6)};
  \node[acc, anchor=south] at (-3.4,1.34) {critical};
  % captured ray (b small): straight into hole
  \draw[black, very thick] (-4.2,0.30) -- (-0.42,0.14);
  \node[black, anchor=south] at (-3.4,0.32) {capture};
  \draw[->, black] (-0.95,0.24) -- (-0.5,0.18);
\end{tikzpicture}
$$

## The deflection of a grazing ray

For a ray that stays far from the mass, $b \gg r_s$, the deflection is small and
computable in closed form. The total change in $\phi$ as the ray comes from
infinity, reaches its turning point $r_0$, and returns to infinity is

$$
\Delta\phi_{\rm total} = 2\int_{r_0}^{\infty}\frac{\d r}{r^2\sqrt{\dfrac{1}{b^2} - \dfrac{1}{r^2}\left(1 - \dfrac{r_s}{r}\right)}} .
$$

A straight line accumulates exactly $\pi$, so the **deflection angle** is
$\delta = \Delta\phi_{\rm total} - \pi$. Expanding the integrand to first order in
the small quantity $r_s/r$ and carrying out the integral gives

$$
\delta = \frac{2 r_s}{b} = \frac{4GM}{c^2 b} .
$$

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 $2GM/(c^2 b)$.
The extra factor of two comes from the curvature of space — the $g_{rr}$ 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.

> **Worked example (Deflection at the solar limb).** A ray grazing the Sun's
> surface has $b = R_\odot = 6.96\times10^8\ \text{m}$ and $r_s = 2.95\times10^3\
> \text{m}$, so
>
> $$
> \delta = \frac{2 r_s}{b} = \frac{2 \times 2.95\times10^3}{6.96\times10^8}
> = 8.5\times10^{-6}\ \text{rad} = 1.75'' .
> $$
>
> The 1919 eclipse measurement of this $1.75''$ shift, twice the Newtonian
> $0.87''$, confirmed general relativity over the Newtonian alternative. The
> [lensing lesson](/relativity/tests-of-general-relativity/deflection-of-light-and-gravitational-lensing)
> develops the measurement and its extension to gravitational lensing.

$$
% caption: A light ray with impact parameter b passing a mass M is deflected by
% delta = 4 GM / (c-squared b). The incoming and outgoing straight asymptotes
% (dashed) meet at the deflection angle; the actual ray (solid) bends toward the
% mass.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw (0,0) circle (0.4);
  \node[font=\scriptsize] at (0,0) {M};
  % impact parameter (undeflected asymptote height)
  \draw[black, dashed] (-4.5,1.4) -- (4.5,1.4);
  \draw[<->, black] (-3.6,0) -- (-3.6,1.4);
  \node[black, anchor=east] at (-3.65,0.7) {b};
  % bent ray
  \draw[acc, very thick] plot coordinates
    {(-4.5,1.4)(-2.5,1.35)(-0.9,1.15)(0.9,0.9)(2.6,0.55)(4.5,0.15)};
  % outgoing straight asymptote
  \draw[black, dashed] (0.9,0.9) -- (4.7,0.14);
  % deflection angle arc between the two dashed asymptotes near right
  \draw[->, black] (3.8,1.4) arc (0:-11:3.0);
  \node[black, anchor=west] at (3.9,1.2) {bend angle};
\end{tikzpicture}
$$

## 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

$$
\theta_{\rm sh} = \frac{b_c}{D} = \frac{3\sqrt{3}\,GM}{c^2 D}
$$

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 $r_{\rm ph}$ 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.

$$
% caption: The black-hole shadow. Lines of sight with impact parameter below b-c
% end on the horizon and return no light, forming a dark disk of apparent radius
% b-c = 3 sqrt-3 GM/c-squared, larger than the horizon because the photon sphere
% bends grazing rays around the hole.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % the hole and photon sphere
  \fill[black!80] (0,0) circle (0.55);
  \draw[acc, dashed] (0,0) circle (0.85);
  % shadow edge radius b_c
  \draw[black] (0,0) circle (1.45);
  \node[acc, anchor=south east] at (-0.6,0.6) {r$_{\rm ph}$};
  % rays that graze and escape, defining edge
  \draw[black, very thick] plot coordinates
    {(-4.0,1.45)(-2.2,1.4)(-1.0,1.2)(0.2,1.3)(1.6,1.75)(3.2,2.4)};
  \draw[black, very thick] plot coordinates
    {(-4.0,-1.45)(-2.2,-1.4)(-1.0,-1.2)(0.2,-1.3)(1.6,-1.75)(3.2,-2.4)};
  % captured line of sight
  \draw[black, very thick] (-4.0,0.3) -- (-0.5,0.15);
  \draw[->, black] (-1.0,0.24) -- (-0.55,0.18);
  \node[black, anchor=west] at (-4.0,0.55) {into horizon};
  % shadow radius label
  \draw[<->, black] (0,0) -- (1.45,0);
  \node[black, anchor=north] at (0.9,-0.08) {b$_c$};
\end{tikzpicture}
$$

The three lengths of the Schwarzschild geometry now stand in a fixed ratio:
the horizon at $r_s$, the photon sphere at $\tfrac{3}{2}r_s$, 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](/relativity/tests-of-general-relativity/perihelion-precession-of-mercury)
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](/relativity/black-holes/horizons-and-coordinate-singularities)
follows the captured rays across the horizon.

[^hartle-light]: **Hartle**, _Gravity_, Ch. 9–10 — null geodesics in the Schwarzschild geometry, the photon effective potential and the unstable photon sphere at $3GM/c^2$, the deflection of light, and the impact-parameter picture of capture and escape.

[^carroll-null]: **Carroll**, _Lecture Notes on General Relativity_, §7 — null geodesics of Schwarzschild, the critical impact parameter $b_c = 3\sqrt{3}\,GM/c^2$, and the deflection angle $4GM/(c^2 b)$ as twice the Newtonian value.