Photon Orbits and the Photon Sphere

Send a beam of light past a black hole and it bends; send it closer and it bends more. Keep closing in and the ordinary story of a deflected ray breaks down: there is one radius at which a photon, given exactly the right aim, neither escapes nor plunges through the horizon but circles the hole forever on a closed null geodesic. That radius is the photon sphere, and in Schwarzschild geometry it sits at $r_{\mathrm{ph}}=3GM/c^2$ — precisely one and a half times the horizon radius, fixed by the geometry alone, independent of the photon's colour or energy. This article takes the machinery the preceding chapter built — the Schwarzschild metric, its two Killing symmetries, the conserved energy $E$ and angular momentum $L$ , and the reduction of geodesic motion to a one-dimensional effective potential — and asks what it says about light. The answer differs from the massive-particle case in one decisive way: light has no rest-mass term, so its effective potential has a single extremum, a maximum. A maximum means the circular orbit is unstable, and it means there is exactly one such orbit — not the family of stable and unstable circular orbits massive particles enjoy. Everything observable flows from that maximum. Its height fixes a single number, the critical impact parameter $b_{\mathrm{crit}}=3\sqrt3\,GM/c^2\approx5.196\,GM/c^2$ , which separates capture from escape and draws the dark disk we call the black-hole shadow. Rays that just miss it wind around the hole an unbounded number of times, stacking into the thin photon ring the Event Horizon Telescope resolved around M87 and Sgr A, while the same instability sets the pitch and decay of a black hole's gravitational-wave ringdown. The photon sphere is where strong-field gravity becomes something you can point a telescope at.

Key takeaways

In Schwarzschild spacetime a photon can orbit on a single unstable circular null geodesic at $r_{\mathrm{ph}}=3GM/c^2=\tfrac32 r_s$ — the photon sphere.
It comes from the photon effective potential $V_{\mathrm{eff}}=\dfrac{L^2}{r^2}\left(1-\dfrac{r_s}{r}\right)$ , which has a single maximum; there is no stable light orbit, in sharp contrast to massive particles.
The height of that maximum fixes the critical impact parameter $b_{\mathrm{crit}}=3\sqrt3\,GM/c^2\approx5.196\,GM/c^2$ : rays with $b<b_{\mathrm{crit}}$ are captured, $b>b_{\mathrm{crit}}$ escape, $b=b_{\mathrm{crit}}$ spirals onto the sphere.
$b_{\mathrm{crit}}$ is the radius of the black-hole shadow; its angular size $b_{\mathrm{crit}}/D$ is tens of microarcseconds for Sgr A and M87, and was imaged by the EHT.
As $b\to b_{\mathrm{crit}}^{+}$ the deflection diverges logarithmically, producing the photon ring and an infinite tower of relativistic lensing images.
Key idea: null geodesics in Schwarzschild obey $\dot r^2=E^2-V_{\mathrm{eff}}(r)$ with $V_{\mathrm{eff}}=\dfrac{L^2}{r^2}\!\left(1-\dfrac{r_s}{r}\right)$ ; this potential has a unique maximum at $r=3GM/c^2$ , the unstable photon sphere, whose height sets $b_{\mathrm{crit}}=3\sqrt3\,GM/c^2$ .

The core insight

Null geodesics in Schwarzschild obey $\dot r^2=E^2-V_{\mathrm{eff}}(r)$ with $V_{\mathrm{eff}}=\dfrac{L^2}{r^2}\!\left(1-\dfrac{r_s}{r}\right)$ . This potential has a unique maximum at $r=3GM/c^2$ , the unstable photon sphere; the height of the maximum fixes the critical impact parameter $b_{\mathrm{crit}}=3\sqrt3\,GM/c^2$ , which draws the black-hole shadow.

Where this lives today

The photon sphere stopped being a textbook abstraction on 10 April 2019, when the Event Horizon Telescope — a planet-sized array of radio dishes synthesised into one interferometer — published the first image of a black hole, the shadow of M87, followed in 2022 by Sgr A. The bright ring is lensed emission from the hot accretion flow; the dark centre is the shadow of radius $\sqrt{27}\,GM/c^2$ . Just inside it is the far sharper photon ring, the exponentially thin stack of sub-images ( $n=1,2,\dots$ ) from light that looped the photon sphere before escaping. Because each sub-ring's diameter is set almost purely by the metric — nearly independent of the messy astrophysics — resolving its $e^{-\pi}$ -spaced substructure is the primary science case for space-based VLBI such as the Black Hole Explorer. The same instability governs the time domain: a merger remnant rings down in quasinormal modes whose eikonal frequency and damping are the orbital frequency and Lyapunov exponent of the photon sphere, so every LIGO–Virgo–KAGRA ringdown is, in the high-frequency limit, light orbiting a black hole.

The physical picture

Picture a photon as something that always tries to go straight — but in curved spacetime "straight" means "geodesic," and near a black hole the geometry is bent so severely that at one radius a photon's straight-ahead tendency curves it back onto itself. The one-dimensional radial problem makes this precise: light rolls like a particle of pseudo-energy $E^2$ in the potential $V_{\mathrm{eff}}(r)$ , and a circular orbit sits where $V_{\mathrm{eff}}$ is flat. The first figure is that potential, with its single, telltale maximum.

Figure

The photon effective potential V_eff = (L²/r²)(1 − r_s/r) has one maximum, at r = 1.5 r_s = 3GM/c². Its height equals 1/b_crit² with b_crit = 3√3 GM/c² ≈ 5.196 GM/c². A photon with b < b_crit clears the barrier and is captured; b > b_crit turns back and escapes; b = b_crit asymptotes onto the photon sphere

Because the orbit sits on a hilltop, the balance is precarious. Nudge the photon inward and it accelerates through the horizon; nudge it outward and it rolls to infinity. This is why the photon sphere is not a shell that traps light: no real photon lingers there unless aimed with impossible precision. What the instability does give is a threshold — the barrier height decides whether an incoming ray clears the crest (capture) or turns back (escape), and the threshold aim is the single number $b_{\mathrm{crit}}=3\sqrt3\,GM/c^2$ . The second figure places the three characteristic radii — horizon, photon sphere, and apparent shadow — to scale, and shows the three fates of a ray.

Figure

Three radii to scale for a Schwarzschild hole: the event horizon at r_s = 2GM/c², the photon sphere at r_ph = 3GM/c², and the apparent shadow of radius b_crit = 3√3 GM/c² ≈ 5.196 GM/c². A ray with b = 2.5 GM/c² is captured, one with b = 5.196 GM/c² spirals onto the photon sphere, and one with b = 7 GM/c² is deflected and escapes

Two facts organize what follows. First, the shadow ( $\sim5.2\,GM/c^2$ ) is markedly larger than the horizon ( $2GM/c^2$ ) or photon sphere ( $3GM/c^2$ ), because strong bending funnels large-offset rays into capture. Second, none of these radii depends on the photon's energy: $E$ and $L$ enter only through $b=L/E$ , so the geometry alone — the mass $M$ — fixes where light orbits.

The core equations

We work in the equatorial plane $\theta=\pi/2$ (allowed by spherical symmetry) and adopt geometric units $c=1$ for the dynamics, restoring $c$ in the boxed results. The stage is the Schwarzschild line element, whose single function $1-r_s/r$ carries all the strong-field physics.

\mathrm{d}s^2 = -\left(1-\frac{r_s}{r}\right)c^2\,\mathrm{d}t^2 + \left(1-\frac{r_s}{r}\right)^{-1}\!\mathrm{d}r^2 + r^2\,\mathrm{d}\Omega^2 \qquad (1)

where: $r_s=2GM/c^2$ — Schwarzschild radius (horizon); $r$ — areal radial coordinate (a sphere at $r$ has area $4\pi r^2$ ); $\mathrm{d}\Omega^2=\mathrm{d}\theta^2+\sin^2\theta\,\mathrm{d}\phi^2$ — unit-sphere element; $c$ — speed of light. Reading: the geometry is static and spherically symmetric, so $\partial_t$ and $\partial_\phi$ are Killing vectors — the seeds of two conserved quantities. Those Killing vectors give a conserved energy and angular momentum along every geodesic, collapsing the motion to a one-dimensional radial problem.

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

where: $\lambda$ — affine parameter (not proper time, which vanishes for light); $E$ — conserved energy; $L$ — conserved angular momentum; overdots denote $\mathrm{d}/\mathrm{d}\lambda$ . Reading: only the ratio $b\equiv L/E$ survives into the observable physics — light carries no length scale of its own. Feeding the two constants into the null condition $g_{\mu\nu}\dot x^\mu\dot x^\nu=0$ gives the radial energy equation, which looks like a nonrelativistic particle of energy $E^2$ in a potential.

\left(\frac{\mathrm{d}r}{\mathrm{d}\lambda}\right)^2 = E^2 - \frac{L^2}{r^2}\left(1-\frac{r_s}{r}\right) \qquad (3)

where: the right side is $E^2-V_{\mathrm{eff}}(r)$ ; turning points are where $\dot r=0$ . Reading: the missing rest-mass term is the whole difference from the massive case — it is why the potential below has no Newtonian well to build a stable orbit. Reading the potential off equation (3) isolates the object that controls everything: a centrifugal $L^2/r^2$ barrier, cut down at small $r$ by the factor $1-r_s/r$ .

V_{\mathrm{eff}}(r) = \frac{L^2}{r^2}\left(1-\frac{r_s}{r}\right) \qquad (4)

where: $V_{\mathrm{eff}}$ — photon effective potential; capture versus escape depends on whether $E^2$ exceeds the barrier peak. Reading: it falls as $L^2/r^2$ at large $r$ and is dragged to zero as $r\to r_s$ , so something between must be a maximum — the photon sphere. Extremizing $V_{\mathrm{eff}}$ locates that maximum — a single root, which the second-derivative test brands unstable.

r_{\mathrm{ph}} = \frac{3}{2}\,r_s = \frac{3GM}{c^2}, \qquad V_{\mathrm{eff}}''(r_{\mathrm{ph}})<0 \qquad (5)

where: $r_{\mathrm{ph}}$ — photon-sphere radius, the unique unstable circular null orbit. Reading: $1.5$ horizon radii, independent of $E$ , $L$ , and frequency — a pure property of $M$ . The observable born from the barrier height is the critical impact parameter — the capture threshold and, equivalently, the radius of the shadow on the sky.

b\equiv\frac{L}{E}, \qquad b_{\mathrm{crit}} = 3\sqrt3\,\frac{GM}{c^2} \approx 5.196\,\frac{GM}{c^2} \qquad (6)

where: $b$ — impact parameter, the perpendicular offset of an incoming ray's asymptote; $b_{\mathrm{crit}}$ — threshold separating capture from escape. Reading: equivalently $b_{\mathrm{crit}}=\tfrac{3\sqrt3}{2}\,r_s=\sqrt{27}\,GM/c^2$ ; the $\sqrt{27}$ is the fingerprint of the Schwarzschild shadow. Finally, changing to $u=1/r$ casts the orbit in Binet form, whose turning-point structure controls the bending of light near $b_{\mathrm{crit}}$ .

\left(\frac{\mathrm{d}u}{\mathrm{d}\phi}\right)^2 = \frac{1}{b^2} - u^2\left(1-r_s u\right), \qquad \Delta\phi = 2\!\int_0^{u_{\max}}\!\frac{\mathrm{d}u}{\sqrt{1/b^2 - u^2 + r_s u^3}} - \pi \qquad (7)

where: $u=1/r$ — reciprocal radius; $u_{\max}=1/r_0$ — reciprocal of the closest-approach radius; $\Delta\phi$ — net deflection angle. Reading: the cubic $r_s u^3$ is the relativistic correction to the Newtonian conic; as $b\to b_{\mathrm{crit}}$ the integrand develops a double root and $\Delta\phi$ diverges logarithmically.

Derivation AThe photon sphere from the effective potential

The goal is to reduce null-geodesic motion in Schwarzschild to a one-dimensional radial problem, locate the single circular light orbit, and prove it unstable — establishing $r_{\mathrm{ph}}=3GM/c^2$ as a maximum of the effective potential, which is why light, unlike matter, has no bound circular orbit anywhere in this geometry. We exploit the two Killing symmetries and the null normalization, then classify the extremum with the second-derivative test.

Step 1. The Killing vectors $\partial_t$ and $\partial_\phi$ give the conserved $E=(1-r_s/r)\dot t$ and $L=r^2\dot\phi$ (equation 2), and spherical symmetry lets us set $\theta=\pi/2$ . Why: two constants of motion turn four coupled equations into one equation for $r(\lambda)$ , as in the Kepler problem.

Step 2. Impose the null condition, which for equatorial motion reads $-(1-r_s/r)\dot t^2+(1-r_s/r)^{-1}\dot r^2+r^2\dot\phi^2=0$ . Why: a photon's four-velocity has zero length, which removes the rest-mass term a massive particle carries as $g_{\mu\nu}\dot x^\mu\dot x^\nu=-c^2$ .

Step 3. Substitute $\dot t=E/(1-r_s/r)$ and $\dot\phi=L/r^2$ , multiply by $(1-r_s/r)$ , and isolate the radial term: $\dot r^2=E^2-\tfrac{L^2}{r^2}(1-r_s/r)=E^2-V_{\mathrm{eff}}$ . Why: this is the one-dimensional energy equation; the photon "rolls" in $V_{\mathrm{eff}}$ with pseudo-energy $E^2$ , and every feature of the orbit follows from one curve.

Step 4. A circular orbit needs $\dot r=0$ and $\ddot r=0$ , which force $V_{\mathrm{eff}}'(r)=0$ . Differentiating, $V_{\mathrm{eff}}'=L^2\!\left(-\tfrac{2}{r^3}+\tfrac{3r_s}{r^4}\right)=0$ , so $-2r+3r_s=0$ and $r_{\mathrm{ph}}=\tfrac32 r_s=3GM/c^2$ . Why: a single root — the geometry admits exactly one photon orbit, not a family, because the potential has one extremum.

Step 5. Classify: $V_{\mathrm{eff}}''=L^2\!\left(\tfrac{6}{r^4}-\tfrac{12 r_s}{r^5}\right)$ , which at $r=\tfrac32 r_s$ equals $-\tfrac{2L^2}{r^4}<0$ . The extremum is a maximum, so the orbit is unstable — any radial perturbation grows exponentially in $\lambda$ . Why: this negative curvature is the "hilltop," directly responsible for the shadow's sharp edge, the photon ring, and the ringdown damping. *Note: the peak height evaluates to $V_{\mathrm{eff}}(r_{\mathrm{ph}})=L^2/(3\cdot(3r_s/2)^2)=4L^2/(27r_s^2)$ , which equals $1/b_{\mathrm{crit}}^2$ — confirming that the barrier height and the critical impact parameter are two faces of the same number.*

Go deeper · Derivation B · 1 minThe critical impact parameter and the black-hole shadow

Derivation BThe critical impact parameter and the black-hole shadow

The goal is to convert the barrier height into the number a distant observer measures — the angular radius of the dark shadow — by finding the impact parameter $b_{\mathrm{crit}}$ that just lets a photon reach the top of the potential, then projecting it onto the sky. This turns an abstract feature of $V_{\mathrm{eff}}$ into the testable microarcsecond prediction the EHT set out to measure.

Step 1. Divide the radial equation by $E^2$ and use $b=L/E$ : $\dot r^2/E^2=1-\tfrac{b^2}{r^2}(1-r_s/r)$ . A photon turns around where the bracket vanishes; it is captured if the bracket stays positive all the way in. Why: in terms of $b$ alone, a ray's fate depends only on its aim, not its energy — the geometry is scale-free for light.

Step 2. The barrier the ray must clear is $b^2\,W(r)$ with $W(r)=(1-r_s/r)/r^2$ , and capture happens when the peak of $b^2 W$ reaches unity. The peak of $W$ sits at the photon sphere (the same extremum as before), so threshold is $b_{\mathrm{crit}}^2\,W(r_{\mathrm{ph}})=1$ . Why: at threshold the turning point merges with the circular orbit — the ray asymptotically spirals onto the photon sphere.

Step 3. Evaluate at $r_{\mathrm{ph}}=\tfrac32 r_s$ : with $1-r_s/r_{\mathrm{ph}}=\tfrac13$ , $W(r_{\mathrm{ph}})=\tfrac13\big/(\tfrac32 r_s)^2=\tfrac{4}{27r_s^2}$ , so $b_{\mathrm{crit}}^2=27r_s^2/4$ and $b_{\mathrm{crit}}=\tfrac{3\sqrt3}{2}r_s=3\sqrt3\,GM/c^2\approx5.196\,GM/c^2$ . Why: the clean $\sqrt{27}$ is the model-independent signature — measure the shadow, and with an independent mass you test the metric in the strong field.

Step 4. Project onto the sky. A captured ray had asymptotic offset $b$ , so the boundary of captured directions is a circle of angular radius $\theta_{\mathrm{sh}}=b_{\mathrm{crit}}/D$ for $D\gg r_s$ — the shadow. Why: no light reaches the observer from inside $\theta_{\mathrm{sh}}$ ; the shadow is the capture cross-section, lensed beyond the horizon's own angular size.

\theta_{\mathrm{sh}} \approx \frac{b_{\mathrm{crit}}}{D} = \frac{3\sqrt3\,GM}{c^2 D} \qquad (\text{shadow angular radius})

Go deeper · Derivation C · 1 minLogarithmic divergence of the strong-field deflection

Derivation CLogarithmic divergence of the strong-field deflection

The goal is to show that as the impact parameter approaches its critical value the bending angle grows without bound, so a single black hole can produce an infinite sequence of "relativistic images" and a stack of photon-ring sub-images. We expand the deflection integral near the photon sphere and expose the double root that makes it logarithmically divergent — the analytic heart of black-hole imaging.

Step 1. Write $\Delta\phi+\pi=2\int_{r_0}^{\infty}\mathrm{d}r\,\big[r^2\sqrt{1/b^2-(1-r_s/r)/r^2}\,\big]^{-1}$ , where $r_0(b)$ is the closest-approach radius from $1/b^2=(1-r_s/r_0)/r_0^2$ . Why: what the observer sees is encoded in how many radians the ray swept — $\Delta\phi=\pi$ undeflected, $2\pi$ one full loop, and so on.

Step 2. As $b\to b_{\mathrm{crit}}^{+}$ , $r_0\to r_{\mathrm{ph}}$ , where both $\dot r=0$ and $V_{\mathrm{eff}}'=0$ hold, so the radicand develops a double zero, not the simple zero of weak lensing. Why: a simple zero gives a finite integral (weak-field $\Delta\phi=4GM/c^2 b$ ); a double zero pushes the deflection to infinity.

Step 3. Expand the radicand locally, $1/b^2-(1-r_s/r)/r^2\approx \alpha\,(r-r_0)^2+\beta\,(b-b_{\mathrm{crit}})$ with $\alpha,\beta>0$ set by $V_{\mathrm{eff}}''(r_{\mathrm{ph}})$ . The integral then behaves as $\int \mathrm{d}r/\sqrt{\alpha(r-r_0)^2+\cdots}\sim -\tfrac{1}{\sqrt\alpha}\ln(b/b_{\mathrm{crit}}-1)$ . Why: the coefficient $\sqrt\alpha$ is fixed by the orbit's Lyapunov exponent — the same negative $V_{\mathrm{eff}}''$ that made the orbit unstable now controls how the images pile up.

Step 4. Assemble the strong-deflection limit $\Delta\phi\simeq-\ln\!\big(b/b_{\mathrm{crit}}-1\big)+\text{const}$ . Each extra half-loop requires $b/b_{\mathrm{crit}}-1$ to shrink by $e^{-\pi}\approx0.0432$ , so successive images crowd exponentially onto the shadow edge. Why: this $e^{-\pi}$ demagnification is the self-similarity of the photon ring — the $n=1,2,3,\dots$ sub-rings are copies of the accretion flow, each fainter and thinner by a fixed ratio.

\Delta\phi \simeq -\ln\!\left(\frac{b}{b_{\mathrm{crit}}}-1\right) + \text{const} \qquad (\text{strong-field limit})

Limiting cases

Case	Result	Meaning
Newtonian / weak field	$r\gg r_s$	No photon sphere in Newtonian gravity; it needs the full $1-r_s/r$ factor. The weak-field deflection $4GM/c^2 b$ is the far tail of the same integral.
Ratio to the horizon	$r_{\mathrm{ph}}=\tfrac32 r_s$	The photon sphere always sits at exactly $1.5$ horizon radii for Schwarzschild — it never touches the horizon.
Critical aim	$b\to b_{\mathrm{crit}}=3\sqrt3\,GM/c^2$	Deflection diverges as $-\ln(b/b_{\mathrm{crit}}-1)$ ; the ray loops arbitrarily many times, generating the photon ring.
Massless limit of the orbit family	$m\to0$	The marginally bound orbit ( $4GM/c^2$ ) and the ISCO ( $6GM/c^2$ ) of massive particles migrate down and merge onto the single unstable null orbit at $3GM/c^2$ .
Rotating (Kerr) generalization	$a\neq0$	The one radius splits into prograde and retrograde equatorial photon orbits; for extremal Kerr the prograde orbit reaches $GM/c^2$ (Boyer–Lindquist), and the shadow becomes a D-shape.

Common trapThe photon sphere is the horizon, and it traps light like a shell

Both halves of that sentence are wrong, and both are common. First, the photon sphere ( $3GM/c^2$ ) and the horizon ( $2GM/c^2$ ) are distinct radii; a static observer can hover just outside the photon sphere, whereas nothing can hover at or inside the horizon. Second, the circular photon orbit is unstable — it is not a reflective shell that stores light. A photon dwells near $r_{\mathrm{ph}}$ only if its impact parameter is tuned to $b_{\mathrm{crit}}$ to exponential precision; any real bundle falls in or leaks out within a few orbits, which is why the photon ring is thin and faint rather than a bright halo. A third, quieter trap: $r$ is an areal radius (fixed by sphere area $4\pi r^2$ ), not a proper radial distance — do not integrate $\Delta r$ and call it a physical length near the hole.

Worked examples

1 · The shadow of Sgr A* in microarcseconds (fully worked)

The Galactic-centre black hole has mass $M\approx4.3\times10^{6}\,M_\odot$ at distance $D\approx8.3\ \mathrm{kpc}$ . Predict its shadow's angular radius and compare with the EHT.

Step 1. Convert the mass to a length. With $GM_\odot/c^2=1476.6\ \mathrm m$ , $GM/c^2=4.3\times10^{6}\times1476.6\ \mathrm m=6.35\times10^{9}\ \mathrm m$ — the yardstick for every radius here. $GM/c^2=6.35\times10^{9}\ \mathrm m$ .

Step 2. Build the shadow radius $b_{\mathrm{crit}}=3\sqrt3\,GM/c^2=5.196\times6.35\times10^{9}\ \mathrm m=3.30\times10^{10}\ \mathrm m$ (for reference, horizon $r_s=1.27\times10^{10}\ \mathrm m$ and photon sphere $r_{\mathrm{ph}}=1.90\times10^{10}\ \mathrm m$ ). $b_{\mathrm{crit}}=3.30\times10^{10}\ \mathrm m$ .

Step 3. Project onto the sky. With $D=8.3\ \mathrm{kpc}=2.56\times10^{20}\ \mathrm m$ , $\theta_{\mathrm{sh}}=b_{\mathrm{crit}}/D=1.29\times10^{-10}\ \mathrm{rad}$ . Using $1\ \mathrm{rad}=2.063\times10^{11}\ \mu\mathrm{as}$ gives $\theta_{\mathrm{sh}}\approx\mathbf{26.6\ \mu as}$ radius, a shadow diameter of about $\mathbf{53\ \mu as}$ .

Step 4. Compare. The EHT measured a ring diameter of $51.8\pm2.3\ \mu\mathrm{as}$ for Sgr A (2022), matching the $\sqrt{27}\,GM/c^2$ prediction within the mass and distance uncertainties; the same computation for M87 ( $6.5\times10^{9}\,M_\odot$ , $16.8\ \mathrm{Mpc}$ ) gives $\approx40\ \mu\mathrm{as}$ against the measured $42\pm3\ \mu\mathrm{as}$ . A direct sighting of the photon sphere.

2 · From the photon sphere to a LIGO ringdown (numeric)

A merger leaves a remnant of mass $M\approx62\,M_\odot$ (the GW150914 value). Estimate its gravitational-wave ringdown frequency from the photon-sphere orbital frequency, treating the remnant as Schwarzschild.

Step 1. In the eikonal (large- $\ell$ ) limit a black hole's quasinormal modes are tied to the photon sphere: the real frequency is $\ell\,\Omega_c$ , where $\Omega_c$ is the orbital angular velocity of a photon at $r_{\mathrm{ph}}$ , and the damping is the orbit's Lyapunov rate. For Schwarzschild $\Omega_c=c^3/(3\sqrt3\,GM)=c/b_{\mathrm{crit}}$ .

Step 2. With $GM/c^2=62\times1476.6\ \mathrm m=9.16\times10^{4}\ \mathrm m$ , $b_{\mathrm{crit}}=5.196\times9.16\times10^{4}=4.76\times10^{5}\ \mathrm m$ , so $\Omega_c=c/b_{\mathrm{crit}}=\mathbf{630\ rad/s}$ and the bare photon-orbit frequency is $f_{\mathrm{orb}}=\Omega_c/2\pi\approx100\ \mathrm{Hz}$ .

Step 3. The dominant radiative mode is $\ell=2$ , so the ringdown tone is $f_{\mathrm{R}}\approx\Omega_c/\pi\approx\mathbf{200\ Hz}$ , damped on $\tau\approx3.2\ \mathrm{ms}$ (quality factor $\sim2$ ).

Step 4. The observed GW150914 remnant rings near $250\ \mathrm{Hz}$ ; our estimate is lower because the remnant spins ( $a\approx0.67$ ), shrinking the prograde photon orbit and raising $\Omega_c$ . Even so, the Schwarzschild value lands in the right band — the pitch of a black hole's "bell" is the frequency at which light orbits it.

Physics of the portrait

The photon sphere lives inside a geometry written down in weeks, on a battlefield, by a man who did not live to see what it contained — the circular light orbit, the capture cross-section, and the shadow all latent in the algebra Karl Schwarzschild sent Einstein at the end of 1915.

Origins · 1 minKarl Schwarzschild and how this was discovered

Karl Schwarzschild

1873–1916 · the exact vacuum geometry that hosts the photon sphere

Karl Schwarzschild was already a leading astronomer — director of the Potsdam Observatory, a pioneer of photographic photometry and stellar-atmosphere theory — when the First World War sent him to the Russian front as an artillery officer. There, within weeks of Einstein presenting the field equations of general relativity in November 1915, Schwarzschild found their first exact solution: the spacetime outside a spherical mass. He mailed it to Einstein that December, who read it to the Prussian Academy in his name. Schwarzschild was already ill with pemphigus, contracted at the front, and he died in May 1916, aged 42, having glimpsed only the barest edge of what his metric held. Its deep structure — the horizon at $2GM/c^2$ , the unstable photon orbit at $3GM/c^2$ first identified by David Hilbert in 1917, the coordinate versus curvature singularities untangled decades later — unfolded over the following half-century. The geometry behind every result in this article, the shadow the EHT photographed in 2019, and the ringdown LIGO hears all descend from a few pages written under fire by an astronomer with months to live. Einstein, who had doubted an exact solution was even possible, wrote that he "had not expected that one could formulate the exact solution of the problem in such a simple way."

History

1916

— Schwarzschild publishes the exact vacuum metric of a spherical mass

1917

— Hilbert identifies the unstable circular photon orbit at r = 3GM/c²

1966

— Synge computes the photon capture cross-section and escape cones

1973

— Bardeen derives the shadow of a rotating black hole

2019

— the Event Horizon Telescope images the shadow and photon ring of M87*

Going further

Texts. Misner, Thorne & Wheeler, Gravitation (1973), §25.5–25.6 · Wald, General Relativity (1984), §6.3 · Carroll, Spacetime and Geometry (2004), §5.4 · Hartle, Gravity (2003), Ch. 9 · Schutz, A First Course in General Relativity (2009), §11.2 · Chandrasekhar, The Mathematical Theory of Black Holes (1983), Ch. 3.

Papers. Synge, MNRAS 131, 463 (1966) · Bardeen, in Black Holes (Les Houches, 1973) · Bozza, PRD 66, 103001 (2002) · Cardoso, Miranda, Berti, Witek & Zanchin, PRD 79, 064016 (2009) · Event Horizon Telescope Collaboration, ApJL 875, L1 (2019); ApJL 930, L12 (2022). The key results are $r_{\mathrm{ph}}=3GM/c^2$ , $b_{\mathrm{crit}}=3\sqrt3\,GM/c^2=\sqrt{27}\,GM/c^2\approx5.196\,GM/c^2$ , with $V_{\mathrm{eff}}'=0\Rightarrow r=\tfrac32 r_s$ , $V_{\mathrm{eff}}''=-2L^2/r^4<0$ , and $V_{\mathrm{eff}}(r_{\mathrm{ph}})=4L^2/(27r_s^2)=1/b_{\mathrm{crit}}^2$ . The predicted shadow diameters are $\approx53\ \mu$ as for Sgr A and $\approx40\ \mu$ as for M87, measured by the EHT (2019, 2022).

Prerequisites. The Schwarzschild Metric · Geodesics in Schwarzschild · Killing Vectors and Conserved Quantities · Effective Potential for Massive-Particle Orbits · Schwarzschild Radius. Leads to. Circular Orbits and Their Stability · The ISCO · Perihelion Precession in Schwarzschild · Coordinate vs Curvature Singularities.

Ask the tutor: Want to see how $r_{\mathrm{ph}}=3GM/c^2$ splits into prograde and retrograde orbits once the hole spins, or how the $e^{-\pi}$ photon-ring spacing is measured — just ask.

Is the photon sphere the same thing as the event horizon?

No. They are at different radii: the horizon at $r_s=2GM/c^2$ and the photon sphere at $r_{\mathrm{ph}}=3GM/c^2$ , exactly $1.5$ times larger. The horizon is the surface of no return — cross it and every future-directed path leads inward. The photon sphere is outside the horizon, in a region you can still leave; it is simply the one radius where a photon travels on a circular orbit. A rocket could hover at the photon sphere indefinitely and watch light orbit it, which is impossible at the horizon. They are confused because both are "special radii," but physically they are very different objects.

Why doesn't the photon sphere trap light and glow like a bright shell?

Because the orbit is unstable — it sits at a maximum of the effective potential, not a minimum. A photon stays near $r_{\mathrm{ph}}$ only if its impact parameter is tuned to $b_{\mathrm{crit}}$ with exponential precision; any real photon is slightly off and, after at most a few loops, either spirals in or escapes. This is why images show a thin, faint photon ring rather than a bright trapped halo: the ring is light that lingered for one or two extra half-orbits, each contribution dimmer than the last by about $e^{-\pi}$ . A stable orbit would accumulate light, but Schwarzschild offers no stable circular orbit for photons anywhere.

Why is the shadow (

\sim5.2\,GM/c^2

) bigger than the photon sphere (

3GM/c^2

The shadow radius is not a radius in space — it is the impact parameter $b_{\mathrm{crit}}$ , the perpendicular offset a captured ray had while still far away. Strong gravity bends inbound rays toward the hole, so a photon aimed to miss the centre by as much as $5.196\,GM/c^2$ can still be curved in enough to cross the photon sphere and be captured. Projected onto the sky, the boundary of captured directions is a circle of angular radius $b_{\mathrm{crit}}/D$ . The shadow is thus a lensed, magnified image of the capture region, larger than both the photon sphere and the horizon that actually swallow the light.

Does the photon-sphere radius depend on the photon's energy or colour?

Not at all — one of its most useful features. In the orbit condition $V_{\mathrm{eff}}'=0$ , $E$ and $L$ appear only through an overall $L^2$ , which cancels, leaving $r_{\mathrm{ph}}=3GM/c^2$ regardless of frequency or angular momentum. Only the ratio $b=L/E$ matters, and for the exact circular orbit it is pinned to $b_{\mathrm{crit}}$ . Physically, light has no rest mass to set a scale, so the geometry alone — the mass $M$ — fixes where light can orbit. A gamma ray and a radio photon orbit at precisely the same radius.

How does spin (Kerr) change the picture?

Rotation breaks the single radius into a range. For equatorial photons around a Kerr hole there are two circular orbits — a prograde one at smaller radius and a retrograde one at larger — because frame dragging helps light orbiting with the spin and hinders light orbiting against it. For a maximally spinning hole the prograde orbit reaches $GM/c^2$ in Boyer–Lindquist coordinates while the retrograde one sits at $4GM/c^2$ . Off the equator the orbits fill a three-dimensional region rather than a sphere, and the shadow becomes an asymmetric "D" whose flattening encodes the spin — a shape the EHT hopes to resolve.

What is the connection between the photon sphere and gravitational-wave ringdown?

In the eikonal (large- $\ell$ ) limit, a black hole's quasinormal modes map onto photon-sphere geodesics: the frequency equals $\ell$ times the orbital angular velocity $\Omega_c$ of light at $r_{\mathrm{ph}}$ , and the damping equals the Lyapunov exponent of the orbit — the instability we proved via $V_{\mathrm{eff}}''<0$ . So a merger remnant's ringdown pitch is set by how fast light would circle it and the decay by how unstable that orbit is. This is why a photon-sphere calculation estimates ringdown frequencies (Worked Example 2), tying the EHT imaging channel and the LIGO–Virgo channel to one piece of geometry.