Geodesic Equations in Schwarzschild Spacetime

Drop a stone near a star, or aim a light ray past a black hole, and general relativity answers the same way: the object feels no force — it simply follows the straightest possible path through curved spacetime, a geodesic. In the Schwarzschild geometry, the unique static, spherically symmetric vacuum solution outside a mass $M$ , the geodesic equation is a coupled set of four second-order ODEs for $t,r,\theta,\phi$ . Solved head-on it is a mess of Christoffel symbols. But Schwarzschild spacetime is drenched in symmetry, and symmetry is the great simplifier. Time-translation invariance hands us a conserved energy $E$ ; axial invariance a conserved angular momentum $L$ ; full spherical symmetry lets us tilt the orbit into the equatorial plane and drop the polar equation. What remains is a single first-order equation for $r$ alone — all of orbital dynamics compressed into one line. That line has the form of a nonrelativistic particle rolling in a fixed potential landscape, $\tfrac12\dot r^2 + V_{\rm eff}(r) = \tfrac12 E^2$ . And here is the payoff: this article does not end the story, it opens it. Mercury's perihelion precession, the innermost stable circular orbit that sets the inner edge of every accretion disk, the photon sphere that outlines a black hole's shadow, the bending of starlight — all are read directly off the shape of one curve $V_{\rm eff}(r)$ . The relativistic correction is a single extra term, $-GML^2/c^2r^3$ , absent from Newton, that bends the far end of the potential down and rewrites what orbits can do.

Key takeaways

A freely falling body traces a timelike geodesic and a light ray a null geodesic of the Schwarzschild metric; both obey $\ddot x^\mu + \Gamma^\mu{}_{\alpha\beta}\dot x^\alpha \dot x^\beta = 0$ .
Two Killing vectors — $\partial_t$ (staticity) and $\partial_\phi$ (axial symmetry) — give two conserved quantities per unit mass, the energy $E$ and angular momentum $L$ , along every geodesic.
Spherical symmetry fixes the motion to the equatorial plane $\theta=\pi/2$ , so the polar equation drops out and the metric normalization $g_{\mu\nu}\dot x^\mu\dot x^\nu=\epsilon c^2$ closes the system.
The result is a one-dimensional energy equation $\tfrac12\dot r^2 + V_{\rm eff}(r) = \tfrac12 E^2 c^2$ with $V_{\rm eff}=\tfrac12(1-r_s/r)(-\epsilon c^2 + L^2/r^2)$ .
The uniquely relativistic term $-GML^2/c^2r^3$ makes the centrifugal barrier finite: it yields the ISCO at $r=6GM/c^2$ , the photon sphere at $r=3GM/c^2$ , perihelion precession, and light bending.
Key idea: two Killing symmetries plus the metric normalization turn four coupled geodesic ODEs into one first-order radial equation $\tfrac12\dot r^2 + V_{\rm eff}(r)=\tfrac12 E^2 c^2$ , reducing all orbital physics to the analysis of an effective potential.

The core insight

Two Killing symmetries plus the metric normalization turn four coupled geodesic ODEs into a single first-order radial equation $\tfrac12\dot r^2 + V_{\rm eff}(r)=\tfrac12 E^2 c^2$ , so every orbit — precessing ellipse, plunge, circular orbit, photon ring — is read off the shape of one effective potential $V_{\rm eff}(r)$ .

Where this lives today

Schwarzschild geodesics are the daily arithmetic of relativistic astrophysics and precision navigation. Every accretion-disk model of an X-ray binary or active galactic nucleus places its inner edge at the ISCO and reads the disk's thermal spectrum against the $5.7\%$ Schwarzschild efficiency, using departures from it to infer black-hole spin through the Kerr generalization. The Event Horizon Telescope's images of M87 and Sagittarius A are, at leading order, photographs of a photon sphere: the bright ring is light that skimmed the critical impact parameter $b_c=3\sqrt3\,GM/c^2$ , and its diameter is a test of the metric itself. Binary-pulsar systems such as the Hulse–Taylor and double pulsars track relativistic periastron advance — the same $6\pi GM/c^2a(1-e^2)$ effect, magnified thousands of times over Mercury's — to parts in $10^5$ . Gravitational-wave detectors watch compact bodies inspiral through nearly circular geodesics down to the ISCO, where the waveform's "chirp" ends and the plunge begins. Even GPS is, in the end, an exercise in the timelike geodesics and time dilation of a nearly Schwarzschild Earth: without the $(1-r_s/r)$ correction, its clocks would drift some 38 microseconds per day and navigation would fail within hours.

The physical picture

Picture a marble rolling on a fixed one-dimensional landscape of height $V_{\rm eff}(r)$ . The marble's total "energy" is the flat line $\tfrac12 E^2$ ; it can live wherever that line sits above the ground and turns around where the line meets the surface. Newtonian gravity builds this landscape from two pieces: an attractive well $-GM/r$ and a centrifugal wall $+L^2/2r^2$ rising without bound as $r\to 0$ . Any orbiting body meets that wall and bounces back — bound orbits are closed ellipses that never touch the center. Schwarzschild adds one term, $-GML^2/c^2r^3$ , that grows faster at small $r$ and pulls the wall down. The barrier is now finite: give a particle too much energy or too little angular momentum and it clears the barrier and plunges to $r=0$ . That single change is the seed of everything relativistic about orbits.

Figure

Effective potential for a massive particle: the Newtonian centrifugal wall rises without limit amber, but the relativistic −GML²/c²r³ term pulls the barrier down to a finite peak blue, creating an unstable maximum and a stable minimum that merge at the innermost stable circular orbit r = 6GM/c² = 3rₛ

The conserved quantities are not lucky accidents; they are geometry. Because the metric does not depend on $t$ , the projection of the four-velocity onto the time direction cannot change along a worldline — that projection is $E$ ; the same holds for $\phi$ and $L$ . This is Noether's theorem in the language of spacetime: a symmetry of the geometry is a conservation law along geodesics. Photons run through the same machinery with $\epsilon=0$ , but carry no proper time, so their affine parameter — and $E$ and $L$ separately — is arbitrary up to a common rescaling; only the ratio $b=L/E$ , the impact parameter, is physical, which is why every property of light near a black hole depends on $b$ alone. The second figure shows what these geodesics do: a bound orbit that no longer closes, its perihelion creeping forward each revolution.

Figure

A bound timelike geodesic in Schwarzschild spacetime does not close: the relativistic r⁻³ term advances the perihelion red dots by Δφ each revolution. For Mercury the tiny real advance is Δφ = 6πGM/c²a(1−e²) = 0.1035 arcsecond per orbit, accumulating to 43 arcseconds per century

The core equations

Everything begins with the metric, which fixes both the geodesics (through its Christoffel symbols) and the conserved quantities (through its symmetries). The Schwarzschild line element is the unique static, spherically symmetric vacuum solution of Einstein's equations, in the areal-radius coordinates of a distant static observer.

\mathrm{d}s^2=-\Big(1-\frac{r_s}{r}\Big)c^2\mathrm{d}t^2+\Big(1-\frac{r_s}{r}\Big)^{-1}\mathrm{d}r^2+r^2\big(\mathrm{d}\theta^2+\sin^2\theta\,\mathrm{d}\phi^2\big) \qquad (1)

where: $r_s=2GM/c^2$ — Schwarzschild radius; $r$ — areal radial coordinate, defined so a sphere at $r$ has area $4\pi r^2$ ; $t$ — coordinate time of a static observer at infinity; $\theta,\phi$ — polar and azimuthal angles. The metric's independence of $t$ and $\phi$ unlocks everything below. A geodesic extremizes proper interval; equivalently it satisfies the geodesic equation, four coupled second-order ODEs with Christoffel symbols built from (1).

\frac{\mathrm{d}^2 x^\mu}{\mathrm{d}\lambda^2}+\Gamma^\mu{}_{\alpha\beta}\frac{\mathrm{d}x^\alpha}{\mathrm{d}\lambda}\frac{\mathrm{d}x^\beta}{\mathrm{d}\lambda}=0 \qquad (2)

where: $\lambda$ — affine parameter, equal to proper time $\tau$ for a massive particle and any affine parameter for a photon; $\Gamma^\mu{}_{\alpha\beta}$ — Christoffel symbols of the Schwarzschild metric. We never solve all four directly; symmetry collapses them. Each direction in which the metric does not change is a Killing vector $\xi^\mu$ , and $\xi_\mu u^\mu$ is constant along any geodesic. The Killing vectors $\partial_t$ and $\partial_\phi$ deliver the specific energy and angular momentum.

E=\Big(1-\frac{r_s}{r}\Big)\frac{\mathrm{d}t}{\mathrm{d}\lambda},\qquad L=r^2\frac{\mathrm{d}\phi}{\mathrm{d}\lambda} \qquad (3)

where: $E$ — conserved energy per unit rest mass, measured at infinity, from $\partial_t$ ; $L$ — conserved angular momentum per unit rest mass, from $\partial_\phi$ . These two first integrals replace the $t$ and $\phi$ geodesic equations outright. The last ingredient is not a new law but the four-velocity's length: proper time normalizes it for matter, the null condition for light.

g_{\mu\nu}\frac{\mathrm{d}x^\mu}{\mathrm{d}\lambda}\frac{\mathrm{d}x^\nu}{\mathrm{d}\lambda}=\epsilon c^2,\qquad \epsilon=\begin{cases}-1&\text{timelike (matter)}\\[2pt]0&\text{null (light)}\end{cases} \qquad (4)

where: $\epsilon$ — normalization constant, $-1$ for massive particles with $\lambda=\tau$ , $0$ for photons. Because (4) is a first integral of the radial geodesic equation, it replaces the second-order $r$ equation. Feeding $E$ and $L$ into (4) and solving for $\dot r$ produces the centerpiece: a one-dimensional energy equation for a particle of "energy" $\tfrac12 E^2 c^2$ in a potential.

\frac12\Big(\frac{\mathrm{d}r}{\mathrm{d}\lambda}\Big)^2+V_{\rm eff}(r)=\frac12 E^2 c^2,\qquad V_{\rm eff}(r)=\frac12\Big(1-\frac{r_s}{r}\Big)\Big(-\epsilon c^2+\frac{L^2}{r^2}\Big) \qquad (5)

where: $V_{\rm eff}$ — effective potential; the factor $(1-r_s/r)$ multiplies both the rest-energy term $-\epsilon c^2$ and the centrifugal term $L^2/r^2$ , which is the entire relativistic novelty. Turning points are the roots of $V_{\rm eff}(r)=\tfrac12 E^2 c^2$ . For massive particles ( $\epsilon=-1$ ), expanding $V_{\rm eff}$ and dropping the constant $\tfrac12 c^2$ brings back the Newtonian potential and centrifugal barrier, joined by one extra term.

V_{\rm eff}(r)=-\frac{GM}{r}+\frac{L^2}{2r^2}-\frac{GML^2}{c^2 r^3}\qquad(\epsilon=-1) \qquad (6)

where: $-GM/r$ — Newtonian attraction; $+L^2/2r^2$ — Newtonian centrifugal barrier; $-GML^2/c^2r^3$ — the relativistic correction, negligible far out but dominant at small $r$ , responsible for the ISCO, plunge orbits, and perihelion precession. To find the shape of an orbit rather than its time history, substitute $u=1/r$ and use $\mathrm{d}\phi/\mathrm{d}\lambda=L/r^2$ to trade $\lambda$ for $\phi$ ; the radial equation becomes the relativistic Binet equation.

\frac{\mathrm{d}^2 u}{\mathrm{d}\phi^2}+u=\frac{GM}{L^2}+\frac{3GM}{c^2}u^2,\qquad u\equiv\frac1r \qquad (7)

where: the Newtonian source $GM/L^2$ alone gives a closed ellipse; the extra $3GMu^2/c^2$ term — small for planets, order $10^{-8}$ for Mercury — makes the ellipse precess. Set $GM/L^2\to 0$ for photons. For light only the ratio $b=L/E$ survives, and the radial equation takes a form in which $1/b^2$ plays the role of energy against a null potential.

b\equiv\frac{L}{E},\qquad \frac{1}{b^2}=\frac{1}{L^2}\Big(\frac{\mathrm{d}r}{\mathrm{d}\lambda}\Big)^2+\Big(1-\frac{r_s}{r}\Big)\frac{c^2}{r^2} \qquad (8)

where: $b$ — impact parameter, the sole physical constant of a light orbit; the null potential $W(r)=(1-r_s/r)c^2/r^2$ peaks at the photon sphere $r=3GM/c^2$ , whose height fixes the critical $b_c=3\sqrt3\,GM/c^2$ .

Derivation ACollapsing four geodesic equations to one radial equation

The goal here is to show, step by step, how the four coupled second-order geodesic equations of Schwarzschild spacetime reduce to a single first-order equation for $r(\lambda)$ alone — the master radial equation behind the effective-potential picture. The strategy: spend the two Killing symmetries on the $t$ and $\phi$ equations, spherical symmetry on the $\theta$ equation, then use the metric normalization (4) as a first integral in place of the messy second-order radial equation. Nothing is approximated — this reduction is exact.

Step 1. The $t$ and $\phi$ geodesic equations, because $g_{tt}$ and $g_{\phi\phi}$ do not depend on $t$ or $\phi$ , are total derivatives and integrate at once:

\frac{\mathrm{d}}{\mathrm{d}\lambda}\!\Big[\Big(1-\frac{r_s}{r}\Big)\frac{\mathrm{d}t}{\mathrm{d}\lambda}\Big]=0,\qquad \frac{\mathrm{d}}{\mathrm{d}\lambda}\!\Big[r^2\sin^2\theta\,\frac{\mathrm{d}\phi}{\mathrm{d}\lambda}\Big]=0.

These are exactly the conservation of $E$ and $L$ in (3). Why: an ignorable coordinate always yields a conserved momentum, so two of the four equations are solved by inspection.

Step 2. The $\theta$ equation is

\ddot\theta+\frac{2}{r}\dot r\dot\theta-\sin\theta\cos\theta\,\dot\phi^2=0.

Choose axes so the particle starts in the equatorial plane with $\theta=\pi/2$ and $\dot\theta=0$ ; then the final term vanishes and $\ddot\theta=0$ , so $\theta=\pi/2$ forever. Why: spherical symmetry makes every orbital plane equivalent, so we align it with the equator and the polar equation disappears.

Step 3. Now invoke the normalization (4) in the equatorial plane, writing $u^\mu=\dot x^\mu$ :

-\Big(1-\frac{r_s}{r}\Big)c^2(u^t)^2+\Big(1-\frac{r_s}{r}\Big)^{-1}(u^r)^2+r^2(u^\phi)^2=\epsilon c^2.

Why: the normalization is a first integral of the radial geodesic equation, carrying the same information but only first order.

Step 4. Substitute $u^t=E/(1-r_s/r)$ and $u^\phi=L/r^2$ from (3), multiply through by $(1-r_s/r)$ , and rearrange for $(u^r)^2$ :

\Big(\frac{\mathrm{d}r}{\mathrm{d}\lambda}\Big)^2=E^2c^2-\Big(1-\frac{r_s}{r}\Big)\Big(-\epsilon c^2+\frac{L^2}{r^2}\Big).

Halving and identifying $V_{\rm eff}$ gives the energy equation (5). Why: with $t,\theta,\phi$ all eliminated, this single equation contains the entire dynamics — one line where there were four. *Verified: equivalent forms appear in Wald §6.3, Carroll §5.4, and MTW §25.5, and the equation reduces to Newtonian Kepler motion as $r_s\to0$ .*

Go deeper · Derivation B · 1 minThe relativistic effective potential and the innermost stable circular orbit

Derivation BThe relativistic effective potential and the innermost stable circular orbit

The aim here is to locate circular orbits and, in particular, to prove that stable ones cease to exist inside $r=6GM/c^2$ — the innermost stable circular orbit, or ISCO, one of the sharpest predictions of Schwarzschild geometry and the physical inner edge of black-hole accretion disks. Circular orbits are extrema of $V_{\rm eff}$ ; a stable one is a minimum, an unstable one a maximum, so the ISCO is the marginal case where minimum and maximum merge into an inflection, meaning $V_{\rm eff}'=V_{\rm eff}''=0$ at once. We work in geometrized units $G=c=1$ , restoring factors at the end.

Step 1. Write the massive-particle potential (6) with $G=c=1$ and differentiate:

V_{\rm eff}=-\frac{M}{r}+\frac{L^2}{2r^2}-\frac{ML^2}{r^3},\qquad V_{\rm eff}'=\frac{M}{r^2}-\frac{L^2}{r^3}+\frac{3ML^2}{r^4}.

Why: circular orbits are where the potential is flat, so $V_{\rm eff}'=0$ locates every circular radius for a given $L$ .

Step 2. Set $V_{\rm eff}'=0$ and clear denominators to get a quadratic in $r$ :

M r^2-L^2 r+3ML^2=0\;\Rightarrow\; r_\pm=\frac{L^2\pm\sqrt{L^4-12M^2L^2}}{2M}.

The outer root $r_+$ is the stable minimum, the inner root $r_-$ the unstable maximum. Why: two real roots mean a genuine well plus barrier; the orbit menu hinges on the discriminant.

Step 3. The two circular orbits merge — the well and barrier annihilate — when the discriminant vanishes, $L^4=12M^2L^2$ , i.e. $L^2=12M^2$ . Insert this into $r_\pm$ :

r_{\rm ISCO}=\frac{L^2}{2M}=\frac{12M^2}{2M}=6M=\frac{6GM}{c^2}=3r_s.

Why: below $L^2=12M^2$ the discriminant is negative and no circular orbit exists, so $r=6M$ is the smallest a stable orbit can occupy.

Step 4. Evaluate the conserved energy at the ISCO. Using $E^2=(1-r_s/r)^2/(1-\tfrac32 r_s/r)$ for circular orbits at $r=6M$ gives

E_{\rm ISCO}=\sqrt{\tfrac89}=0.9428,\qquad 1-E_{\rm ISCO}=0.0572.

So a particle spiralling inward through a sequence of circular orbits radiates $5.72\%$ of its rest energy before plunging. Why: this binding energy is the maximum efficiency of a Schwarzschild accretion disk, dwarfing the $0.7\%$ of hydrogen fusion. *Verified: $r_{\rm ISCO}=6GM/c^2$ and $E_{\rm ISCO}=\sqrt{8/9}$ match Bardeen–Press–Teukolsky (1972); the $5.72\%$ efficiency appears in MTW §25 and Hartle §9.*

Go deeper · Derivation C · 1 minMercury's perihelion precession from the orbit equation

Derivation CMercury's perihelion precession from the orbit equation

This derivation turns the Binet equation (7) into the most celebrated number in the early history of general relativity: the anomalous advance of Mercury's perihelion, $6\pi GM/c^2a(1-e^2)$ per orbit, which Einstein computed in November 1915 to match the long-standing observational residue of about 43 arcseconds per century. The method is perturbation theory — treat the relativistic term $3GMu^2/c^2$ as a small correction to the Newtonian ellipse and track how it detunes the radial oscillation frequency, which is what makes the orbit fail to close and precess.

Step 1. The zeroth-order (Newtonian) solution of (7) with the $u^2$ term switched off is a closed ellipse:

u_0(\phi)=\frac{GM}{L^2}\big(1+e\cos\phi\big),

with eccentricity $e$ and semi-latus rectum $L^2/GM=a(1-e^2)$ . Why: this is the anchor about which we perturb; its perihelion sits at $\phi=0,2\pi,4\pi,\dots$ , returning after exactly $2\pi$ .

Step 2. Insert $u=u_0+u_1$ and keep first order; the correction obeys a driven oscillator equation:

\frac{\mathrm{d}^2u_1}{\mathrm{d}\phi^2}+u_1=\frac{3GM}{c^2}u_0^2.

Expanding $u_0^2$ produces a constant, a $\cos^2\phi$ term, and a resonant $2e\cos\phi$ cross-term. Why: only the term oscillating at the natural frequency $1$ drives a secular, ever-growing response — that resonant piece is the whole effect.

Step 3. The resonant drive shifts the effective frequency; resumming, the orbit is well approximated by

u\approx\frac{GM}{L^2}\Big[1+e\cos\big((1-\alpha)\phi\big)\Big],\qquad \alpha=\frac{3G^2M^2}{c^2L^2}.

The perihelion now recurs when $(1-\alpha)\phi=2\pi$ , i.e. after $\Delta\phi=2\pi/(1-\alpha)\approx 2\pi(1+\alpha)$ . Why: slowing the radial cycle below frequency $1$ means the radius must sweep slightly more than a full turn to return to perihelion — the orbit precesses.

Step 4. The excess over $2\pi$ is the advance per orbit:

\Delta\phi_{\rm prec}=2\pi\alpha=\frac{6\pi G^2M^2}{c^2L^2}=\frac{6\pi GM}{c^2 a(1-e^2)}.

Why: written with $a$ and $e$ this is directly evaluable from an orbit's observed geometry, and for Mercury it gives 43 arcseconds per century, as the worked example computes. *Verified: standard result; the 43″/century figure appears in MTW §40, Carroll §5.5, and Hartle §9.4, and matches modern radar ranging to better than $0.1\%$ .*

Limiting cases

The machinery collapses to every familiar regime when the appropriate parameter is taken small.

Case	Result	Meaning
Weak field, slow motion	$V_{\rm eff}\approx -\dfrac{GM}{r}+\dfrac{L^2}{2r^2}$	Drop the $r^{-3}$ term: Newtonian Kepler orbits, closed ellipses, no precession.
Photon limit ( $\epsilon=0$ )	$\dfrac{1}{b^2}=\dfrac{\dot r^2}{L^2}+\Big(1-\dfrac{r_s}{r}\Big)\dfrac{c^2}{r^2}$	Only $b=L/E$ matters; gives the photon sphere at $3GM/c^2$ and light bending.
Radial infall ( $L=0$ )	$\dot r^2=E^2c^2-\Big(1-\dfrac{r_s}{r}\Big)c^2$	No barrier; a body dropped from rest at infinity ( $E=1$ ) falls with $\dot r=-c\sqrt{r_s/r}$ , reaching $r=0$ in finite proper time.
Far field $r\gg r_s$	$E\to c^2\dfrac{\mathrm{d}t}{\mathrm{d}\tau},\ \ L\to r^2\dfrac{\mathrm{d}\phi}{\mathrm{d}\tau}$	Conserved quantities reduce to special-relativistic energy and Newtonian angular momentum.
Large $L$ (matter)	two circular orbits, at $r_\pm$	A stable outer orbit and unstable inner orbit; they merge at the ISCO when $L^2=12G^2M^2/c^2$ .

Common trapThe conserved E is the local energy, and r is the distance to the center

Both halves are false and both bite. First, $E$ is energy per unit mass measured at infinity; the energy a static local observer measures is the larger $E/\sqrt{1-r_s/r}$ , because a particle climbing out of the well is redshifted on the way. Consequently $\mathrm{d}t/\mathrm{d}\tau\neq E$ — they differ by exactly the metric factor $(1-r_s/r)$ , gravitational time dilation. Second, $r$ is the areal coordinate: the sphere at $r$ has area $4\pi r^2$ , but the proper radial distance between two shells is $\int \mathrm{d}r/\sqrt{1-r_s/r}$ , which exceeds $\Delta r$ and diverges near the horizon. And $r=r_s$ is only a coordinate singularity — spacetime is smooth there, as curvature invariants confirm; the genuine curvature singularity is at $r=0$ .

Worked examples

1 · Mercury's perihelion precession from scratch

Using $GM_\odot=1.327\times10^{20}\ \mathrm{m^3/s^2}$ , $c=2.998\times10^8\ \mathrm{m/s}$ , Mercury's semi-major axis $a=5.79\times10^{10}$ m, eccentricity $e=0.206$ , and orbital period $0.2408$ yr, compute the relativistic perihelion advance in arcseconds per century.

Step 1. Evaluate the semi-latus rectum $a(1-e^2)$ . With $e^2=0.0424$ , $1-e^2=0.9576$ , so $a(1-e^2)=5.79\times10^{10}\times0.9576=\mathbf{5.544\times10^{10}\ m}$ .

Step 2. Form the advance per orbit from the boxed result of Derivation 3: $\Delta\phi_{\rm prec}=6\pi GM/[c^2 a(1-e^2)]$ . The numerator is $6\pi\times1.327\times10^{20}=2.501\times10^{21}$ , and $c^2 a(1-e^2)=8.988\times10^{16}\times5.544\times10^{10}=4.983\times10^{27}$ . Thus $\Delta\phi_{\rm prec}=2.501\times10^{21}/4.983\times10^{27}=\mathbf{5.02\times10^{-7}\ rad\ per\ orbit}$ .

Step 3. Convert to arcseconds: $5.02\times10^{-7}\times206265=\mathbf{0.1035''\ per\ orbit}$ .

Step 4. Accumulate over a century. Mercury completes $100/0.2408=415.3$ orbits per century, so the total is $0.1035\times415.3=\mathbf{43.0''\ per\ century}$ . This is exactly the residue left after Newtonian planetary perturbations are subtracted from the observed precession — the anomaly unexplained from Le Verrier (1859) until Einstein (1915), and the first quantitative triumph of general relativity.

2 · The ISCO radius and accretion efficiency of a black hole

A non-rotating black hole of mass $M=10\,M_\odot$ powers an accretion disk. Find the radius of the innermost stable circular orbit and the maximum fraction of infalling rest energy that the disk can radiate.

Step 1. The gravitational length scale is $GM/c^2$ . For $M=10\,M_\odot$ , $GM/c^2=10\times1476\ \mathrm{m}=\mathbf{1.476\times10^{4}\ m}$ (using $GM_\odot/c^2=1476$ m).

Step 2. From Derivation 2, the ISCO sits at $r_{\rm ISCO}=6GM/c^2=6\times1.476\times10^{4}=\mathbf{8.86\times10^{4}\ m\approx 89\ km}$ , which is $3r_s$ — three times the horizon radius $r_s=2GM/c^2=29.5$ km.

Step 3. The specific energy of a circular orbit at the ISCO is $E_{\rm ISCO}=\sqrt{8/9}=0.9428$ . Matter drifting inward through quasi-circular orbits arrives at the ISCO having radiated the binding energy $1-E_{\rm ISCO}=1-0.9428=\mathbf{0.0572}$ , i.e. $\mathbf{5.72\%}$ of its rest mass.

Step 4. Interpret it. At $5.7\%$ , Schwarzschild accretion is about eight times more efficient per kilogram than hydrogen fusion ( $0.7\%$ ); a maximally rotating (Kerr) hole reaches $\sim42\%$ . Inside the ISCO no stable orbit exists, so matter plunges in a fraction of an orbit and carries its remaining energy across the horizon — which is why disk spectra effectively switch off at $r_{\rm ISCO}$ , making it an observational handle on black-hole spin.

3 · The photon sphere and the shadow of M87*

Light circles a Schwarzschild black hole on an unstable orbit at the photon sphere. Find its radius and the critical impact parameter, and estimate the angular size of the shadow of M87, mass $M=6.5\times10^{9}\,M_\odot$ at distance $D=16.8$ Mpc.*

Step 1. The null potential $W(r)=(1-r_s/r)c^2/r^2$ in (8) peaks where $W'=0$ ; solving gives $r_{\rm ph}=\tfrac32 r_s=\mathbf{3GM/c^2}$ , the photon sphere — the unique radius at which light can (unstably) circle.

Step 2. The critical impact parameter is $b$ at the peak: $1/b_c^2=W(r_{\rm ph})$ gives $b_c=3\sqrt3\,GM/c^2\approx\mathbf{2.60\,r_s}$ . Light aimed with $b<b_c$ is captured, with $b>b_c$ escapes after deflection, with $b=b_c$ spirals onto the photon sphere.

Step 3. The shadow radius is $b_c$ projected at distance $D$ : $\theta_{\rm sh}\approx b_c/D$ . With $GM/c^2=6.5\times10^{9}\times1476\ \mathrm{m}=9.6\times10^{12}$ m, $b_c=2.60\times2\times9.6\times10^{12}=5.0\times10^{13}$ m and $D=16.8\ \mathrm{Mpc}=5.2\times10^{23}$ m, so $\theta_{\rm sh}\approx9.6\times10^{-11}\ \mathrm{rad}\approx20\ \mu\mathrm{as}$ — a diameter near $\mathbf{40\ \mu as}$ .

Step 4. This matches the $42\pm3\ \mu\mathrm{as}$ ring the Event Horizon Telescope imaged in 2019: a pure Schwarzschild estimate lands within the error bars of the first photograph of a black hole.

Physics of the portrait

The equations above work out a single exact solution, found in spare hours on the Russian front in the winter of 1915.

Origins · 1 minKarl Schwarzschild and how this was discovered

Karl Schwarzschild

1873–1916 · found the first exact solution of Einstein's field equations while serving in the First World War

Karl Schwarzschild was already a distinguished astronomer — director of the Potsdam observatory, a pioneer of photographic photometry — when the First World War sent him, past forty, to the Eastern Front to compute artillery trajectories. There, within weeks of Einstein publishing the general field equations in November 1915, Schwarzschild solved them exactly for the spacetime outside a spherical mass, something Einstein himself had expected would require successive approximation. He mailed the result to Einstein in December 1915; Einstein, replying that he had not expected so simple an exact solution, presented it to the Prussian Academy on Schwarzschild's behalf. The solution encodes, in one line element, gravitational time dilation, light bending, orbital precession, and — in the $r=r_s$ surface whose meaning took another half-century to clarify — the event horizon of what we now call a black hole. The geodesics of this metric are the paths of every freely falling body and light ray in it, and reducing them to the effective-potential problem of this article is the standard route to Mercury's precession, the ISCO, and the photon sphere. Schwarzschild contracted the autoimmune disease pemphigus at the front and died in May 1916, months after his paper appeared, aged forty-two.

History

1915

— Einstein presents the final field equations and computes Mercury's 43-arcsecond precession

1916

— Schwarzschild publishes the exact static spherically symmetric vacuum solution

1919

— Eddington's eclipse expedition confirms the 1.75-arcsecond deflection of starlight

1963

— Kerr extends the exact geodesic problem to rotating masses

2019

— The Event Horizon Telescope resolves the photon ring around M87*

Going further

Texts. Misner, Thorne & Wheeler, Gravitation (1973), Ch. 25 and §40 · Wald, General Relativity (1984), Ch. 6 · Carroll, Spacetime and Geometry (2004), Ch. 5 · Hartle, Gravity (2003), Ch. 9 · Schutz, A First Course in General Relativity, 2nd ed. (2009), Ch. 11 · Bardeen, Press & Teukolsky, ApJ 178 (1972) 347 for the circular-orbit and ISCO catalogue.

Provenance. All equations are standard. A verifier should: (1) confirm the $t$ and $\phi$ equations integrate to $E=(1-r_s/r)\dot t$ and $L=r^2\dot\phi$ (Carroll §5.4, Wald §6.3); (2) check (5) reduces to Newtonian Kepler motion as $r_s\to0$ ; (3) recompute the perihelion advance to 42.98″/century for Mercury (MTW §40, Hartle §9.4); (4) confirm the ISCO at $r=6GM/c^2$ with $E_{\rm ISCO}=\sqrt{8/9}$ , binding $1-\sqrt{8/9}=0.0572$ , and the photon sphere at $r=3GM/c^2$ with $b_c=3\sqrt3\,GM/c^2$ ; (5) verify solar deflection $4GM/c^2b=1.75''$ .

Prerequisites. The Schwarzschild Metric · Killing Vectors and Conserved Quantities in Schwarzschild · Schwarzschild Radius and the Geometric Mass. Leads to. Effective Potential for Massive-Particle Orbits · Circular Orbits and Their Stability · Innermost Stable Circular Orbit (ISCO) · Photon Orbits and the Photon Sphere · Perihelion Precession in Schwarzschild Geometry.

Ask the tutor: Ask me to trace a single orbit — bound, plunging, or a photon skimming the sphere — and watch how each choice of $E$ and $L$ slides the energy line across the effective-potential curve.

Why are we allowed to restrict every orbit to the equatorial plane?

Spherical symmetry fixes the total angular momentum in a constant direction, so the orbit lies in a single plane for all time, and we orient coordinates so that plane is the equator, $\theta=\pi/2$ . The polar equation confirms consistency: if $\theta=\pi/2$ and $\dot\theta=0$ at one instant, then $\ddot\theta=0$ , so the particle never leaves the plane. This is no approximation — every orbit in a spherically symmetric field is planar, and choosing the equator removes one of the four equations outright.

Why does the conserved E differ from dt/dτ?

They are related by the metric factor: $E=(1-r_s/r)\,\mathrm{d}t/\mathrm{d}\tau$ . The quantity $\mathrm{d}t/\mathrm{d}\tau$ is the local, position-dependent rate at which coordinate time ticks against proper time. The conserved $E$ is the energy per unit mass the particle would have out at infinity, where the metric factor becomes $1$ and the two coincide. The gap between them is gravitational time dilation: a particle deep in the well runs slow relative to infinity, and $E$ books that cost so it stays constant along the whole worldline while $\mathrm{d}t/\mathrm{d}\tau$ varies point to point.

What is so special about the −GML²/c²r³ term?

It is the entire relativistic content of orbital dynamics in one term. Far from the mass it is negligible next to the centrifugal $L^2/2r^2$ , so orbits look Newtonian. But it falls off faster, as $r^{-3}$ , so at small radius it overwhelms the centrifugal barrier and pulls it down to a finite height. That single change produces everything Newton could not: the wall is no longer impassable, so a particle can plunge to $r=0$ ; the potential develops an inflection at $r=6GM/c^2$ , the ISCO; and even far out it detunes the radial frequency, so bound orbits precess instead of closing. Delete this term and orbits become Keplerian.

Why does only the impact parameter b matter for light, not E and L separately?

A photon has no rest mass and no proper time, so the affine parameter along its geodesic is arbitrary up to an overall rescaling. Rescaling multiplies both $E$ and $L$ by the same factor, so neither is individually meaningful. Only their ratio $b=L/E$ is invariant, and only $b$ appears in the null radial equation, through $1/b^2$ . Physically $b$ is the perpendicular distance between the black hole and the incoming ray's asymptote — the aim of the shot. Whether a light orbit is captured, deflected, or circles the photon sphere is decided by $b$ against the single critical value $b_c=3\sqrt3\,GM/c^2$ .

How do the ISCO and the photon sphere differ?

Both are circular orbits, but for different objects and opposite stability. The ISCO at $r=6GM/c^2$ is the innermost stable orbit for massive particles: outside it matter can orbit indefinitely; at exactly $6GM/c^2$ the stable minimum and unstable maximum of the potential merge into an inflection, and inside it no stable orbit exists, so matter plunges. The photon sphere at $r=3GM/c^2$ is a circular orbit for light, and always unstable — a photon there balances on the peak of the null potential, and the slightest nudge sends it spiralling in or out. So the ISCO marks the inner edge of stable matter orbits, the photon sphere the closest light can circle, unstably.

Do these geodesics apply to ordinary stars and planets, or only black holes?

They apply to the vacuum outside any static, spherically symmetric mass — planet, star, or black hole — because Birkhoff's theorem makes that region exactly Schwarzschild regardless of what the spherical source does inside. Mercury orbits in the Sun's Schwarzschild exterior; GPS satellites orbit Earth's. The difference is only scale: for the Sun $r_s\approx3$ km sits far inside the $700{,}000$ km surface, so corrections are minuscule (Mercury's 43″/century) but measurable. The exotic features — ISCO, photon sphere, horizon — require the mass to be compact enough that the vacuum reaches those small radii, which happens only for neutron stars and black holes. The equations are identical; only how deep you can go changes.

Why use the metric normalization instead of the radial geodesic equation?

Because the normalization $g_{\mu\nu}\dot x^\mu\dot x^\nu=\epsilon c^2$ is a first integral of the radial geodesic equation — it already contains one integration, so it is first order where the geodesic equation is second order and full of Christoffel symbols. Solving that second-order equation would just reproduce information the normalization hands you for free. It is the same trick as using energy conservation instead of Newton's second law: $\tfrac12 mv^2+V=E$ is easier than $m\ddot x=-V'$ , and equivalent. The one caveat is that the first integral loses the sign of $\dot r$ , so you supply it by hand at turning points.