Schwarzschild Geodesics and Effective Potential

Every classical test of general relativity — Mercury's extra perihelion swing, the bending of starlight, the Shapiro radar delay, the dark shadow at a galaxy's heart — is at bottom a statement about how one test body falls freely through the geometry outside a spherical mass. That geometry is the Schwarzschild solution, the free-fall paths its geodesics, and written in full a geodesic obeys four coupled second-order equations. But the metric is drenched in symmetry, and symmetry means conservation laws: time-translation invariance gives a conserved energy, rotational invariance a conserved angular momentum, spherical symmetry a single orbital plane. What remains is one equation for the radial coordinate — with exactly the form of a nonrelativistic particle of energy $\tfrac12 E^2/c^2$ rolling in a one-dimensional potential $V_{\mathrm{eff}}(r)$ . That is the great simplification. Bound orbits live in the well between two turning points, plunge orbits clear the barrier, circular orbits sit at the extrema. The whole difference from Newton hides in one term: Newton's centrifugal barrier $L^2/2r^2$ climbs to infinity as $r\to0$ , always turning a body back, but relativity adds an attractive $-GML^2/c^2r^3$ that beats the barrier at small $r$ and pulls its peak to a finite height. That single cubic term is why a black hole has a last stable orbit, why photons can circle on an unstable sphere, and why matter can plunge through a horizon — none of which Newton allows.

Key takeaways

The Schwarzschild metric has two Killing vectors, $\partial_t$ and $\partial_\phi$ , giving each geodesic a conserved specific energy $E$ and specific angular momentum $L$ ; spherical symmetry confines the motion to a plane.
The four-velocity normalization then collapses the radial motion to a first integral $\tfrac12\dot r^2 + V_{\mathrm{eff}}(r) = \tfrac12 E^2/c^2$ , formally the Newtonian energy equation.
The relativistic potential $V_{\mathrm{eff}} = \tfrac{c^2}{2} - \tfrac{GM}{r} + \tfrac{L^2}{2r^2} - \tfrac{GML^2}{c^2r^3}$ differs from Newton only by the last, cubic term — but that term rewrites the physics of small $r$ .
Timelike orbits acquire an innermost stable circular orbit at $r = 6GM/c^2$ with binding energy $1 - 2\sqrt2/3 \approx 5.7\%$ , and their bound ellipses precess by $6\pi GM/c^2a(1-e^2)$ per revolution.
Null geodesics feel a potential with a single maximum at the photon sphere $r = 3GM/c^2$ ; light aimed inside the critical impact parameter $b_c = 3\sqrt3\,GM/c^2$ is captured, setting the edge of the black-hole shadow.
The same $V_{\mathrm{eff}}$ delivers Mercury's 43″/century, the 1.75″ solar deflection of light, and the Shapiro radar delay — the classical tests, all as corollaries.
Key idea: two Killing vectors and the four-velocity normalization reduce the Schwarzschild geodesic equations to a single first integral $\tfrac12\dot r^2 + V_{\mathrm{eff}}(r) = \tfrac12 E^2/c^2$ ; all orbital phenomenology follows from the shape of $V_{\mathrm{eff}}$ , whose relativistic $1/r^3$ term creates the ISCO and the photon sphere.

The core insight

Two Killing vectors and the four-velocity normalization turn the Schwarzschild geodesic equations into a single first integral $\tfrac12\dot r^2 + V_{\mathrm{eff}}(r) = \tfrac12 E^2/c^2$ ; all orbital phenomenology follows from the shape of $V_{\mathrm{eff}}$ , whose relativistic $1/r^3$ term creates the ISCO and the photon sphere.

Where this lives today

The effective-potential picture is the working language of modern strong-gravity astrophysics. The ISCO at $6GM/c^2$ sets the inner edge of accretion disks, and its radius shrinks for spinning holes, so X-ray spectroscopists fit the broadened iron K $\alpha$ line and the disk continuum to measure black-hole spin — the ISCO is a ruler read from a spectrum. In gravitational-wave astronomy it marks where a compact binary's inspiral gives way to plunge and merger, and LIGO and Virgo templates are stitched across that transition. The photon sphere became a photograph when the Event Horizon Telescope imaged M87 in 2019 and Sagittarius A in 2022, the bright ring tracing light that skimmed the $3GM/c^2$ orbit. And the GRAVITY interferometer has watched the star S2 whip around Sgr A* and detected its Schwarzschild perihelion precession directly. Every one is a feature of the one-dimensional landscape above.

The physical picture

Picture the radial coordinate $r$ as a ball rolling in a one-dimensional landscape $V_{\mathrm{eff}}(r)$ , released with a fixed "total energy" $\tfrac12 E^2/c^2$ . Wherever that energy line meets the landscape the ball has $\dot r = 0$ — a turning point. A ball oscillating between two turning points is a bound orbit (an ellipse in Newton, a slow rosette in Schwarzschild); one that comes in, turns once, and returns is a scattering orbit; one energetic enough to clear the inner barrier rolls over the top and plunges to $r = 0$ , swallowed. Every qualitative feature is read off the graph, with no differential equation left to solve. The physics lives in the height of the inner barrier. In Newtonian gravity that barrier is the centrifugal term $L^2/2r^2$ , climbing to infinity as $r\to0$ : however energetic the incoming body, if it carries angular momentum it is always turned back before the origin. Relativity adds a purely attractive $-GML^2/c^2r^3$ ; because it rises faster than the centrifugal term at small $r$ , it drags the barrier's peak down from infinity to a finite value — now a hill a body can be thrown over.

Figure

Timelike effective potential V_eff(r) for L = 4.3 GM/c, with r in units GM/c² and V in units c². The finite inner barrier peaks at the unstable circular orbit near r ≈ 3.8; the well bottoms at the stable circular orbit near r ≈ 14.7. An energy line at E²/2c² = 0.475 traps a bound orbit between perihelion r ≈ 9.7 and aphelion r ≈ 27.5, and the dashed line marks the rest-energy asymptote c²/2

Circular orbits are the stationary points of this landscape: a minimum is stable (nudge the body and it oscillates about the circle), a maximum unstable (the slightest nudge sends it spiralling in or out). As $L$ is lowered the well and barrier slide together and shrink; at a critical $L$ they merge into a single inflection point. For matter that merger is the innermost stable circular orbit, the ISCO; for photons the null potential has only a maximum, at the photon sphere. These radii, with the horizon, organize the whole near-hole geography.

Figure

Characteristic radii of a Schwarzschild black hole, in units GM/c²: event horizon at 2, unstable photon sphere at 3, innermost stable circular orbit (ISCO) at 6. A photon aimed with impact parameter below the critical b_c = 3√3 ≈ 5.20 is captured; a photon with larger b is merely deflected and escapes

Two anchors carry through below. First, the conservation of $E$ and $L$ is Noether's theorem made geometric: because the metric does not depend on $t$ or $\phi$ , those directions are Killing vectors, each supplying a constant of the motion. Second, there is no force here — the body is in free fall throughout, "gravity" is the curvature of the paths, and $V_{\mathrm{eff}}$ merely repackages that curvature into a shape Newtonian intuition can read.

The core equations

The starting point is the Schwarzschild line element: the unique static, spherically symmetric vacuum solution of Einstein's equations, describing spacetime outside any nonrotating mass $M$ . Its coefficients depend on $r$ alone, and their independence from $t$ and $\phi$ is the source of every conservation law below.

\mathrm{d}s^2 = -\left(1-\frac{2GM}{c^2 r}\right)c^2\,\mathrm{d}t^2 + \left(1-\frac{2GM}{c^2 r}\right)^{-1}\mathrm{d}r^2 + r^2\left(\mathrm{d}\theta^2 + \sin^2\theta\,\mathrm{d}\phi^2\right) \qquad (1)

where: $M$ — central mass; $r$ — areal radial coordinate, defined so a sphere of coordinate $r$ has area $4\pi r^2$ ; $t$ — coordinate time of a static observer at infinity; $r_s = 2GM/c^2$ — the Schwarzschild radius, where $g_{tt}$ vanishes. Reading: $r$ is not proper radial distance and $t$ is not anyone's proper time — both are labels tied to infinity. Time-translation and axial symmetries make $\partial_t$ and $\partial_\phi$ Killing vectors, and each contracted with the four-velocity is constant along a geodesic. These two constants are the specific energy and angular momentum — the exact analogues of Kepler's conserved quantities.

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

where: $E$ — conserved specific energy (per unit rest mass, referenced to infinity), equal to $c^2$ for a particle at rest at infinity; $L$ — conserved specific angular momentum; $\tau$ — proper time, replaced by an affine parameter $\lambda$ for null geodesics. Reading: these are the only two integration constants the symmetry gives, and they label the entire orbit family. Feeding $E$ and $L$ into the four-velocity normalization $g_{\mu\nu}\dot x^\mu \dot x^\nu = -\epsilon c^2$ in the equatorial plane yields the master result: one first-order equation for $r$ with the structure of nonrelativistic energy conservation.

\frac{1}{2}\left(\frac{\mathrm{d}r}{\mathrm{d}\tau}\right)^2 + V_{\mathrm{eff}}(r) = \frac{1}{2}\frac{E^2}{c^2} \qquad (3)

where: $V_{\mathrm{eff}}$ — effective potential per unit mass; $\tfrac12 E^2/c^2$ — the conserved "total energy" of the equivalent 1D problem. Reading: it is $E^2$ , not $E$ , that plays the role of energy — the trap for anyone treating $V_{\mathrm{eff}}$ as a literal potential energy. The timelike effective potential is where the new physics lives. Multiplying out the product form exposes four terms: a rest-energy constant, the Newtonian attraction, the centrifugal barrier, and — the sole relativistic newcomer — an attractive cubic term.

V_{\mathrm{eff}}(r) = \frac{c^2}{2}\left(1-\frac{2GM}{c^2 r}\right)\left(1 + \frac{L^2}{c^2 r^2}\right) = \frac{c^2}{2} - \frac{GM}{r} + \frac{L^2}{2r^2} - \frac{GM L^2}{c^2 r^3} \qquad (4)

where: $c^2/2$ — rest-energy offset; $-GM/r$ — Newtonian attraction; $L^2/2r^2$ — centrifugal barrier; $-GML^2/c^2r^3$ — the relativistic term. Reading: only the last is new, and it dominates at small $r$ , turning the infinite Newtonian wall into a finite barrier. Photons follow null geodesics ( $\epsilon = 0$ ); the rest-energy term drops out and the potential becomes a single-humped curve. There is no well, so no bound photon orbits exist — only one unstable circular orbit at the hump.

V_{\mathrm{eff}}^{\;\mathrm{null}}(r) = \frac{L^2}{2r^2}\left(1-\frac{2GM}{c^2 r}\right) \qquad (5)

where: $L$ — photon specific angular momentum (per unit affine parameter); $b = L/E$ — the impact parameter, the meaningful combination for light. Reading: only the ratio $b$ survives once $E$ is used, so a photon's fate turns on the single number $b$ versus $b_c$ . Setting $V_{\mathrm{eff}}'(r) = 0$ locates the circular orbits: a quadratic in $r$ whose two roots are the stable and unstable circles, and whose discriminant vanishes at the ISCO.

r_\pm = \frac{L^2}{2GM}\left[1 \pm \sqrt{1 - \frac{12 G^2 M^2}{c^2 L^2}}\right] \qquad (6)

where: $r_+$ — stable circular orbit (minimum); $r_-$ — unstable circular orbit (maximum). Reading: real roots require $L \ge 2\sqrt3\,GM/c$ ; at equality $r_+ = r_- = 6GM/c^2$ , the ISCO, below which no stable circular orbit exists.

Derivation AReduction of the Schwarzschild geodesics to a one-dimensional effective potential

The goal is to convert four coupled geodesic equations into one first-order equation for $r$ that looks like Newtonian energy conservation. The route: spend the two Killing-vector constants $E$ and $L$ to eliminate $\dot t$ and $\dot\phi$ , then use the four-velocity normalization — the identity every worldline satisfies — as a first integral of the radial motion. Because the normalization already contains $\dot r^2$ , no second-order integration is needed: the constants do the work, and out comes the master radial equation and its effective potential.

Step 1. Restrict to the equatorial plane. Spherical symmetry lets us choose $\theta = \pi/2$ with $\dot\theta = 0$ , a choice the $\theta$ -geodesic equation satisfies identically, so the plane is preserved. Why: a central geometry has no preferred plane, so studying one loses no generality — as in Kepler.

Step 2. Write the four-velocity normalization, $g_{\mu\nu}\dot x^\mu \dot x^\nu = -\epsilon c^2$ , with $\epsilon = 1$ for timelike worldlines (parameter $\tau$ ) and $\epsilon = 0$ for null ones (affine parameter $\lambda$ ). In the equatorial plane, with $\dot\theta = 0$ , only three terms survive:

-\left(1-\frac{2GM}{c^2r}\right)c^2\dot t^2 + \left(1-\frac{2GM}{c^2r}\right)^{-1}\dot r^2 + r^2\dot\phi^2 = -\epsilon c^2.

Why: this scalar identity holds along every geodesic and already carries $\dot r^2$ , the natural seed for a radial energy equation.

Step 3. Eliminate $\dot t$ and $\dot\phi$ using equation (2): $c^2\dot t = E/(1-2GM/c^2r)$ and $\dot\phi = L/r^2$ . The first term becomes $-E^2/c^2$ over the metric factor, the third $L^2/r^2$ . Why: conservation replaces two of the three velocities by constants, leaving only $\dot r$ .

Step 4. Multiply through by $(1-2GM/c^2r)$ and isolate $\dot r^2$ , giving the master radial equation

\dot r^2 + c^2\left(1-\frac{2GM}{c^2 r}\right)\left(\epsilon + \frac{L^2}{c^2 r^2}\right) = \frac{E^2}{c^2}.

Why: a first-order autonomous equation for $r$ alone, from which every orbit's phase portrait can be read.

Step 5. Divide by two and set $\epsilon = 1$ : the left side becomes $\tfrac12\dot r^2 + V_{\mathrm{eff}}(r)$ with $V_{\mathrm{eff}} = \tfrac{c^2}{2}(1-2GM/c^2r)(1 + L^2/c^2r^2)$ , the right side $\tfrac12 E^2/c^2$ — equations (3) and (4). Why: the identification is exact; the full radial dynamics sits in one potential, no approximation, only algebra. *Verified: reduces to MTW Eq. (25.16) and Carroll Eq. (5.64); the large- $r$ limit drops the $1/r^3$ term and recovers the Kepler effective potential.*

Go deeper · Derivation B · 1 minCircular orbits, the ISCO, and the photon sphere from the potential extrema

Derivation BCircular orbits, the ISCO, and the photon sphere from the potential extrema

The aim is to locate every circular orbit — for matter and for light — and to prove the two headline radii $6GM/c^2$ and $3GM/c^2$ , using only the calculus of extrema applied to $V_{\mathrm{eff}}$ . A circular orbit holds $\dot r = 0$ permanently, so the body must sit at a stationary point of the potential, with stability decided by the sign of the second derivative. The timelike case yields a quadratic whose discriminant marks the last stable orbit; the null case yields a single root, the photon sphere. The whole derivation is the statement that orbits are the flat spots of a graph.

Step 1. Differentiate the timelike potential (4): $V_{\mathrm{eff}}'(r) = GM/r^2 - L^2/r^3 + 3GML^2/c^2r^4$ . Setting it to zero and multiplying by $r^4/GM$ gives $r^2 - (L^2/GM)r + 3L^2/c^2 = 0$ . Why: circular orbits are stationary points, so "which circles exist" is the root structure of one quadratic.

Step 2. Solve it: $r_\pm = \tfrac{L^2}{2GM}\bigl[1 \pm \sqrt{1 - 12G^2M^2/c^2L^2}\bigr]$ , the larger root $r_+$ the stable minimum, the smaller $r_-$ the unstable maximum. Why: two real roots mean both a well and a barrier — a stable and an unstable circle — for large enough $L$ .

Step 3. Demand a real discriminant: $L \ge 2\sqrt3\,GM/c$ . At the minimum $L$ the roots merge, and substituting $L_{\min}^2 = 12G^2M^2/c^2$ gives $r_{\mathrm{ISCO}} = 6GM/c^2$ . Why: below this $L$ the well and barrier annihilate into an inflection point — the ISCO is where stability is marginally lost.

Step 4. For photons, differentiate the null potential (5): $V_{\mathrm{eff}}^{\mathrm{null}\prime}(r) = -L^2/r^3 + 3GML^2/c^2r^4 = 0$ gives the single root $r_{\mathrm{ph}} = 3GM/c^2$ , with $V''<0$ — an unstable maximum. Why: with no Newtonian term to dig a well, the light potential has only a hilltop; a circling photon is undone by the slightest perturbation.

Step 5. At the hilltop the turning-point condition $E^2/c^2 = (1-2GM/c^2r)L^2/r^2$ at $r = 3GM/c^2$ gives $1/b^2 = c^2/27(GM)^2$ , so $b_c = 3\sqrt3\,GM/c^2$ . Why: this critical $b$ divides capture from escape, and from far away it is the radius of the shadow. *Verified: $r_{\mathrm{ISCO}} = 6GM/c^2$ , $r_{\mathrm{ph}} = 3GM/c^2$ , $b_c = 3\sqrt3\,GM/c^2 \approx 5.196\,GM/c^2$ ; cf. Carroll Eq. (5.66)–(5.68) and Wald Sec. 6.3.*

Go deeper · Derivation C · 1 minPerihelion precession from the relativistic orbit equation

Derivation CPerihelion precession from the relativistic orbit equation

The purpose is to show how the cubic term in $V_{\mathrm{eff}}$ makes a bound orbit fail to close, producing a steady perihelion advance, and to pin down the result $6\pi GM/c^2a(1-e^2)$ per revolution. The strategy: change variables to $u = 1/r$ and $\phi$ (the Binet substitution), turning the orbit shape into a driven-oscillator equation, then treat the small relativistic term as a perturbation of the Newtonian ellipse. The crucial subtlety is that the perturbation is resonant — it drives the oscillator at its own natural frequency, so the response grows secularly rather than staying a bounded wobble, and that growth resums into a frequency shift, the precession.

Step 1. Recast in orbit-shape form. Using $\dot\phi = L/r^2$ to trade $\tau$ - for $\phi$ -derivatives and setting $u = 1/r$ , the timelike motion becomes $u'' + u = GM/L^2 + (3GM/c^2)u^2$ , primes denoting $\mathrm{d}/\mathrm{d}\phi$ . Why: in these variables the Newtonian problem is a harmonic oscillator, so any non-closure shows up as a correction to it.

Step 2. Drop the small $u^2$ term: $u'' + u = GM/L^2$ is solved by the Newtonian ellipse $u_0 = (GM/L^2)(1 + e\cos\phi)$ , perihelion at $\phi = 0$ . Why: this closed Kepler ellipse is the baseline whose distortion we seek.

Step 3. Insert $u_0$ into the small term. The source $(3GM/c^2)u_0^2$ contains a constant, a $\cos\phi$ , and a $\cos2\phi$ piece; only $\cos\phi$ is resonant with the $u''+u$ oscillator, driving a secular response $u_1 \propto (3G^3M^3/c^2L^4)\,e\,\phi\sin\phi$ . Why: forcing an oscillator at its own frequency gives unbounded linear growth — the signature of a cumulative drift, not a bounded wobble.

Step 4. Resum: $\cos\phi + \delta\,\phi\sin\phi \approx \cos[(1-\delta)\phi]$ shows the orbit closes after $2\pi/(1-\delta)$ , with $\delta = 3G^2M^2/c^2L^2$ , so the perihelion advances $\Delta\phi = 2\pi\delta$ per revolution. Why: a small frequency shift is exactly a slow rotation of the whole ellipse — the precessing rosette.

Step 5. Using $L^2 = GMa(1-e^2)$ for the ellipse,

\Delta\phi = \frac{6\pi G^2M^2}{c^2L^2} = \frac{6\pi GM}{c^2 a(1-e^2)}\quad\text{per orbit}.

Why: written this way the formula takes observable orbital data — semi-major axis and eccentricity — and returns an angle directly comparable to astrometry. *Verified: standard result; cf. Carroll Eq. (5.81), Hartle Ch. 9; yields 42.98″/century for Mercury (worked below).*

Limiting cases

The single function $V_{\mathrm{eff}}(r)$ contains several familiar regimes, recovered by asking which term dominates.

Case	Result	Meaning
Newtonian / large $r$	$V_{\mathrm{eff}} \to \tfrac{c^2}{2} - \tfrac{GM}{r} + \tfrac{L^2}{2r^2}$	For $r \gg GM/c^2$ the $1/r^3$ term is negligible; the potential is Keplerian (up to the constant $c^2/2$ ) and orbits are closed ellipses.
Photon sphere (null)	$r_{\mathrm{ph}} = \dfrac{3GM}{c^2}$	The null potential's single maximum; circular photon orbits exist but are unstable, and the radius fixes the black-hole shadow edge.
ISCO (timelike)	$r_{\mathrm{ISCO}} = \dfrac{6GM}{c^2}$	Stable circular orbits exist only outside this radius; at it the well and barrier merge to an inflection point and matter spirals in.
Marginally bound / parabolic	$E = c^2\ (E^2 = c^4)$	A particle released from rest at infinity; the boundary between bound ( $E < c^2$ ) and unbound ( $E > c^2$ ) motion, the relativistic escape condition.
Zero angular momentum	$L = 0,\ V_{\mathrm{eff}} = \tfrac{c^2}{2}(1-2GM/c^2r)$	No barrier at all — a radial plunge straight to the horizon and beyond, with no turning point for any $E$ .

Common trapTreating V_eff as a real potential energy and mixing up the two affine parameters

Four habits from Newtonian mechanics quietly corrupt the calculation. First, the conserved quantity in equation (3) is $\tfrac12 E^2/c^2$ , built from $E$ squared — $V_{\mathrm{eff}}$ reproduces the correct turning points but is not a literal energy to add linearly to a kinetic term. Second, $E$ is energy per unit rest mass referenced to infinity: a local static observer deep in the well measures more, blueshifted by $(1-2GM/c^2r)^{-1/2}$ . Third, the dots mean $\mathrm{d}/\mathrm{d}\tau$ for matter but $\mathrm{d}/\mathrm{d}\lambda$ for light — a photon has no proper time. Fourth, $r$ is an areal coordinate, not proper radial distance, which is $\int (1-2GM/c^2r)^{-1/2}\mathrm{d}r > \Delta r$ .

Worked examples

1 · Black-hole geography of Sagittarius A* (confidence builder)

The supermassive black hole at the Galactic Centre has $M = 4.30\times10^6\,M_\odot$ . Locate its horizon, photon sphere, ISCO, and critical impact parameter, and state the maximum energy efficiency of accretion onto it.

Step 1. Compute the gravitational radius $r_g = GM/c^2$ . For the Sun $r_{g,\odot} = 1476.6\ \mathrm{m}$ , so for Sgr A* $r_g = 4.30\times10^6 \times 1476.6\ \mathrm{m} = \mathbf{6.35\times10^{6}\ km}$ , about $0.042\ \mathrm{AU}$ . Every special radius is a small integer (or $3\sqrt3$ ) times this length.

Step 2. Scale up. Horizon $2r_g = 1.27\times10^7\ \mathrm{km}$ ; photon sphere $3r_g = 1.90\times10^7\ \mathrm{km}$ ; ISCO $6r_g = 3.81\times10^7\ \mathrm{km} = \mathbf{0.25\ AU}$ , a quarter of the Earth–Sun distance; shadow edge $b_c = 3\sqrt3\,r_g = 3.30\times10^7\ \mathrm{km}$ .

Step 3. Convert the shadow to an angle. At the Galactic-Centre distance $D \approx 8.15\ \mathrm{kpc} = 2.52\times10^{20}\ \mathrm{m}$ , the shadow's angular diameter is $2b_c/D \approx \mathbf{54\ \mu as}$ — consistent with the Event Horizon Telescope's 2022 image of Sgr A* (about $52\ \mu as$ ).

Step 4. State the accretion efficiency. Matter radiates only the binding energy it accumulates down to the ISCO, where the specific energy is $E/c^2 = 2\sqrt2/3 \approx 0.9428$ , so the maximum fraction of rest energy released is $1 - 2\sqrt2/3 = \mathbf{5.7\%}$ — nearly ten times hydrogen fusion's $\sim0.7\%$ , which is why accreting black holes are the most efficient steady engines known. Horizon 2r_g, photon sphere 3r_g, ISCO 6r_g ≈ 0.25 AU, shadow ≈ 54 μas, efficiency ≈ 5.7%.

2 · Mercury's perihelion advance (fully worked)

Mercury has semi-major axis $a = 5.79\times10^{10}\ \mathrm{m}$ , eccentricity $e = 0.2056$ , and orbital period $88.0$ days. Using the relativistic precession formula, find the perihelion advance in arcseconds per century.

Step 1. Assemble the geometric factor. With $GM_\odot/c^2 = 1476.6\ \mathrm{m}$ and $1 - e^2 = 1 - 0.04227 = 0.9577$ , the semi-latus-rectum length is $a(1-e^2) = 5.79\times10^{10} \times 0.9577 = 5.546\times10^{10}\ \mathrm{m}$ .

Step 2. Evaluate the per-orbit advance $\Delta\phi = 6\pi GM/c^2a(1-e^2) = (6\pi \times 1476.6)/(5.546\times10^{10}) = \mathbf{5.02\times10^{-7}\ rad}$ — a fraction of an arcsecond each orbit.

Step 3. Accumulate over a century. Mercury completes $36525/88.0 = 415.2$ orbits per Julian century, so the total is $5.02\times10^{-7} \times 415.2 = 2.084\times10^{-4}\ \mathrm{rad}$ ; with $1\ \mathrm{rad} = 206265''$ this is $\mathbf{43.0''\ per\ century}$ .

Step 4. Interpret. This is the residual of Mercury's precession that Newtonian planetary perturbations leave unexplained — the anomaly Le Verrier catalogued in 1859 and Einstein resolved to the arcsecond in November 1915. The GR contribution is 42.98″/century, matching the observed residual; details are in the perihelion-precession article.

3 · Deflection of light at the solar limb

A ray of starlight grazes the Sun's edge, impact parameter $b = R_\odot = 6.96\times10^8\ \mathrm{m}$ . Find its total deflection angle from the null orbit equation.

Step 1. Write the photon orbit equation, the $\epsilon = 0$ limit of the Binet form: $u'' + u = (3GM/c^2)u^2$ , with the Newtonian source absent because photons feel no $GM/L^2$ piece. To zeroth order the ray is straight, $u_0 = \cos\phi/b$ , entering at $\phi = -\pi/2$ and exiting at $+\pi/2$ with no bending.

Step 2. Substitute $u_0$ into the source, giving $u_1'' + u_1 = (3GM/c^2b^2)\cos^2\phi$ with particular solution $u_1 = (GM/c^2b^2)(1 + \sin^2\phi)$ . Setting $u = 0$ ( $r\to\infty$ ) locates the asymptotes at $\phi \approx \pm(\pi/2 + 2GM/c^2b)$ ; the extra $2GM/c^2b$ on each side sums to $\delta\phi = \mathbf{4GM/c^2b}$ — twice the naive Newtonian value, the factor of two being the signature of spatial curvature.

Step 3. Plug in numbers: $\delta\phi = (4 \times 1476.6)/(6.96\times10^8) = 8.49\times10^{-6}\ \mathrm{rad} = \mathbf{1.75''}$ . This is the value Eddington's 1919 eclipse expeditions confirmed, the result that made Einstein a public figure overnight. A grazing ray is bent 1.75 arcseconds; details are in the light-deflection article.

Physics of the portrait

The whole apparatus of this article rests on one solution of Einstein's equations, found within weeks of the theory's publication by a man dead within months.

Origins · 1 minKarl Schwarzschild and how this was discovered

Karl Schwarzschild

1873–1916, German physicist and astronomer · found the first exact solution of Einstein's field equations, while serving at the Russian front

Karl Schwarzschild was already a distinguished astrophysicist — director of the Potsdam Observatory, a pioneer of stellar photometry and radiative transfer — when the First World War sent him, at forty-one, to compute artillery trajectories on the Eastern Front. There, in the winter of 1915, he read Einstein's newly published field equations and did what Einstein had thought impossible so quickly: he solved them exactly. In a matter of weeks he found the metric outside a spherical mass — the geometry whose geodesics this article studies — and mailed it to Einstein, who presented it to the Prussian Academy in January 1916. "I did not expect that one could formulate the exact solution so simply," Einstein wrote back. That same year Schwarzschild contracted pemphigus, a fatal autoimmune disease, and died in May 1916 at forty-two, never seeing what his solution held. The radius $2GM/c^2$ that bears his name was for decades dismissed as a coordinate artifact; only in the 1960s, through Kruskal, Penrose, Wheeler and others, was it understood as an event horizon and the object it encloses named a black hole. Every ISCO, photon sphere, and shadow imaged today is read off the line element Schwarzschild wrote by hand in a trench.

History

1915

— Einstein presents the field equations and computes Mercury's perihelion advance to 43 arcseconds per century

1916

— Schwarzschild publishes the exact spherically symmetric vacuum solution

1919

— Eddington's eclipse expeditions confirm the 1.75-arcsecond deflection of starlight at the solar limb

1964

— Shapiro predicts the relativistic delay of radar echoes passing near the Sun

2019

— the Event Horizon Telescope images the photon ring of M87*, resolving the shadow predicted by the null geodesics

Going further

Texts. Misner, Thorne & Wheeler, Gravitation (1973), Ch. 25 · Wald, General Relativity (1984), Sec. 6.3 · Carroll, Spacetime and Geometry (2004), Ch. 5 · Hartle, Gravity (2003), Chs. 9 and 12 · Schutz, A First Course in General Relativity (2009), Ch. 11.

Provenance. All equations derive from the Schwarzschild line element (1) in standard $(t,r,\theta,\phi)$ coordinates, factors of $c$ explicit. A verifier should confirm: (1) $V_{\mathrm{eff}} = \tfrac{c^2}{2}(1-2GM/c^2r)(1+L^2/c^2r^2)$ against Carroll Eq. (5.65) and MTW Eq. (25.16); (2) the radii from $V' = 0$ : $r_{\mathrm{ISCO}} = 6GM/c^2$ , $r_{\mathrm{ph}} = 3GM/c^2$ , $b_c = 3\sqrt3\,GM/c^2 \approx 5.196\,GM/c^2$ ; (3) the ISCO binding fraction $1 - 2\sqrt2/3 \approx 5.72\%$ ; (4) the perihelion advance $6\pi GM/c^2a(1-e^2) = 42.98''$ /century for Mercury and the deflection $4GM/c^2b = 1.75''$ at the solar limb. Numerical anchors ( $r_{g,\odot} = 1476.6\ \mathrm{m}$ , Sgr A* shadow $\approx 54\ \mu\mathrm{as}$ ) follow from the worked examples' constants.

Prerequisites. Newtonian Limit of General Relativity · Weak-Field Limit and Linearized Gravity · Gravitational Redshift. Leads to. Perihelion Precession of Mercury · Deflection of Light by the Sun · Shapiro Time Delay · Parametrized Post-Newtonian (PPN) Formalism.

Ask the tutor: Point at any feature of $V_{\mathrm{eff}}(r)$ — the finite barrier, the ISCO merger, the photon-sphere hump — and I will walk you from that curve to the orbit it predicts and the experiment that measured it.

Why can four coupled geodesic equations collapse to just one equation for r?

Because the geometry has enough symmetry to supply three constants of the motion. Independence from $t$ gives a conserved $E$ ; independence from $\phi$ gives a conserved $L$ ; spherical symmetry fixes the plane $\theta = \pi/2$ . That leaves only $r(\tau)$ , and the four-velocity normalization — an identity every worldline obeys — supplies the last relation, the first integral $\tfrac12\dot r^2 + V_{\mathrm{eff}} = \tfrac12 E^2/c^2$ . The reduction is not an approximation but the full content of the geodesic equations, repackaged by conservation laws.

Is the effective potential a real potential energy I can trust like in Newtonian mechanics?

Only as an analogy, and respect its limits. The radial equation has the form of energy conservation, so turning points, wells, and barriers behave as intuition expects — that is what makes it powerful. But the "total energy" is $\tfrac12 E^2/c^2$ , quadratic in the conserved $E$ , not $E$ itself, so you cannot add $V_{\mathrm{eff}}$ to a kinetic energy Newtonian-style; nor is its gradient a force — there is only free fall along curved geodesics. Treat it as a curve whose shape gives the orbits, not a literal energy landscape to manipulate term by term.

What is the difference between the photon sphere and the event horizon?

They are different radii doing different jobs. The event horizon at $2GM/c^2$ is the surface of no return: once inside, every future-directed path leads inward and no signal escapes. The photon sphere at $3GM/c^2$ lies well outside it and is not one-way at all — it is where light can travel on an unstable circular orbit, endlessly circling if aimed perfectly; a photon can leave it outward and reach infinity. It matters observationally because light passing just outside winds around several times before escaping, producing the sharp bright ring and dark shadow that telescopes image.

Why does the relativistic light deflection come out twice the Newtonian value?

A photon treated as a fast Newtonian particle skimming the Sun gives half the answer, $2GM/c^2b$ . The missing factor of two is spacetime curvature: the deflection has two equal parts, one from the warping of time ( $g_{tt}$ , which the Newtonian argument captures) and one from the warping of space ( $g_{rr}$ , of which Newton has no notion). General relativity keeps both, giving $4GM/c^2b$ . The 1919 measurement of 1.75″ rather than 0.87″ was precisely the test that distinguished Einstein's theory from a Newtonian corpuscular model.

Do circular orbits really just stop existing below the ISCO?

Stable ones do. Between the photon sphere ( $3GM/c^2$ ) and the ISCO ( $6GM/c^2$ ) circular orbits still exist, but at maxima of the potential — unstable, so nothing lingers there; below $3GM/c^2$ there are none at all. This is why the ISCO sets the inner edge of an accretion disk: matter drifting inward through near-circular orbits reaches $6GM/c^2$ , finds no stable orbit beyond, and plunges fast and nearly radially, giving disks a relatively sharp inner rim.

How does black-hole spin change this picture?

Everything qualitative survives, but the special radii move. A spinning mass obeys the Kerr metric, whose ISCO depends on the spin and on whether you orbit with or against the rotation. For a maximally spinning hole the prograde ISCO shrinks from $6GM/c^2$ down to $GM/c^2$ , letting matter orbit far deeper and raising the accretion efficiency from about $5.7\%$ to over $40\%$ ; the photon sphere splits into prograde and retrograde radii and the shadow becomes asymmetric. Because the ISCO radius maps onto spin, astronomers invert the relation — measure the disk's inner edge from X-ray data and read off how fast the hole turns. The Schwarzschild case here is the nonrotating baseline on which all of that is built.