Orbits in the Schwarzschild Geometry

A planet or a test particle in the field of a spherical star follows a timelike geodesic of the Schwarzschild metric. Rather than integrate the four geodesic equations directly, the two continuous symmetries of the metric supply two constants of motion, and the normalization of the four-velocity supplies a third relation. Together they collapse the problem to a single first-order equation for $r (τ)$ , formally identical to a Newtonian particle in an effective potential. The differences from Newton's potential are one extra term, and that term drives every relativistic orbital effect the solar system measures.

Conserved quantities from the Killing vectors

The Schwarzschild metric components depend on neither $t$ nor $ϕ$ . Each ignored coordinate corresponds to a Killing vector — a symmetry direction along which the geometry is unchanged — and each Killing vector gives a quantity conserved along any geodesic. The relevant fact from the geodesics lesson is that if $ξ^{μ}$ is a Killing vector and $u^{μ} = d x^{μ} / d τ$ is the four-velocity, then $ξ_{μ} u^{μ}$ is constant along the geodesic.

Working in the equatorial plane $θ = π /2$ (spherical symmetry lets any orbit be rotated into this plane, where it stays), the time-translation Killing vector $ξ = \partial_{t}$ gives the conserved energy per unit mass

E = - c^{2} ξ_{μ} u^{μ} = (1 - \frac{r _{s}}{r}) c^{2} \frac{d t}{d τ},

and the rotational Killing vector $η = \partial_{ϕ}$ gives the conserved angular momentum per unit mass

L = η_{μ} u^{μ} = r^{2} \frac{d ϕ}{d τ} .

Far from the mass $E \to c^{2} + \frac{1}{2} v^{2} + \dots$ , the rest energy plus the Newtonian kinetic and potential energy per unit mass; $L$ is the familiar Newtonian specific angular momentum $r^{2} \dot{ϕ}$ . Both are exact constants of the full relativistic motion, not approximations.

Each ignored coordinate of the metric is a Killing symmetry direction. Time-translation invariance conserves the energy per unit mass; rotational invariance conserves the angular momentum per unit mass. Both are constant along the orbit.

The effective potential

The third relation is the normalization of the four-velocity for a massive particle, $g_{μν} u^{μ} u^{ν} = - c^{2}$ . Written out in the equatorial plane,

- (1 - \frac{r _{s}}{r}) c^{2} (\frac{d t}{d τ})^{2} + (1 - \frac{r _{s}}{r})^{- 1} (\frac{d r}{d τ})^{2} + r^{2} (\frac{d ϕ}{d τ})^{2} = - c^{2} .

Substituting $d t / d τ = E / [c^{2} (1 - r_{s} / r)]$ and $d ϕ / d τ = L / r^{2}$ and multiplying through by $(1 - r_{s} / r)$ removes every reference to $t$ and $ϕ$ , leaving one equation for $r (τ)$ :

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

Grouping the $r$ -dependent terms into a potential casts this in the form of a one-dimensional energy equation,

\frac{1}{2} (\frac{d r}{d τ})^{2} + V_{eff} (r) = \frac{E ^{2} - c ^{4}}{2 c ^{2}}, V_{eff} (r) = - \frac{GM}{r} + \frac{L ^{2}}{2 r ^{2}} - \frac{GM L ^{2}}{c ^{2} r ^{3}},

using $r_{s} = 2 GM / c^{2}$ . The right-hand side is a constant fixed by $E$ , playing the role of the total energy of a fictitious unit-mass particle moving in one dimension. The radial motion matches that of a Newtonian particle in the potential $V_{eff}$ , and the orbit is read off the same turning-point analysis as any central-force problem.

The three terms name themselves against the Newtonian case.

$- GM / r$ — the Newtonian gravitational attraction, unchanged.
$+ L^{2} /2 r^{2}$ — the centrifugal barrier from angular momentum, unchanged.
$- GM L^{2} / (c^{2} r^{3})$ — the general-relativistic correction, absent from Newton's potential. It is attractive and, because it falls off as $1/ r^{3}$ , it dominates at small $r$ and overwhelms the centrifugal barrier there.

In Newtonian gravity the $1/ r^{2}$ centrifugal term wins as $r \to 0$ and the potential rises to $+ \infty$ , so any particle with $L \neq = 0$ is thrown back out — nothing with angular momentum reaches the center. In Schwarzschild the $1/ r^{3}$ term eventually wins, so the barrier has a finite maximum and turns over to $- \infty$ . A particle with enough energy passes the top of the barrier and is captured. Capture of orbiting matter is impossible in Newtonian gravity and generic in general relativity.

The Schwarzschild effective potential (solid) against the Newtonian one (dotted) at the same angular momentum. Newton's centrifugal barrier rises without bound; the relativistic barrier has a finite maximum and plunges to minus infinity at small r, so a sufficiently energetic particle is captured.

Circular orbits and the ISCO

Circular orbits sit at extrema of $V_{eff}$ , where $V_{eff}^{'} (r) = 0$ :

\frac{GM}{r ^{2}} - \frac{L ^{2}}{r ^{3}} + \frac{3 GM L ^{2}}{c ^{2} r ^{4}} = 0 ⟺ GM r^{2} - L^{2} r + \frac{3 GM L ^{2}}{c ^{2}} = 0.

For fixed $L$ this quadratic in $r$ has two roots,

r_{\pm} = \frac{L ^{2} \pm L ^{4} - 12 G ^{2} M ^{2} L ^{2} / c ^{2}}{2 GM} .

The larger root $r_{+}$ is a minimum of $V_{eff}$ , a stable circular orbit; the smaller root $r_{-}$ is a maximum, an unstable circular orbit that a small inward push sends into the plunge. Newton's potential has only the single stable minimum; the unstable inner orbit is a relativistic feature, the balance point on top of the finite barrier.

The two roots exist only when the discriminant is non-negative, $L^{2} \geq 12 G^{2} M^{2} / c^{2}$ . At equality the roots merge: the stable and unstable orbits coincide and the last stable circular orbit sits at

L^{2} = \frac{12 G ^{2} M ^{2}}{c ^{2}} ⟹ r_{ISCO} = \frac{L ^{2}}{2 GM} = \frac{6 GM}{c ^{2}} = 3 r_{s} .

The ISCO has a direct astrophysical role: it sets the inner edge of an accretion disk around a black hole, and the binding energy released by matter spiraling to the ISCO fixes the efficiency of accretion power. For the Schwarzschild ISCO the orbital energy is $E / c^{2} = 8/9 \approx 0.943$ , so about $5.7%$ of the rest energy is radiated on the way in — far above the fraction released in nuclear fusion.

As the angular momentum decreases toward the marginal value, the well and barrier of the effective potential merge. At L = 2 sqrt-3 GM/c the minimum and maximum coincide at r = 6 GM/c-squared, the innermost stable circular orbit.

Radial infall in proper and coordinate time

Set $L = 0$ for a particle falling straight in, released from rest at infinity so $E = c^{2}$ . The radial equation collapses to

(\frac{d r}{d τ})^{2} = \frac{E ^{2}}{c ^{2}} - c^{2} (1 - \frac{r _{s}}{r}) = \frac{c ^{2} r _{s}}{r},

which is identical in form to Newtonian free fall. Integrating from radius $R$ inward gives a finite proper time to reach any radius, including the horizon and the center:

τ = \frac{2}{3 c r _{s}} (R^{3/2} - r^{3/2}) .

The infalling observer crosses $r = r_{s}$ and reaches $r = 0$ in finite time by their own clock, feeling nothing special at the horizon. Coordinate time tells a different story. From $d t / d τ = E / [c^{2} (1 - r_{s} / r)] = 1/ (1 - r_{s} / r)$ ,

\frac{d t}{d r} = \frac{d t / d τ}{d r / d τ} = - \frac{1}{c} \frac{r}{r _{s}} \frac{1}{1 - r _{s} / r},

which diverges as $r \to r_{s}$ . The coordinate time integral goes as $- ln (r - r_{s})$ near the horizon: an observer using the far-away time $t$ sees the infalling body slow, freeze, and never quite reach $r_{s}$ . Both descriptions are correct; they refer to different clocks. The divergence is a property of the Schwarzschild coordinate $t$ at $r_{s}$ , not of the geometry, a point the horizon lesson resolves with better-behaved coordinates.

A body falling from rest reaches the horizon r-s in finite proper time tau (solid), crossing smoothly. In coordinate time t (dotted) it asymptotically approaches r-s and never crosses, appearing to freeze to a distant observer.

Precession from the extra term

For a bound orbit the same $1/ r^{3}$ term that reshapes the barrier makes the orbit fail to close. Changing the independent variable from $τ$ to $ϕ$ with $u = 1/ r$ and $d r / d τ = - L d u / d ϕ$ , the radial equation differentiates into the orbit equation

\frac{d ^{2} u}{d ϕ ^{2}} + u = \frac{GM}{L ^{2}} + \frac{3 GM}{c ^{2}} u^{2} .

Without the final term this is the Newtonian orbit equation, whose bound solutions are closed ellipses $u = (GM / L^{2}) (1 + e cos ϕ)$ that return to the same perihelion every $2 π$ . The term $3 GM u^{2} / c^{2}$ is the relativistic correction. It is small in the solar system — its ratio to the leading term is $3 L^{2} / (c^{2} r^{2}) \sim 3 (v / c)^{2} \sim 1 0^{- 8}$ for Mercury — but it does not average to zero over an orbit. Its effect is to make the ellipse advance: the perihelion rotates by a small angle each revolution, and the orbit traces a slowly turning rosette that never closes.

The relativistic 1/r-cubed term makes a bound orbit precess. The ellipse advances by a small angle each revolution, so the perihelion rotates and the orbit fills an annulus rather than closing on itself.

The precession rate integrates to a per-orbit advance of $6 π GM / (c^{2} a (1 - e^{2}))$ for a nearly Newtonian ellipse of semimajor axis $a$ and eccentricity $e$ . Applied to Mercury it gives the residual precession that Newtonian perturbation theory could not account for; the perihelion lesson carries the calculation to the observed $4 3^{''}$ per century. The same term, at smaller $r$ where $(v / c)^{2}$ is not small, produces the strong-field orbits near black holes, whose light-ray counterpart the next lesson takes up.