Geodesics and Orbits
Conserved quantities, effective potentials, and orbital dynamics in Schwarzschild spacetime
Killing Vectors and Conserved Quantities
The Schwarzschild metric is independent of both t and φ, which implies the existence of two Killing vector fields. By Noether's theorem, each Killing vector generates a conserved quantity along geodesics.
$$\xi^\mu_{(t)} = (1, 0, 0, 0) \quad \Longrightarrow \quad E = -g_{\mu\nu}\xi^\mu_{(t)} \frac{dx^\nu}{d\tau} = \left(1 - \frac{2M}{r}\right)\frac{dt}{d\tau}$$
The quantity E is the specific energy (energy per unit rest mass) of the test particle. For a particle at rest at infinity, E = 1. For bound orbits E < 1, and for unbound (scattering) orbits E > 1.
$$\xi^\mu_{(\phi)} = (0, 0, 0, 1) \quad \Longrightarrow \quad L = g_{\mu\nu}\xi^\mu_{(\phi)} \frac{dx^\nu}{d\tau} = r^2 \frac{d\phi}{d\tau}$$
The quantity L is the specific angular momentum. Since the metric has full spherical symmetry (not just axial symmetry), we can always orient coordinates so the motion lies in the equatorial plane θ = π/2 without loss of generality.
The Radial Geodesic Equation
For a massive particle, the normalization condition gμν(dxμ/dτ)(dxν/dτ) = -1 provides the radial equation of motion. Substituting the conserved quantities:
$$-\left(1 - \frac{2M}{r}\right)\left(\frac{dt}{d\tau}\right)^2 + \frac{1}{1 - \frac{2M}{r}}\left(\frac{dr}{d\tau}\right)^2 + r^2\left(\frac{d\phi}{d\tau}\right)^2 = -1$$
Substituting E and L and rearranging:
Radial Energy Equation
$$\frac{1}{2}\left(\frac{dr}{d\tau}\right)^2 + V_{\text{eff}}(r) = \frac{1}{2}(E^2 - 1)$$
This has exactly the form of a one-dimensional energy conservation law, with an effective potential:
The Effective Potential
Effective Potential (Massive Particles)
$$V_{\text{eff}}(r) = -\frac{M}{r} + \frac{L^2}{2r^2} - \frac{ML^2}{r^3}$$
Compare with the Newtonian effective potential VNewton = -M/r + L²/(2r²). The crucial difference is the relativistic correction term -ML²/r³, which dominates at small r. In Newtonian gravity, the centrifugal barrier L²/(2r²) always prevents a particle with L ≠ 0 from reaching r = 0. In GR, the additional attractive -ML²/r³ term overwhelms the centrifugal barrier, allowing capture.
For massless particles (photons), the normalization condition becomes gμνkμkν = 0, and the effective potential is different:
Effective Potential (Massless Particles)
$$V_{\text{eff}}^{\text{photon}}(r) = \frac{L^2}{r^2}\left(1 - \frac{2M}{r}\right)$$
Circular Orbits
Circular orbits occur at extrema of the effective potential: V'eff(r) = 0. Stability requires V''eff(r) > 0 (a minimum). The conditions are:
$$V'_{\text{eff}} = 0 \quad \Longrightarrow \quad \frac{M}{r^2} - \frac{L^2}{r^3} + \frac{3ML^2}{r^4} = 0$$
$$\Longrightarrow \quad L^2 = \frac{Mr^2}{r - 3M}$$
This immediately shows that circular orbits exist only for r > 3M (since L² must be positive).
Innermost Stable Circular Orbit (ISCO)
The marginally stable orbit occurs where V'eff = 0 and V''eff = 0 simultaneously (the minimum and maximum of Veff merge into an inflection point):
$$r_{\text{ISCO}} = 6M \qquad (= 3r_s)$$
$$E_{\text{ISCO}} = \frac{2\sqrt{2}}{3} \approx 0.9428 \qquad L_{\text{ISCO}} = 2\sqrt{3}\,M$$
The binding energy at the ISCO is 1 - EISCO ≈ 5.72% of the rest mass energy. This is the maximum energy extractable from matter falling into a Schwarzschild black hole via an accretion disk — setting the fundamental efficiency limit for non-rotating black hole accretion.
Orbit classification by radius:
- r > 6M: Stable circular orbits exist (bound elliptical orbits possible)
- 3M < r < 6M: Unstable circular orbits (any perturbation leads to plunge or escape)
- r = 3M: Photon sphere (light can orbit, unstably)
- r < 3M: No circular orbits possible for any particle
- r < 2M: Inside the event horizon — all trajectories terminate at r = 0
Photon Sphere at r = 3M
For null geodesics, the effective potential has a maximum at:
$$\frac{dV_{\text{eff}}^{\text{photon}}}{dr} = 0 \quad \Longrightarrow \quad r_{\text{photon}} = 3M$$
This is an unstable orbit: any radial perturbation sends the photon either into the black hole or out to infinity. The photon sphere determines the apparent angular size (the "shadow") of the black hole as seen by distant observers, and played a key role in the Event Horizon Telescope image of M87*.
The critical impact parameter for photon capture is:
$$b_{\text{crit}} = \frac{L}{E}\bigg|_{r=3M} = 3\sqrt{3}\,M \approx 5.196\,M$$
Photons with impact parameter b < bcrit are captured; those with b > bcritare deflected and escape.
Precession of Perihelion
One of the great triumphs of GR was explaining the anomalous precession of Mercury's orbit. For a nearly circular orbit at radius r₀, the angular displacement per orbit differs from 2π by:
Perihelion Precession per Orbit
$$\Delta\phi = \frac{6\pi GM}{c^2 a(1 - e^2)}$$
where a is the semi-major axis and e is the eccentricity. For Mercury:
Semi-major axis: a = 5.79 × 10¹⁰ m
Eccentricity: e = 0.2056
Orbital period: T = 87.97 days
Precession per orbit: Δφ = 0.1038"
Precession per century: 42.98"/century
Observed anomaly: 43.11 ± 0.45"/century
The derivation proceeds by substituting u = 1/r into the orbit equation and treating the relativistic correction as a perturbation:
$$\frac{d^2u}{d\phi^2} + u = \frac{M}{L^2} + 3Mu^2$$
The 3Mu² term is the GR correction to the Newtonian orbit equation
Solving perturbatively, the orbit is no longer a closed ellipse — the perihelion advances by Δφ per revolution, accumulating over time to produce the observed precession.
Complete Geodesic Equations
For completeness, the full set of geodesic equations in Schwarzschild spacetime (equatorial plane, θ = π/2) are:
$$\frac{dt}{d\tau} = \frac{E}{1 - 2M/r}$$
$$\frac{d\phi}{d\tau} = \frac{L}{r^2}$$
$$\left(\frac{dr}{d\tau}\right)^2 = E^2 - \left(1 - \frac{2M}{r}\right)\left(1 + \frac{L^2}{r^2}\right)$$
$$\frac{d^2r}{d\tau^2} = -\frac{M}{r^2} + \frac{L^2}{r^3}\left(1 - \frac{3M}{r}\right) - \frac{M}{r^2}\left(\frac{dr}{d\tau}\right)^2 \frac{1}{1 - 2M/r}$$
For radial free fall (L = 0), these simplify dramatically. A particle dropped from rest at infinity (E = 1, L = 0) falls with:
$$\frac{dr}{d\tau} = -\sqrt{\frac{2M}{r}}, \qquad \tau = -\frac{2}{3}\frac{r^{3/2}}{\sqrt{2M}} + \text{const}$$
The proper time to fall from r = 2M to r = 0 is τ = (4M/3)π — a finite, short time. An observer at infinity, however, sees the infalling object asymptotically approach r = 2M, never quite reaching it (infinite coordinate time t).