Geodesics Near Black Holes
Free particles and light rays follow geodesicsāthe straightest possible paths in curved spacetime. We derive the geodesic equations in Schwarzschild geometry and analyze orbital motion, including the innermost stable circular orbit (ISCO) and radial infall.
1. The Geodesic Equation
The geodesic equation describes the motion of a free particle with four-velocity u^Ī¼ = dx^Ī¼/dĻ:
$\frac{du^\mu}{d\tau} + \Gamma^\mu_{\;\nu\rho} u^\nu u^\rho = 0$
For the Schwarzschild metric, spherical symmetry allows us to restrict motion to the equatorial plane (Īø = Ļ/2) without loss of generality. The metric in this plane is:
$ds^2 = -\left(1 - \frac{r_s}{r}\right) c^2 dt^2 + \left(1 - \frac{r_s}{r}\right)^{-1} dr^2 + r^2 d\phi^2$
2. Conserved Quantities from Killing Vectors
The Schwarzschild spacetime has two Killing vectors corresponding to symmetries:
- Time translation symmetry (ā/āt): conserved energy per unit mass E
- Rotational symmetry (ā/āĻ): conserved angular momentum per unit mass L
Energy (per unit mass)
$E = \left(1 - \frac{r_s}{r}\right) c^2 \frac{dt}{d\tau}$
Angular Momentum (per unit mass)
$L = r^2 \frac{d\phi}{d\tau}$
These are constants of motion along any geodesic, drastically simplifying the equations of motion.
3. Effective Potential for Radial Motion
The normalization condition for massive particles is u^Ī¼ u_Ī¼ = -cĀ². Substituting our conserved quantities and rearranging gives:
$\left(\frac{dr}{d\tau}\right)^2 + V_{\text{eff}}(r) = E^2$
where the effective potential is:
$V_{\text{eff}}(r) = c^2 \left(1 - \frac{r_s}{r}\right) \left(1 + \frac{L^2}{r^2 c^2}\right)$
This reduces the problem to 1D motion in an effective potential! The structure has three terms:
- cĀ²: rest mass energy
- -r_s cĀ²/r: gravitational attraction (Newtonian-like)
- LĀ²/rĀ² (1 - r_s/r): centrifugal barrier with GR correction
Critical Points
Circular orbits occur at extrema of V_eff. Taking dV_eff/dr = 0:
$\frac{r_s}{r^2} \left(1 + \frac{L^2}{r^2 c^2}\right) = \frac{2L^2}{r^3 c^2} \left(1 - \frac{r_s}{r}\right)$
Simplifying yields:
$r^2 - 3r_s r + \frac{2L^2}{c^2} = 0$
Solving for r gives the radii of circular orbits. Stability requires dĀ²V_eff/drĀ² > 0.
4. Innermost Stable Circular Orbit (ISCO)
The ISCO is the smallest radius at which a stable circular orbit exists. For r < r_ISCO, orbits are unstableāsmall perturbations cause the particle to spiral inward.
The ISCO occurs where the maximum and minimum of V_eff merge (inflection point). This requires:
$\frac{dV_{\text{eff}}}{dr} = 0 \quad \text{and} \quad \frac{d^2 V_{\text{eff}}}{dr^2} = 0$
Solving these conditions simultaneously yields:
$r_{\text{ISCO}} = 6GM/c^2 = 3r_s$
At the ISCO, the angular momentum and energy are:
$L_{\text{ISCO}} = 2\sqrt{3} \frac{GM}{c}, \quad E_{\text{ISCO}} = \frac{2\sqrt{2}}{3} mc^2 \approx 0.9428 mc^2$
Binding Energy
The binding energy at the ISCO is:
$\frac{E_{\text{ISCO}} - mc^2}{mc^2} = 1 - \frac{2\sqrt{2}}{3} \approx 5.72\%$
About 6% of the rest mass can be radiated as a particle spirals from infinity to the ISCOāfar more efficient than nuclear fusion (~0.7%)! This powers accretion disks around black holes.
5. Radial Infall (L = 0)
For purely radial motion (zero angular momentum), the effective potential simplifies:
$V_{\text{eff}}(r) = c^2 \left(1 - \frac{r_s}{r}\right)$
The radial equation becomes:
$\left(\frac{dr}{d\tau}\right)^2 = E^2 - c^2 \left(1 - \frac{r_s}{r}\right)$
Infall from Rest at Infinity
For a particle dropped from rest at infinity, E = cĀ². Then:
$\frac{dr}{d\tau} = -c \sqrt{\frac{r_s}{r}}$
(Negative sign for infall). Integrating gives proper time to fall from rā to r:
$\Delta\tau = \frac{2}{3c} \left(\sqrt{r_0^3/r_s} - \sqrt{r^3/r_s}\right)$
The proper time to reach r = 0 from r = r_s is:
$\Delta\tau = \frac{2r_s}{3c} = \frac{4GM}{3c^3} \approx 10 \text{ Ī¼s} \left(\frac{M}{M_\odot}\right)$
An infalling observer crosses the horizon and reaches the singularity in finite proper time!
6. Gravitational Time Dilation and Redshift
A static observer at radius r has proper time related to coordinate time by:
$d\tau = \sqrt{1 - \frac{r_s}{r}} \, dt$
Time runs slower closer to the black hole. For a photon emitted at rā and received at rā:
$\frac{\nu_2}{\nu_1} = \sqrt{\frac{1 - r_s/r_2}{1 - r_s/r_1}}$
As rā ā r_s, the frequency at infinity (rā ā ā) approaches zeroāinfinite gravitational redshift.
Schwarzschild Coordinate Time to Horizon
For an observer at infinity watching a particle fall, the coordinate time diverges as r ā r_s:
$t \to \infty \text{ as } r \to r_s$
The distant observer never sees the particle cross the horizonāit appears frozen, asymptotically approaching r_s while becoming infinitely redshifted and dimmed.
Key Takeaways
- Killing vectors give conserved energy E and angular momentum L along geodesics
- Radial motion reduces to 1D motion in effective potential V_eff(r)
- Innermost stable circular orbit (ISCO) is at r = 3r_s = 6GM/cĀ²
- ~6% of rest mass can be radiated by accretion from infinity to ISCO
- Radial infall from rest: finite proper time (~10 Ī¼s for solar mass) to singularity
- Infinite gravitational redshift at horizon in Schwarzschild coordinates
- Distant observers never see horizon crossing; infalling observers cross smoothly