# Effective potential
In general relativity, the **effective potential** is a function of the radial coordinate that condenses the orbital motion of a test particle in the Schwarzschild geometry into a one-dimensional energy equation. The two Killing symmetries of the metric, time translation and rotation, give conserved energy and angular momentum per unit mass, and the normalization of the four-velocity supplies a third relation; together they reduce the four geodesic (A freely falling particle's straightest possible path through curved spacetime) equations to a single first-order equation for r(τ), formally identical to the Newtonian central-force problem.[1](https://notes.amittai.studio/relativity/the-schwarzschild-solution/geodesics-and-orbits-in-schwarzschild)
| Key fact | Detail |
|---|---|
| Form (timelike geodesic) | \( V_{\mathrm{eff}}(r) = -\dfrac{GM}{r} + \dfrac{L^2}{2r^2} - \dfrac{GM L^2}{c^2 r^3} \), the Newtonian Kepler potential plus one extra attractive term[1](https://notes.amittai.studio/relativity/the-schwarzschild-solution/geodesics-and-orbits-in-schwarzschild) |
| Conserved quantities | Specific energy \( \mathcal{E} = (1 - r_s/r)c^2\,\mathrm{d}t/\mathrm{d}\tau \) and specific angular momentum \( L = r^2\,\mathrm{d}\varphi/\mathrm{d}\tau \), from the two Killing vectors[2](https://www.nbow.io/en/products/ai-powered-research/treatise-of-physics/chapter/05-general-relativity-cosmology/04-schwarzschild-black-holes.html) |
| Structure vs. angular momentum | Both a maximum and a minimum for \( L > \sqrt{12}\,GM/c \), an inflection point at equality, strictly monotonic below[3](https://cosmo.nyu.edu/yacine/teaching/GR_2019/lectures/lecture20.pdf) |
| ISCO | \( r_{\mathrm{ISCO}} = 6GM/c^2 = 3r_s \), reached at the minimum angular momentum \( L_{\min} = 2\sqrt{3}\,GM/c \)[1](https://notes.amittai.studio/relativity/the-schwarzschild-solution/geodesics-and-orbits-in-schwarzschild) |
| Circular-orbit angular momentum | \( l_o = \sqrt{M r_o^2/(r_o - 3M)} \) in geometrized units \( G = c = 1 \); circular orbits require \( r_o > 3M \)[4](https://link.springer.com/article/10.1140/epjc/s10052-021-09623-6) |
| Accretion efficiency | Orbital energy at the Schwarzschild ISCO is \( E/c^2 = 8/9 \approx 0.943 \), so about 5.7% of rest energy is radiated as matter spirals in[1](https://notes.amittai.studio/relativity/the-schwarzschild-solution/geodesics-and-orbits-in-schwarzschild) |
| Perihelion advance | \( 6\pi GM/(c^2 a(1-e^2)) \) per orbit for a nearly Newtonian ellipse of semimajor axis a and eccentricity e; 43″ per century for Mercury[1](https://notes.amittai.studio/relativity/the-schwarzschild-solution/geodesics-and-orbits-in-schwarzschild) |
## Derivation from the conserved quantities
The [Schwarzschild metric](https://www.edgepedia.org/schwarzschild-metric) components depend on neither t nor φ. For every coordinate that has been dropped, there is a Killing vector—a direction of symmetry in which the geometry stays the same—and from each such Killing vector one can derive a conserved quantity valid along any geodesic: given a Killing vector \( \xi^\mu \) and the four-velocity \( u^\mu = \mathrm{d}x^\mu/\mathrm{d}\tau \), the product \( \xi^\mu u_\mu \) remains constant as the geodesic is followed.[1](https://notes.amittai.studio/relativity/the-schwarzschild-solution/geodesics-and-orbits-in-schwarzschild) Spherical symmetry confines each orbit to a plane; with initial conditions \( \theta = \pi/2 \) and \( \mathrm{d}\theta/\mathrm{d}\tau = 0 \), the θ geodesic equation gives \( \mathrm{d}^2\theta/\mathrm{d}\tau^2 = 0 \), so the motion stays equatorial.[2](https://www.nbow.io/en/products/ai-powered-research/treatise-of-physics/chapter/05-general-relativity-cosmology/04-schwarzschild-black-holes.html)
Far from the mass, \( \mathcal{E} \to c^2 + \tfrac{1}{2}v^2 + \ldots \), the rest energy plus the Newtonian kinetic and potential energy per unit mass, and L is the familiar Newtonian specific angular momentum \( r^2\dot{\varphi} \). Both are exact constants of the full relativistic motion, not approximations.[1](https://notes.amittai.studio/relativity/the-schwarzschild-solution/geodesics-and-orbits-in-schwarzschild)
When the conserved quantities are inserted into the normalization condition \( g_{\mu\nu}u^\mu u^\nu = -c^2 \), and both sides are then scaled by \( (1 - r_s/r) \), the coordinates t and φ drop out entirely, yielding a single equation that governs r as a function of τ:[1](https://notes.amittai.studio/relativity/the-schwarzschild-solution/geodesics-and-orbits-in-schwarzschild)
\[ \frac{1}{2}\left(\frac{\mathrm{d}r}{\mathrm{d}\tau}\right)^2 + V_{\mathrm{eff}}(r) = \frac{c^2\mathcal{E}^2 - c^4}{2c^2}, \qquad V_{\mathrm{eff}}(r) = -\frac{GM}{r} + \frac{L^2}{2r^2} - \frac{GM L^2}{c^2 r^3}, \]
using \( r_s = 2GM/c^2 \). The right-hand side is a constant fixed by the energy, playing the role of the total energy of a fictitious unit-mass particle moving in one dimension, and the orbit is read off by the same turning-point analysis as any central-force problem.[1](https://notes.amittai.studio/relativity/the-schwarzschild-solution/geodesics-and-orbits-in-schwarzschild) An equivalent convention writes the radial equation as \( (\mathrm{d}r/\mathrm{d}\tau)^2 = (E/m)^2 - (1 - 2M/r)(1 + L^2/(m^2 r^2)) \), with the square root of the second term plotted as an effective potential energy.[5](https://phys.libretexts.org/Courses/Skidmore_College/Introduction_to_General_Relativity/03%3A_Schwarzschild_Orbits/3.03%3A_Effective_Potential)
## The extra term and capture
The three terms name themselves against the Newtonian case: \( -GM/r \) is the Newtonian gravitational attraction, \( +L^2/(2r^2) \) is the centrifugal barrier, and \( -GM L^2/(c^2 r^3) \) is the general-relativistic correction, absent from Newton's potential. This term pulls inward, and since it decays only as \( 1/r^3 \), it becomes the leading contribution near small r, swamping the centrifugal barrier in that region.[1](https://notes.amittai.studio/relativity/the-schwarzschild-solution/geodesics-and-orbits-in-schwarzschild)
**Capture of orbiting matter.** In Newtonian gravity the \( 1/r^2 \) centrifugal term wins as \( r \to 0 \) and the potential rises without bound, so any particle with \( L \neq 0 \) is thrown back out. In Schwarzschild the \( 1/r^3 \) term eventually wins, so the barrier has a finite maximum and turns over toward \( -\infty \); 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.[1](https://notes.amittai.studio/relativity/the-schwarzschild-solution/geodesics-and-orbits-in-schwarzschild) A particle that does not clear the barrier may still have enough energy to fall to the [Schwarzschild radius](https://www.edgepedia.org/schwarzschild-radius); as seen from infinity it takes an infinite coordinate time to reach \( r_s \), but in the particle's own frame it quickly reaches \( r = 0 \).[6](https://hepweb.ucsd.edu/ph110b/110b_notes/node80.html)
## Circular orbits and the ISCO
Circular orbits sit at extrema of \( V_{\mathrm{eff}} \), where \( V_{\mathrm{eff}}'(r) = 0 \). For fixed L this condition is a quadratic in r with two roots,[1](https://notes.amittai.studio/relativity/the-schwarzschild-solution/geodesics-and-orbits-in-schwarzschild)
\[ r_\pm = \frac{L^2}{2GM} \pm \frac{1}{2}\sqrt{\frac{L^4}{G^2M^2} - \frac{12 L^2}{c^2}}.[7](https://phys.libretexts.org/Courses/Skidmore_College/Introduction_to_General_Relativity/03%253A_Schwarzschild_Orbits/3.03%253A_Effective_Potential) \]
At the greater root, \( r_+ \), the potential has a minimum corresponding to a stable circular orbit; at the lesser root, \( r_- \), it reaches a maximum, giving an unstable circular orbit where even a slight nudge inward makes the body fall in. In the Newtonian case just one stable minimum exists; the unstable inner orbit, resting atop the barrier's peak, arises only from relativity.[1](https://notes.amittai.studio/relativity/the-schwarzschild-solution/geodesics-and-orbits-in-schwarzschild) In geometrized units the two roots exist only for \( L > \sqrt{12}\,M \), merge at an inflection point at equality, and disappear for smaller L, where the potential is strictly monotonic.[3](https://cosmo.nyu.edu/yacine/teaching/GR_2019/lectures/lecture20.pdf)
At the merging point the stable and unstable orbits coincide, giving the innermost stable circular orbit (ISCO), the orbit of minimum angular momentum among all circular orbits:[1](https://notes.amittai.studio/relativity/the-schwarzschild-solution/geodesics-and-orbits-in-schwarzschild) • [4](https://link.springer.com/article/10.1140/epjc/s10052-021-09623-6)
\[ r_{\mathrm{ISCO}} = \frac{6GM}{c^2} = 3r_s, \qquad L_{\min} = \frac{2\sqrt{3}\,GM}{c}. \]
For angular momentum below \( L_{\min} \) no stable circular orbit exists; inside this radius all circular orbits are unstable and matter spirals in.[1](https://notes.amittai.studio/relativity/the-schwarzschild-solution/geodesics-and-orbits-in-schwarzschild) The ISCO 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.[1](https://notes.amittai.studio/relativity/the-schwarzschild-solution/geodesics-and-orbits-in-schwarzschild)
The effective-potential method and the method of varying the conserved orbital angular momentum are equivalent in general static and stationary spacetimes, and the angular-momentum method extends to dynamical spacetimes where no conserved orbital energy exists.[4](https://link.springer.com/article/10.1140/epjc/s10052-021-09623-6)
## 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 \( u = 1/r \), the radial equation becomes the orbit equation[1](https://notes.amittai.studio/relativity/the-schwarzschild-solution/geodesics-and-orbits-in-schwarzschild)
\[ \frac{\mathrm{d}^2 u}{\mathrm{d}\varphi^2} + u = \frac{GM}{L^2} + \frac{3GM}{c^2}u^2. \]
Without the final term this is the Newtonian orbit equation, whose bound solutions are closed ellipses. For Mercury the correction is tiny compared with the leading term, at a ratio of \( 3L^2/(c^2 r^2) \sim 3(v/c)^2 \sim 10^{-8} \), but over a full orbit it does not average out: each revolution swings the perihelion forward slightly, so the path draws a rosette that turns gradually and never closes.[1](https://notes.amittai.studio/relativity/the-schwarzschild-solution/geodesics-and-orbits-in-schwarzschild) The per-orbit advance integrates to \( 6\pi GM/(c^2 a(1-e^2)) \), which applied to Mercury gives the residual precession that Newtonian perturbation theory could not account for, 43″ per century.[1](https://notes.amittai.studio/relativity/the-schwarzschild-solution/geodesics-and-orbits-in-schwarzschild)
## Radial infall
Setting \( L = 0 \) for a particle falling straight in, released from rest at infinity so \( \mathcal{E} = c^2 \), the radial equation collapses to \( (\mathrm{d}r/\mathrm{d}\tau)^2 = c^2 r_s/r \), identical in form to Newtonian free fall. Starting the integration at radius R and working toward smaller radii yields a proper time that remains finite for arrival at every radius, the horizon and the center included. Measured instead by coordinate time, the ratio \( \mathrm{d}t/\mathrm{d}\tau = 1/(1 - r_s/r) \) blows up as \( r \) approaches \( r_s \); from the perspective of a distant observer tracking the far-away time t, the falling object appears to decelerate, stall just outside \( r_s \), and take forever to arrive there. Both descriptions are correct; they refer to different clocks, and the divergence is a property of the Schwarzschild coordinate t at \( r_s \), not of the geometry.[1](https://notes.amittai.studio/relativity/the-schwarzschild-solution/geodesics-and-orbits-in-schwarzschild)
## References
1. [Orbits in the Schwarzschild Geometry, study notes](https://notes.amittai.studio/relativity/the-schwarzschild-solution/geodesics-and-orbits-in-schwarzschild)
2. [Schwarzschild Geometry and Black Holes, Treatise of Physics](https://www.nbow.io/en/products/ai-powered-research/treatise-of-physics/chapter/05-general-relativity-cosmology/04-schwarzschild-black-holes.html)
3. [General Relativity Fall 2019, Lecture 20: Geodesics of Schwarzschild, Yacine Ali-Haïmoud, NYU](https://cosmo.nyu.edu/yacine/teaching/GR_2019/lectures/lecture20.pdf)
4. [The evolutions of the innermost stable circular orbits in dynamical spacetimes, European Physical Journal C (2021)](https://link.springer.com/article/10.1140/epjc/s10052-021-09623-6)
5. [3.3: Effective Potential, Physics LibreTexts](https://phys.libretexts.org/Courses/Skidmore_College/Introduction_to_General_Relativity/03%3A_Schwarzschild_Orbits/3.03%3A_Effective_Potential)
6. [Orbits in the Schwarzschild Metric, UCSD PH110B notes](https://hepweb.ucsd.edu/ph110b/110b_notes/node80.html)
7. [3.3: Effective Potential](https://phys.libretexts.org/Courses/Skidmore_College/Introduction_to_General_Relativity/03%253A_Schwarzschild_Orbits/3.03%253A_Effective_Potential)
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*Topic: Encyclopedia › Physical world and mathematics › Physics › Relativity and gravitation › General relativity and curved spacetime › Foundations and field equations › Geodesic motion › Timelike geodesic orbital dynamics*
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