---
title: Orbits in the Schwarzschild Geometry
module: The Schwarzschild Solution
moduleNumber: 6
lessonNumber: 2
order: 602
summary: >
  The two Killing symmetries of the Schwarzschild metric give a conserved energy
  and angular momentum per unit mass, reducing geodesic motion to a
  one-dimensional problem in an effective potential. The potential carries an
  extra attractive $1/r^3$ term absent from Newton's, which caps the centrifugal
  barrier, produces an innermost stable circular orbit at $6GM/c^2$, and makes
  bound orbits precess instead of closing.
topics: [The Schwarzschild Solution]
draft: false
sources:
  - book: Hartle
    ref: "Gravity, Ch. 9 — Orbits in the Schwarzschild Geometry; Ch. 12"
  - book: Carroll
    ref: "Lecture Notes on General Relativity, §7 — Geodesics of Schwarzschild"
---

A planet or a test particle in the field of a spherical star follows a timelike
geodesic of the [Schwarzschild metric](/relativity/the-schwarzschild-solution/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 $r(\tau)$, 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 $\phi$. 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](/relativity/curved-spacetime/geodesics-and-the-geodesic-equation)
is that if $\xi^\mu$ is a Killing vector and $u^\mu = \d x^\mu/\d\tau$ is the
four-velocity, then $\xi_\mu u^\mu$ is constant along the geodesic.

Working in the equatorial plane $\theta = \pi/2$ (spherical symmetry lets any
orbit be rotated into this plane, where it stays), the time-translation Killing
vector $\xi = \partial_t$ gives the conserved **energy per unit mass**

$$
\mathcal E = -c^2\,\xi_\mu u^\mu = \left(1 - \frac{r_s}{r}\right)c^2\,\frac{\d t}{\d\tau},
$$

and the rotational Killing vector $\eta = \partial_\phi$ gives the conserved
**angular momentum per unit mass**

$$
\mathcal L = \eta_\mu u^\mu = r^2\,\frac{\d\phi}{\d\tau}.
$$

Far from the mass $\mathcal E \to c^2 + \tfrac12 v^2 + \dots$, the rest energy
plus the Newtonian kinetic and potential energy per unit mass; $\mathcal L$ is the
familiar Newtonian specific angular momentum $r^2\dot\phi$. Both are exact
constants of the full relativistic motion, not approximations.

$$
% caption: Each ignored coordinate of the metric is a Killing symmetry direction.
% Time-translation invariance conserves the energy per unit mass; rotational
% invariance conserves the angular momentum per unit mass. Both are constant along
% the orbit.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw (0,0) circle (0.5);
  \node at (0,0) {M};
  % orbit ellipse
  \draw[acc, very thick] (0,0) ++(-0.6,0) ellipse (2.4 and 1.5);
  % particle and velocity
  \fill[black] (1.5,1.18) circle (1.8pt);
  \draw[->, black!70, thick] (1.5,1.18) -- (2.05,0.65);
  \node[black, anchor=south west] at (1.55,1.22) {particle};
  % radius vector
  \draw[->, black] (0,0) -- (1.45,1.14);
  \node[black, anchor=south east] at (0.9,0.75) {r};
  % conserved labels
  \node[anchor=west] at (2.5,0.9) {E const (time symmetry)};
  \node[black, anchor=west] at (2.5,0.3) {L const (rotation symmetry)};
\end{tikzpicture}
$$

## The effective potential

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

$$
-\left(1 - \frac{r_s}{r}\right)c^2\left(\frac{\d t}{\d\tau}\right)^2
+ \left(1 - \frac{r_s}{r}\right)^{-1}\left(\frac{\d r}{\d\tau}\right)^2
+ r^2\left(\frac{\d\phi}{\d\tau}\right)^2 = -c^2 .
$$

Substituting $\d t/\d\tau = \mathcal E/[c^2(1 - r_s/r)]$ and
$\d\phi/\d\tau = \mathcal L/r^2$ and multiplying through by $(1 - r_s/r)$ removes
every reference to $t$ and $\phi$, leaving one equation for $r(\tau)$:

$$
\left(\frac{\d r}{\d\tau}\right)^2 = \frac{\mathcal E^2}{c^2} - \left(1 - \frac{r_s}{r}\right)\left(c^2 + \frac{\mathcal L^2}{r^2}\right).
$$

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

$$
\frac12\left(\frac{\d r}{\d\tau}\right)^2 + V_{\rm eff}(r) = \frac{\mathcal E^2 - c^4}{2c^2},
\qquad
V_{\rm eff}(r) = -\frac{GM}{r} + \frac{\mathcal L^2}{2r^2} - \frac{GM\,\mathcal L^2}{c^2 r^3},
$$

using $r_s = 2GM/c^2$. The right-hand side is a constant fixed by $\mathcal E$,
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 $V_{\rm eff}$, 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.
- **$+\mathcal L^2/2r^2$** — the centrifugal barrier from angular momentum,
  unchanged.
- **$-GM\mathcal L^2/(c^2 r^3)$** — 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 $\mathcal L \ne 0$ 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.

$$
% caption: The Schwarzschild effective potential (solid) against the Newtonian one
% (dotted) at the same angular momentum. Newton's centrifugal barrier rises without
% bound; the relativistic barrier has a finite maximum and plunges to minus
% infinity at small r, so a sufficiently energetic particle is captured.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (-0.2,0) -- (6.0,0) node[right] {r};
  \draw[->, black] (0,-1.5) -- (0,2.4) node[above] {V(r)};
  % relativistic curve (solid, acc)
  \draw[acc, very thick] plot coordinates
    {(0,-0.93)(0.13,-0.14)(0.26,0)(0.52,-0.20)(0.78,-0.46)(1.30,-0.78)(1.82,-0.90)(2.34,-0.93)(3.38,-0.88)(4.42,-0.80)(5.46,-0.72)};
  % plunge arrow at small r
  \draw[->] (0,-0.93) -- (-0.15,-1.35);
  \node[anchor=north] at (0.9,-1.15) {capture};
  \node[acc, anchor=west] at (3.0,-0.42) {Schwarzschild};
  % Newtonian curve (dotted, black!62), starting at r=6
  \draw[black, very thick, densely dotted] plot coordinates
    {(0.78,1.39)(1.30,0)(1.82,-0.5)(2.34,-0.69)(3.38,-0.78)(4.42,-0.75)(5.46,-0.69)};
  \draw[->, black] (0.78,1.39) -- (0.62,2.1);
  \node[black, anchor=west] at (0.7,2.0) {Newton barrier};
  % peak marker
  \fill (0.26,0) circle (1.6pt);
\end{tikzpicture}
$$

## Circular orbits and the ISCO

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

$$
\frac{GM}{r^2} - \frac{\mathcal L^2}{r^3} + \frac{3GM\,\mathcal L^2}{c^2 r^4} = 0
\quad\Longleftrightarrow\quad
GM\,r^2 - \mathcal L^2\,r + \frac{3GM\,\mathcal L^2}{c^2} = 0 .
$$

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

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

The larger root $r_+$ is a minimum of $V_{\rm eff}$, 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,
$\mathcal L^2 \ge 12\,G^2M^2/c^2$. At equality the roots merge: the stable and
unstable orbits coincide and the last stable circular orbit sits at

$$
\mathcal L^2 = \frac{12\,G^2 M^2}{c^2}
\quad\Longrightarrow\quad
r_{\rm ISCO} = \frac{\mathcal L^2}{2GM} = \frac{6GM}{c^2} = 3\,r_s .
$$

> **Definition (Innermost stable circular orbit).** For angular momentum below
> $\mathcal L_{\min} = 2\sqrt{3}\,GM/c$ no stable circular orbit exists; the
> smallest possible stable circular orbit, reached at that angular momentum, lies
> at $r_{\rm ISCO} = 6GM/c^2$. Inside this radius all circular orbits are
> unstable, and matter spirals in.

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 $\mathcal E/c^2 = \sqrt{8/9} \approx 0.943$, so about $5.7\%$ of
the rest energy is radiated on the way in — far above the fraction released in
nuclear fusion.

$$
% caption: As the angular momentum decreases toward the marginal value, the well
% and barrier of the effective potential merge. At L = 2 sqrt-3 GM/c the minimum
% and maximum coincide at r = 6 GM/c-squared, the innermost stable circular orbit.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (-0.2,0) -- (4.2,0) node[right] {r};
  \draw[->, black] (0,-3.1) -- (0,0.6) node[above] {V(r)};
  % high-L curve: stable min + unstable max
  \draw[acc, very thick] plot coordinates
    {(0,-0.93)(0.13,-0.14)(0.26,0)(0.52,-0.20)(0.78,-0.46)(1.30,-0.78)(1.82,-0.90)(2.34,-0.93)(3.38,-0.88)};
  \fill (0.26,0) circle (1.6pt);
  \node[anchor=south west] at (0.34,0.06) {unstable};
  \fill (2.34,-0.93) circle (1.6pt);
  \node[anchor=west] at (2.5,-0.82) {stable};
  \node[anchor=west] at (1.35,-0.5) {large L};
  % marginal-L curve: inflection at r=6 (x=0.78)
  \draw[black, very thick, densely dotted] plot coordinates
    {(0,-2.78)(0.13,-1.90)(0.26,-1.56)(0.52,-1.4)(0.78,-1.39)(1.04,-1.39)(1.30,-1.37)(1.82,-1.30)(2.34,-1.21)(3.38,-1.05)};
  \fill[black] (0.78,-1.39) circle (1.8pt);
  \node[black, anchor=north] at (0.85,-1.62) {ISCO};
  \node[black, anchor=north] at (2.6,-1.5) {marginal L};
\end{tikzpicture}
$$

## Radial infall in proper and coordinate time

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

$$
\left(\frac{\d r}{\d\tau}\right)^2 = \frac{\mathcal E^2}{c^2} - c^2\left(1 - \frac{r_s}{r}\right) = \frac{c^2 r_s}{r},
$$

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:

$$
\tau = \frac{2}{3\,c\sqrt{r_s}}\left(R^{3/2} - r^{3/2}\right).
$$

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 $\d t/\d\tau = \mathcal E/[c^2(1 - r_s/r)] = 1/(1 - r_s/r)$,

$$
\frac{\d t}{\d r} = \frac{\d t/\d\tau}{\d r/\d\tau} = -\frac{1}{c}\sqrt{\frac{r}{r_s}}\,\frac{1}{1 - r_s/r},
$$

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](/relativity/black-holes/horizons-and-coordinate-singularities)
resolves with better-behaved coordinates.

$$
% caption: A body falling from rest reaches the horizon r-s in finite proper time
% tau (solid), crossing smoothly. In coordinate time t (dotted) it asymptotically
% approaches r-s and never crosses, appearing to freeze to a distant observer.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (-0.2,0) -- (6.0,0) node[right] {time};
  \draw[->, black] (0,-0.2) -- (0,3.6) node[above] {r};
  % horizon line
  \draw[black, dashed] (0,0.6) -- (5.8,0.6);
  \node[black, anchor=west] at (5.0,0.42) {r $=$ r$_s$};
  % start radius
  \node[black, anchor=east] at (-0.05,3.1) {R};
  % proper-time curve: crosses horizon in finite time (reaches r=0)
  \draw[acc, very thick] plot coordinates
    {(0,3.1)(0.5,3.0)(1.2,2.7)(2.0,2.1)(2.7,1.4)(3.1,0.6)(3.4,0.0)};
  \node[acc, anchor=west] at (1.2,1.4) {proper time};
  % coordinate-time curve: asymptotes to horizon
  \draw[black, very thick, densely dotted] plot coordinates
    {(0,3.1)(0.5,2.95)(1.2,2.6)(2.2,2.0)(3.2,1.4)(4.2,0.95)(5.2,0.72)(5.8,0.65)};
  \node[black, anchor=west] at (4.35,1.25) {coordinate time};
\end{tikzpicture}
$$

## 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 $\tau$ to $\phi$ with
$u = 1/r$ and $\d r/\d\tau = -\mathcal L\,\d u/\d\phi$, the radial equation
differentiates into the **orbit equation**

$$
\frac{\d^2 u}{\d\phi^2} + u = \frac{GM}{\mathcal L^2} + \frac{3GM}{c^2}\,u^2 .
$$

Without the final term this is the Newtonian orbit equation, whose bound solutions
are closed ellipses $u = (GM/\mathcal L^2)(1 + e\cos\phi)$ that return to the same
perihelion every $2\pi$. The term $3GM u^2/c^2$ is the relativistic correction. It
is small in the solar system — its ratio to the leading term is
$3\mathcal L^2/(c^2 r^2) \sim 3(v/c)^2 \sim 10^{-8}$ 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.

$$
% caption: The relativistic 1/r-cubed term makes a bound orbit precess. The
% ellipse advances by a small angle each revolution, so the perihelion rotates and
% the orbit fills an annulus rather than closing on itself.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw (0,0) circle (0.35);
  \node[font=\scriptsize] at (0,0) {M};
  % first ellipse
  \draw[acc, very thick, rotate=0] (0.9,0) ellipse (2.3 and 1.45);
  % second ellipse, precessed
  \draw[black, very thick, dashed, rotate=42] (0.9,0) ellipse (2.3 and 1.45);
  % perihelion markers (closest points ~ left ends)
  \fill (-1.4,0) circle (1.7pt);
  \fill[black] ({-1.4*cos(42)},{-1.4*sin(42)}) circle (1.7pt);
  % advance arc
  \draw[->, black] (-1.75,0) arc (180:138:1.75);
  \node[black, anchor=east] at (-1.8,0.6) {perihelion advance};
\end{tikzpicture}
$$

The precession rate integrates to a per-orbit advance of $6\pi GM/(c^2 a(1-e^2))$
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](/relativity/tests-of-general-relativity/perihelion-precession-of-mercury)
carries the calculation to the observed $43''$ 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](/relativity/the-schwarzschild-solution/light-bending-and-null-geodesics)
takes up.

[^hartle-orbits]: **Hartle**, _Gravity_, Ch. 9 — conserved energy and angular momentum from the Killing vectors, the effective potential for Schwarzschild orbits, circular orbits and the innermost stable circular orbit at $6GM/c^2$, and radial infall in proper versus coordinate time; Ch. 12 collects the black-hole applications.

[^carroll-geo]: **Carroll**, _Lecture Notes on General Relativity_, §7 — geodesics of the Schwarzschild metric, the effective-potential formulation, the orbit equation with the $3GM u^2/c^2$ term, and perihelion precession.