Schwarzschild orbit

Note

I’m not sure “Schwarzschild orbit” is a real thing, that’s just what I’m calling a body orbiting $r=0$ in a Schwarzschild spacetime.

The Schwarzschild metric has some nice conserved quantities. In particular:

\begin{align*} \hat E &= \left(1-\frac{2GM}{r}\right) \frac{\dt}{\d\tau}, \\ \text{and } \hat L_z &= r^2 \frac{\d\phi}{\d\tau} \end{align*}

We will use the invariant $\fv u \cdot \fv u = -1$ and the above conserved quantities to explore orbits in the $\theta=\pi/2$ plane.

\begin{align*} -1 = - (1-2GM/r) \dot t^2 + \frac{\dot r^2}{1-2GM/r} + r^2 \dot \phi^2. \end{align*}

We rewrite the above as

\begin{align*} \dot r^2 &= \hat E^2 - V_\eff(r) \\ V_\eff(r) &= \left(1-\frac{2GM}{r}\right)\left(1 + \frac{\hat L_z^2}{r^2}\right). \end{align*}

Since $\dot r$ must be real, we require $\hat E > V_\eff$ .

Plotting $V_\eff$ , we see the physically valid regions for some $\hat E$ .

When $\hat E$ is greater than the trough of $V_\eff$ , an eccentric orbit is possible. When it is less, the only solution is a body crashing into $r=0$ . Various energy levels lead to various stable or unstable orbits.

The effective potential depends on the angular momentum. As the angular momentum becomes small, stable orbits are no longer possible.

Light

Light can orbit in a Schwarzschild spacetime, too. The equations are a little different.

Consider a photon moving in from $r=\infty$ towards the center of the Schwarzschild spacetime, separated by a transverse distance $b$ .

We write the Lagrangian in terms of $\lambda$ as we do for light in strong gravity (assuming $\theta = \pi/2$ )

\begin{align*} L &= \frac12 \left[-(1-2GM/r) \dot t^2 + (1-2GM/r)^{-1} \dot r^2 + r^2 \dot\phi^2 \right], \\ \end{align*}

We see that there are two conserved quantities:

\begin{align*} \hat E &= -\left(1 - \frac{2GM}{r}\right) \dot t = g_{tt} p^t = p_t \\ \hat L_z &= r^2 \dot \phi = b \hat E \end{align*}

From $\fv p \cdot \fv p = 0$ we find

\begin{align*} \dot r^2 &= \hat E^2 - \frac{\hat L_z^2}{r^2} \left(1-\frac{2GM}{r}\right) \\ &= \frac{\hat L_z^2}{b^2} - \frac{\hat L_z^2}{r^2} \left(1-\frac{2GM}{r}\right) \\ \hat L_z^2 \dot r^2 &= \frac{1}{b^2} - \frac{1}{r^2} \left(1-\frac{2GM}{r}\right) \\ &=: \frac{1}{b^2} - V_\text{light}(r). \end{align*}

Again, we get certain valid radii given some $b$ and the light potential.

There are three regimes light can be in:

$b>3\sqrt3GM$ : light comes in from $r=\infty$ , bends as it passes

around the center, and then goes out to $\infty$ again.

$v<3\sqrt3GM$ : light comes in from $r=\infty$ and crashes into

$r=0$ . Light disappears forever.

$v=3\sqrt3GM$ : light comes in from $r=\infty$ , then reaches $r=3GM$

and orbits there forever! This is an unstable orbit, though, so the light is likely to crash in or fly away given any perturbation.