Learn Orbital Precession in the Schwarzschild and Kerr Metrics

The Schwarzschild Metric

A Lagrangian that can be used to describe geodesics is $F = g_{\mu\nu}v^\mu v^\mu$, where $v^\mu = dx^\mu/ds$ is the four-velocity. In the equatorial plane of the Schwarzschild metric this is
$$F = (1 – 2m/r)^{-1} (dr/ds)^2 + r^2(d\phi/ds)^2 – (1 – 2m/r)(dt/ds)^2$$
The canonical momenta are $p_\mu = \partial F/\partial v^\nu = g_{\mu\nu}v^\nu$. Since the Lagrangian is independent of $t$ and $\phi$, we have two immediate first integrals
$$\begin{eqnarray*}L & = & g_{\phi \nu}u^\nu = r^2(d\phi/ds)\\ \Gamma & = & g_{t \nu} u^\nu = (1 – 2m/r)(dt/ds)\end{eqnarray*}$$
along with the constraint $F = -1$.

Physically, $L$ represents the angular momentum per unit mass and has dimensions of distance. $\Gamma$
represents the energy per unit mass and is dimensionless. In other words, $\Gamma$ is just the usual relativistic gamma factor. $F$ is the norm of the four-velocity.

Combining all three, we obtain the radial equation of motion:
$$(dr/ds)^2 = \Gamma^2 – (1 – 2m/r) – (1 – 2m/r)(L^2/r^2) \equiv V(r)$$

Circular Orbits

For a circular orbit, $V(r) = V'(r) = 0$. These condiitions can be used to determine the constants of the motion $L^2$ and $\Gamma^2$ in terms of the orbital radius:
$$\begin{eqnarray*}V(r) & = & 0 \Rightarrow \Gamma^2 = 1 – 2m/r + L^2/r^2 – 2mL^2/r^3\\ V'(r) & = & 0 \Rightarrow 0 = -2m/r^2 + 2L^2/r^3 – 6mL^2/r^4\end{eqnarray*}$$
from which we find
\begin{eqnarray*}L^2 & = & \frac{mr}{1 – 3m/r}\\
\Gamma^2 & = & \frac{(1 – 2m/r)^2}{1 – 3m/r}\end{eqnarray*}
Note that both $L^2$ and $\Gamma^2$ go to infinity at $r = 3m$, which is the radius at which a photon can undergo a circular orbit. Inside $r = 3m$ they both become negative, and circular orbits of any kind are not possible.

Radial Oscillations

Small radial oscillations are governed by the second derivative $V”(r)$.

For Schwarzschild,
$$V”(r) = 4m/r^3 – 6L^2/r^4 + 24mL^2/r^5 = – \frac{2m}{r^3} \frac{1-6m/r}{1-3m/r}$$
To obtain the equation of the orbit, let $(d\phi/ds)^2 = L^2/r^4 \equiv D(r)$. Then
$$(dr/d\phi)^2 = \frac{(dr/ds)^2}{(d\phi/ds)^2} = \frac{V(r)}{D(r)}$$
Small oscillations about a circular orbit will be governed by:
$$(d\delta r/d\phi)^2 = \frac{(1/2) V”(r)}{D(r)} (\delta r)^2$$
A radial oscillation takes the form $\delta r \sim \sin(\omega \phi)$, so the above implies
$$ \omega ^2 = – \frac{(1/2)V”(r)}{D(r)} = 1 – 6m/r$$
This shows that all circular orbits inside $r = 6m$ are unstable, since $\omega^2 < 0$ there. Outside $r = 6m$,
oscillations are stable. For a complete orbit, in which $\phi$ increases by $2\pi$, the phase of a radial motion will increase by $2\pi \omega$. The difference between these two quantities represents the advance of the perihelion,
namely $6\pi m/r$ per revolution.

The Kerr Metric

Analysis of the perihelion advance in the case of the Kerr metric is quite similar, although a bit more complicated.
The Kerr metric in Boyer-Lindquist coordinates, restricted to the equatorial plane is:
$$ds^2 = (1 -2m/r) dt^2 – r^2/\Delta dr^2 – (r^2 + a^2 + 2ma^2/r) d\phi^2 + (4ma/r) d\phi dt$$
where
$$\Delta = r^2 – 2mr + a^2$$
The first integrals are:
\begin{eqnarray*}L & = & g_{\phi \mu}v^\mu = (2ma/r) (dt/ds) – (r^2 + a^2 + 2ma^2/r) (d\phi/ds)\\
\Gamma & = & g_{t \mu} v^\mu = (1 – 2m/r) (dt/ds) + (2ma/r) (d\phi/ds)\end{eqnarray*}
or conversely
\begin{eqnarray*}d\phi/ds & = & g^{\phi t} \Gamma + g^{\phi \phi}L = (1/r\Delta)(2ma\Gamma – (r – 2m)L)\\
dt/ds & = & g^{t t}\Gamma + g^{t \phi}L = (1/r\Delta)((r^2 + a^2 + 2ma^2/r)\Gamma + 2ma L)\end{eqnarray*}
The Lagrangian for geodesics:
$$F = g^{\mu\nu}v_\mu v_\nu = g^{tt}\Gamma^2 + 2g^{t \phi} \Gamma L + g^{\phi\phi} L^2 + g_{rr}(dr/ds)^2$$
Combining these:
$$1 = (1/\Delta)(r^2 + a^2 + 2ma^2/r) \Gamma^2 + (4ma/r\Delta) \Gamma L – (1/\Delta)(1 – 2m/r) L^2 – (r^2/\Delta) (dr/ds)^2$$
we get the radial equation of motion:
$$\begin{eqnarray*}(dr/ds)^2 & = & – (\Delta/r^2) + (1/r^2)(r^2 + a^2 + 2ma^2/r) \Gamma^2 + (4ma/r^3)\Gamma L – (1 – 2m/r)L^2/r^2\\ & = & – (\Delta/r^2) + \Gamma^2 (r^2 + a^2)/r^2 – L^2/r^2 + (2m/r^3)(a\Gamma + L)^2 \mathbf{\equiv V(r)}\end{eqnarray*}$$

Circular Orbits

For a circular orbit, $V(r) = V'(r) = 0$.
$$\begin{eqnarray*}V(r) & = & 0 \Rightarrow 0 = \Gamma^2(1 + a^2/r^2) – (r^2 + L^2)/r^2 – a^2/r^2 + 2m/r + (2m/r^3)(a\Gamma + L)^2\\ V'(r) & = & 0 \Rightarrow 0 = \Gamma^2(-2a^2/r^3) + 2L^2/r^3 + 2a^2/r^3 – 2m/r^2 – (6m/r^4)(a\Gamma + L)^2\end{eqnarray*}$$
These can be simplified by taking linear combinations and multiplying by $r^4$:
$$\begin{eqnarray*}3V(r)r^3 + V'(r)r^4 & = & 0 \Rightarrow L^2 = (\Gamma^2 – 1)(3r^2 + a^2) + 4mr\\ 2V(r)r^3 + V'(r)r^4 & = & 0 \Rightarrow m(a\Gamma + L)^2 = r^3(\Gamma^2 – 1) + mr^2\\ & = & ma^2 \Gamma^2 + 2maL \Gamma + m[(\Gamma^2 – 1)(3r^2 + a^2) + 4mr]\\ 2maL \Gamma & = & \Gamma^2 (r^3 – 3mr^2 – 2ma^2) + 4mr^2 – r^3 + ma^2 – 4m^2r\end{eqnarray*}$$
By squaring the latter, $L$ can now be eliminated, leading to a quadratic equation for $\Gamma^2$.

The solutions are:
$$\Gamma = \frac{1 – 2m/r – (a/r^2) \sqrt{mr}}{\sqrt{1 – 3m/r – (2a/r^2) \sqrt{mr}}}$$
$$L = \frac{\sqrt{mr} – 2am/r – (a^2/r^2)\sqrt{mr}}{\sqrt{1 – 3m/r – (2a/r^2)\sqrt{mr}}}$$

As before, circular photon orbits exist where the denominators of $L$ and $\Gamma$ vanish.

Radial Oscillations

Once again, the frequency of small radial oscillations is governed by $V”(r)$:

$$V”(r) = \Gamma^2(6a^2/r^4) – 6L^2/r^4 – 6a^2/r^4 + 4m/r^3 + (24m/r^5)(a\Gamma + L)^2$$
From this point on, we assume that a is small, and keep only terms of first order.

$$\begin{eqnarray*}\Gamma^2 & \approx & \frac{(1 – 2m/r)^2}{1 – 3m/r} + \frac{(2am/r^3)\sqrt{mr}(1 – 2m/r)}{(1 – 3m/r)^2}\\ L^2 & \approx & \frac{mr}{1 – 3m/r} + \frac{(6am/r)\sqrt{mr}(1 – 2m/r)}{(1 – 3m/r)^2}\\ (a\Gamma + L)^2 & \approx & \frac{mr}{1 – 3m/r} + \frac{2a\sqrt{mr}(1 – 2m/r)}{(1 – 3m/r)^2}\\ V”(r) & \approx & – \frac{(2m)(1 – 6m/r)}{r^3(1 – 3m/r)} + \frac{12am\sqrt{mr}}{r^5}\frac{1 – 2m/r}{r^5(1 – 3m/r)^2}\\ (d\phi/ds)^2 & = & D(r) = \frac{L^2}{r^4} – \frac{(4ma/r^5)\Gamma L}{1 – 2m/r} \approx \frac{m}{r^3(1 – 3m/r)} + \frac{2am\sqrt{mr}}{r^5(1 – 3m/r)^2}\\ \frac{V”(r)}{D(r)} & \approx & -2 (1 – 6m/r) + \frac{12a\sqrt{mr}}{r^2}\frac{1 – 2m/r}{1 – 3m/r} + \frac{4a\sqrt{mr}}{r^2}\frac{1 – 6m/r}{1 – 3m/r}\\ & = & -2(1 – 6m/r) + \frac{16a\sqrt{mr}}{r^2}\end{eqnarray*}$$
Following the same reasoning as for Schwarzschild, the rate of small radial oscillations in the Kerr metric is therefore
$$\omega^2 = 1 – 6m/r – 8a\sqrt{m/r^3}$$
indicating that the perihelion advance per orbit is
$$6\pi m/r + 8\pi a\sqrt{m/r^3}$$

— This article was originally part of Physics Forums member Bill_K‘s PF blog. He may not respond to comments.

Next up:

Geodesic Congruences in FRW, Schwarzschild and Kerr Spacetimes