Planetary orbits in Schwarzschild spacetime

6.3.2. Planetary orbits in Schwarzschild spacetime#

Schwarzschild metrics (6.3) provides the metrics induced by a massive object in the vacuum spacetime around. The motion of a light object approximately doesn’t affect the massive object generating the steady Schwarzschild metrics. The light object moves following the geodesics. Let the motion occurs in the equatorial plane, i.e. \(\theta = \frac{\pi}{2}\), and \(d \theta = 0\). Thus

\[ - c^2 d \tau^2 = ds^2 = - \left(1 - \frac{r_s}{r} \right) c^2 dt^2 + \left(1 - \frac{r_s}{r} \right)^{-1} dr^2 + r^2 \, d\phi^2 \ . \]

Geodesics. With \(A(r) = \frac{1}{2} \ln \left( 1 - \frac{r_s}{r} \right)\), \(A'(r) = \frac{1}{2} \frac{1}{1 - \frac{r_s}{r}} \frac{r_s}{r^2}\)

\[0 = \ddot{q}^k + \Gamma_{a b}^k \dot{q}^{a} \dot{q}^b\]

\[\begin{split}\begin{aligned} t : \qquad 0 & = \ddot{t} + \Gamma_{ab}^t \dot{q}^a \dot{q}^b = \\ & = \ddot{t} + 2 \Gamma_{tr}^t \dot{t} \dot{r} = \\ & = \ddot{t} + \frac{1}{1 - \frac{r_s}{r}} \frac{r_s}{r^2} \dot{t} \dot{r} \\ 0 & = \left(1 - \frac{r_s}{r} \right)^{-1} \left[ \left( 1-\frac{r_s}{r}\right) \ddot{t} + \frac{r_s}{r^2} \dot{t} \dot{r} \right] = \\ 0 & = \left(1 - \frac{r_s}{r} \right)^{-1} \dfrac{d}{d \tau} \left[ \left( 1-\frac{r_s}{r} \right) \dot{t} \right] \quad \rightarrow \quad \left( 1-\frac{r_s}{r} \right) \dot{t} =: \frac{E}{c} = \text{const.} \\ r : \qquad 0 & = \ddot{r} + \Gamma_{ab}^r \dot{q}^a \dot{q}^b = \\ & = \ddot{r} + \Gamma_{tt}^r \dot{t}^2 + \Gamma_{rr}^r \dot{r}^2 + \Gamma_{\theta \theta}^{r} \dot{\theta}^2 + \Gamma_{\phi \phi}^{r} \dot{\phi}^2 = && (\dot{\theta} = 0 )\\ & = \ddot{r} + \Gamma_{tt}^r \dot{t}^2 + \Gamma_{rr}^r \dot{r}^2 + \Gamma_{\phi \phi}^{r} \dot{\phi}^2 = \\ & = \ddot{r} + A' e^{4 A} c^2 \dot{t}^2 - A'(r) \dot{r}^2 - r \sin^2 \theta e^{2 A} \dot{\phi}^2 = \\ & = \ddot{r} + A' e^{4 A} c^2 \dot{t}^2 - A'(r) \dot{r}^2 - r \theta e^{2 A} \dot{\phi}^2 = \\ \phi : \qquad 0 & = \ddot{\phi} + \Gamma_{ab}^{\phi} \dot{q}^a \dot{q}^b \\ & = \ddot{\phi} + 2 \Gamma_{r \phi}^\phi \dot{r} \dot{\phi} + 2 \Gamma_{\theta \phi}^\phi \dot{\theta} \dot{\phi} = && ( \dot{\theta} = 0 ) \\ & = \ddot{\phi} + 2 \Gamma_{r \phi}^\phi \dot{r} \dot{\phi} = \\ & = \ddot{\phi} + 2 \frac{1}{r} \dot{r} \dot{\phi} \ , \\ 0 & = \frac{1}{r^2} \dfrac{d}{d \tau} \left( r^2 \dot{\phi} \right) \quad \rightarrow \quad r^2(\tau) \dot{\phi}(\tau) =: L = \text{const.} \\ \theta : \qquad 0 & = \ddot{\theta} + \Gamma_{ab}^{\theta} \dot{q}^a \dot{q}^b \\ & = \ddot{\theta} + 2 \Gamma_{r \phi}^\theta \dot{\theta} \dot{\phi} + \Gamma_{\phi \phi}^\theta \dot{\phi}^2 = && ( \dot{\theta} = 0 ) \\ & = \ddot{\theta} - \sin \theta \cos \theta \Gamma_{\phi \phi}^\theta \dot{\phi}^2 = \\ & = \ddot{\theta} \ . \end{aligned}\end{split}\]

Replacing the expression for \(\dot{t}\) and \(\dot{\phi}\) into the relation \(u^\mu u_{\mu} = - c^2\) (from \(ds^2/d \tau^2\))

\[\begin{split}\begin{aligned} -c^2 & = - \left( 1 - \frac{r_s}{r} \right) c^2 \dot{t}^2 + \left( 1 - \frac{r_s}{r} \right)^{-1} \dot{r}^2 + r^2 \dot{\phi}^2 = \\ & = - \left( \right)^{-1} E^2 + \left( \right)^{-1} \dot{r}^2 + \frac{L}{r^2} \\ \end{aligned}\end{split}\]

and thus

\[ \dot{r}^2 = E^2 - c^2 \left( 1 - \frac{r_s}{r} \right) \left( 1 + \frac{L^2}{r^2 c^2} \right) \ . \]

Now, changing variable \(u := \frac{1}{r}\) and using the relation \(\phi(\tau)\) to write

\[\dot{r} = \frac{d r}{d\tau} = \frac{d \phi}{d \tau} \frac{d}{d \phi} = \frac{L}{r^2} \frac{d r}{d \phi} = L u^2 \left( - \frac{1}{u^2} \right) u'(\phi) = - L u'(\phi) \ ,\]

the equation becomes

\[ L^2 {u'}^2 = E^2 - c^2 ( 1 - r_s u ) \left( 1 + \frac{L^2}{c^2} u^2 \right) \ .\]

Differentiating both sides by \(\phi\)

\[0 = L^2 2 u' u'' - c^2 r_s u' \left( 1 + \frac{L^2}{c^2} u^2 \right) + c^2 ( 1 - r_s u) 2 \frac{L^2}{c^2} u u' \]

or dividing by \(2 L^2 u'\),

\[0 = u'' - \frac{c^2 r_s}{2 L^2} - \frac{3}{2}r_s u^2 + u\]

or, with \(r_s = \frac{2 GM}{c^2}\)

\[u'' + u = \frac{c^2 r_s}{2 L^2} + \frac{3}{2} r_s u^2 = \frac{GM}{L^2} + \frac{3 GM}{c^2} u^2 \ .\]

Solution via perturbation method. As \(\frac{3 GM}{c^2} \to 0\), the equation becomes the equation of the Kepler orbits.

Let the solution be \(u(\phi) = u_0(\phi) + u_1(\phi)\), with \(u_0(\phi)\) the solution of the equations of classical mechanics.

\[u_0(\phi) = \frac{GM}{L^2}(1 + e \cos \phi) \ . \]

As

\[0 = \underbrace{u_0'' + u_0 - \frac{GM}{L^2}}_{=0} + u_1'' + u_1 - \frac{3GM}{c^2} (u_0 + u_1)^2 \ ,\]

the perturbation \(u_1(\phi)\) (of order \(\frac{GM}{c^2}\) must satisfy (approximately, truncation) the equation

\[\begin{split}\begin{aligned} u_1'' + u_1 & \approx \frac{3 GM}{c^2} u_0^2(\phi) = \\ & = \frac{3 (GM)^3}{c^2 L^4} \left( 1 + 2 e \cos \phi + e^2 \cos^2 \phi \right) = && ( \cos 2 x = 2 \cos^2 x - 1 ) = \\ & = 3 \left( \frac{(GM)^{3/2}}{ c L^2} \right)^2 \left( 1 + 2 e \cos \phi + \frac{e^2}{2} \left( 1 + \cos ( 2 \phi ) \right) \right) = \\ & = 3 \left( \frac{(GM)^{3/2}}{ c L^2} \right)^2 \left( 1 + \frac{e}{2} + 2 e \cos \phi + \frac{e^2}{2} \cos ( 2 \phi ) \right) \ , \end{aligned}\end{split}\]

whose solution undergoes resonance, as the forcing \(2 e \cos \phi\) has the same frequency as the natural frequency of the system.The general solution reads

\[u_1(\phi) = A \cos \phi + B \sin \phi + 3 \left( \frac{(GM)^{3/2}}{c L^2} \right)^2 \left[ \left( 1 + \frac{e}{2} \right) + e \phi \sin \phi + \frac{e^2}{6} \cos 2 \phi \right]\]

Only the term \(\propto \phi \sin \phi\) is unbounded. Combining this term with the unperturbed solution gives

\[\begin{aligned} u(\phi) & \approx \frac{GM}{L^2} \left\{ ( 1 + e \cos \phi ) + 3 \left( \frac{GM}{cL} \right)^2 e \phi \sin \phi \right\} \ , \end{aligned}\]

and using small angle approximation \(\cos(\alpha + x ) = \cos \alpha \cos x - \sin \alpha \sin x \approx \cos x - \alpha \sin x \), for \(\alpha \sim 0\),

\[\begin{split}\begin{aligned} u(\phi) & \approx \frac{GM}{L^2} \left\{ 1 + e \cos \phi + \underbrace{ 3 \left( \frac{ GM}{cL} \right)^2 \phi}_{"-\alpha"} e \sin \phi \right\} = \\ & \approx \frac{GM}{L^2} \left\{ 1 + e \cos \left( \phi - \left( \frac{GM}{cL} \right)^2 \phi \right) \right\} = \\ & = \frac{GM}{L^2} \left\{ 1 + e \cos \left[ \left( \phi - 3 \left(\frac{ GM}{c L} \right)^2 \right) \phi \right] \right\} \ . \end{aligned}\end{split}\]

As \(u:= \frac{1}{r}\), the perihelion occurs for \(u(\phi_{perih}) = \max_{\phi} u(\phi)\), i.e. for

\[\left( 1 - 3\left( \frac{GM}{c L} \right)^2 \right) \phi_{perih,n} = n 2 \pi \quad , \quad n \in \mathbb{Z} \ .\]

The precession of the perihelion can be evaluated as the difference between the angular coordinate of two successive revolutions (adding a \(2pi\) to the first angle, mimicing an orbit with no precession) i.e.

\[\begin{split}\begin{aligned} \Delta \phi & = \phi_{perih, n+1} - ( \phi_{perih, n} + 2 \pi ) = \\ & = \frac{2 \pi}{1 - 3 \left(\frac{GM}{cL}\right)^2} - 2 \pi = \\ & \approx 2 \pi \left( 1 + 3 \left( \frac{GM}{cL} \right)^2 - 1 \right) = 6 \pi \left( \frac{GM}{cL} \right)^2 \ . \end{aligned}\end{split}\]