Schwarzschild Geodesics - Orbits of Test Particles

Orbits of Test Particles

Dividing both sides by dτ2, the Schwarzschild metric can be rewritten as

$c^{2} = \left( 1 - \frac{r_{s}}{r} \right) c^{2} \left( \frac{dt}{d\tau} \right)^{2} - \frac{1}{1 - \frac{r_{s}}{r}} \left( \frac{dr}{d\tau} \right)^{2} - r^{2} \left( \frac{d\theta}{d\tau} \right)^{2} - r^{2} \sin^{2} \theta \, \left( \frac{d\varphi}{d\tau} \right)^{2},$

The orbit of a particle in this metric is defined by the geodesic equation, which may be solved by any of several methods (as outlined below). This equation yields three constants of motion. First, the motion of the particle is always in a plane, which is equivalent to fixing θ = π/2. The second and third constants of motion, derived below, are taken as two length-scales, a and b, defined by the equations

$\left( 1 - \frac{r_{s}}{r} \right) \left( \frac{dt}{d\tau} \right) = \frac{a}{b}$

$r^{2} \left( \frac{d\varphi}{d\tau} \right) = a c$

where c represents the speed of light. Incorporating these constants of motion into the metric yields the fundamental equation for the particle's orbit

$\left( \frac{dr}{d\varphi} \right)^{2} = \frac{r^{4}}{b^{2}} - \left( 1 - \frac{r_{s}}{r} \right) \left( \frac{r^{4}}{a^{2}} + r^{2} \right)$

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