## Saturday, 27 June 2020

### Schwarzschild Black Holes - The Geodesic

(1) and (2) are the $t$ and $r$ geodesic equations in the Schwarzschild metric which we found in section  5.4. (2) becomes a bit simpler on a radial path. That's (3). Geodesic equations are meant to give you trajectories of freely falling particles parametrised by $\lambda$. (4) and (5) are the equation of a particle (or beacon) falling along a radius into a black hole from a distance $r_*$. We used them to plot a beacon's path here. They come from (6) which we calculated in exercise 5.5 where we also calculated (7).
$t$ is coordinate time, $r$ is the distance from the centre. They are the coordinates.
$\lambda$ is an affine parameter (it is proportionate to the length along the line).
$R_s$ is the Schwarzschild radius (radius of event horizon).
$\theta,\phi$ are the other spherical polar coordinates (polar and azimuth), which we can ignore.
$r_*$ is the radial distance from which the test particle, or beacon, is dropped.
$\tau$ is the proper time, which can be an affine parameter for a massive particle.

The question is: Is the path given by (4) and (5) a real geodesic? That is, does it satisfy (1) and (3)?