(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)?
And the answer is YES.
I studied these geodesics three months ago and they have never yet been useful. This is the first time that they have even clicked with anything!
Here's how they click: Commentary 5.4#1 Geodesics of Schwarzschild.pdf (only 2 pages really)
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