In General Relativity by Woodhouse there are the three following diagrams in Chapter 9 about Black Holes. Despite a (very brief) description of these diagrams in the book itself, I am struggling to understand what they exactly correspond to? Does anyone have good way of looking at how these diagrams help convey a) the properties of the particular coordinate system and b) a understanding of how black holes work in the particular coordinate system being looked at.
The radial null geodesics for the Schwarzschild metric have the form
$$ t \pm [r + 2 m \ln (r - 2m)] = k \tag{1} $$
where $k$ is a constant. This is a plot for different values of $k$
The plots you showed are just an attempt to represent these geodesics in three spatial coordinates. Unfortunately that cannot be done trivially, so instead Woodhouse decided to ignore one of the coordinates and draw Eq. (1) in the other two. You can do that by setting $x = r\cos\phi$ and $y = r \sin\phi$ for $0\leq \phi \leq 2\pi$. Or equivalently by rotating the curves in the figure above around the $t$ axis. This is an example for $k = 1$
This is just a fancy extension of the first figure above, but it is completely unnecessary, since the problem has spherical symmetry, and adding an extra coordinate just adds to the noise without any benefit.