Why not consider following my advice?
dookie said:
would still very much appreciate understanding it)
Well, my previous advice stands.
Now that you have finally bothered to tell us where you found the problem and what it actually states, it has become clear that my suspicions were correct:
1. the problem you were trying to work does indeed concern the "future interior" of the Schwarzschild vacuum,
2. you entirely missed the point of the problem.
I see that you haven't even yet grasped the geometry of the local light cones inside the event horizon. This suggests to me that you are not ready for the problem book yet, much less Wald, so I advise you to put your advanced books away until you have more geometric insight. At this point they are obviously only confusing you, and possibly allowing you to believe you know much more than you really do. Please don't take offense, and don't be put off by the fact that Geroch's book is high school level--- it is sufficiently challenging to be interesting, and I repeat, at this point in your understanding it is just what you need.
File the following points for future reference, when you revisit this material after learning enough geometric background to try again to start learning those aspects of gtr which involve calculus:
1. Don't confuse metric (as in the chosen metric tensor, which must be valid everywhere on the given Lorentzian manifold, by definition) with "chart",
2. Don't omit coordinate ranges when writing out a line element in a coordinate chart. In particular note that the exterior Schwarzschild chart
ds^2 = -(1-2m/r) \, dt^2 + \frac{dr^2}{1-2m/r} + r^2 \, d\Omega^2, \; -\infty < t < \infty, \; 2m < r < \infty
is valid on a domain which is
disjoint from the interior Schwarzschild chart
ds^2 = -(1-2m/r) \, dt^2 + \frac{dr^2}{1-2m/r} + r^2 \, d\Omega^2, \; -\infty < t < \infty, \; 0 < r < 2m
Figure out why you can and should rewrite the latter as
ds^2 = \frac{-d\tau^2}{1+2m/\tau} + (1+2m/\tau) \, dz^2 + \tau^2 \, d\Omega^2, \; -\infty < z < \infty, \; -2m < \tau < 0
3. If you want to study infall into a black hole, you will need to use a chart which covers both the exterior and future interior regions, such as the Eddington, Painleve, Lemaitre, or Novikov charts. If you want to understand the global structure, you will need to paste together information from various charts without getting confused (which can be hard for novices), or else to use a global chart such as the Kruskal-Szekeres chart or the "compactified" Penrose chart.
I've said all of these things before, maybe even to yourself under a previous handle, so repeating myself it getting a bit tedious. I added you to my ignore list and hope the moderators lock this thread before it becomes even more pointless.
