In the first part of section 5.2 (two posts ago) it seemed to be impossible to get inside the Schwarzschild radius. In the second part we look at other coordinate systems and find the Eddington-Finkelstein coordinates which show us how. The metric then takes a different form (which says something about Birkhoff's theorem) and it does not have an infinity at the Schwarzschild radius (##r=2GM##).
The properties of the function ##1-2GM/r## (which causes the trouble) frequently amaze. It keeps eating itself up which is very satisfactory.
Eddington-Finkelstein coordinates eliminate the time coordinate ##t## and introduce
$$v=t+r+2GM\ln{\left(\frac{r}{2GM}-1\right)}$$
so the relationship between ##v## and ##t## is complicated.
The image shows a light cone at various distances out from the centre at ##r=0## . The ingoing light beam always heads for the centre the outgoing beam can get away when ##r>2GM## but flips towards the centre once it originates at a distance less than the Schwarzschild radius (aka the event horizon). The closer the starting point is to the centre, the less room there is for manoeuvre.
I am slightly dubious about the direction of the ingoing side of a light cone inside the Schwarzschild radius.
See why and check my maths: Commentary 5.6#2 Schwarzschild Black Holes.pdf (6 pages)
No comments:
Post a Comment