BR wrote
Does your comment imply that it is possible that the answer "what is
the distance to a BH?" is "finite" if you do it one way, a different
"finite" if you do it another way, or even "infinite" if you do it
another way? What conditions did you and TR assume when you said it
was finite?
Sorry, badly put question. Of course the distance can change depending
on how you do it. But the problem was like this:
If the distance between A and B is not changing with time (they are
holding a rope, and time-of-flight is constant), then according to A
the distance AB has a unique unambiguous value.
In Euclidean space this is easy, fix the origin at A, and the distance
AB is the root of the sum of the squares of the coordinates of B. But
if B is down some gravitational well, then the distance AB depends on
how you define it??? Even though the clock at B is going at a
different rate than A, it doesn't matter because the distance does not
change with time, so according to A there is a unique unambiguous
space-like interval? In the Schwartzschild metric, fix the coordinates
of the hovering spaceship, t is irrelevant as the coordinates are not
changing with time, so you just integrate r?
If this is the case, does it still apply to the distance to an event
horizon? After all, the definition of "hovering" is that the distance
does not change. Does it still come out finite? (I'm not sure if there
is another conversion factor from coordinate r, to "real" distance, or
maybe it gets screwed up somehow).
thanks again,
Brendan.
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