Quote:
Originally Posted by nauthiz
I've heard that, but that just raises a bunch more questions for me:
What other coordinate systems are used, and what do the equations look like when we use them?
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The
Kruskal-Szekeres coordinates.
Quote:
Originally Posted by nauthiz
The singularity at the Schwarzschild radius doesn't look like a simple coordinate singularity where everything looks fine except that you've got this spot that's undefined. I'm certainly not mathematician, but it seems odd to me that a whole vertical asymptote could be a mere artifact of the notation.
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It can. Mathematically, the same function can be written with many different formulas. Sometimes, a particular formula creates problems at some points (like singularities), but those problems disappear when you find another formula that gives the same results everywhere except at that point. These are just mathematical technicalities.
Quote:
Originally Posted by nauthiz
And wouldn't it be a worse paradox if a mere change in coordinate system gives results that contradict each other?
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You don't really have two different results. It's just that, at the Schwarzschild radius, one formula (the Schwarzschild coordinates) happens to not be applicable; it does not produce any meaningful result. But other formulas do produce a meaningful, and unique result. This is a kind of singularity, but one that is just a mathematical artifact of the formulas we use. It doesn't mean anything physically, since it can be completely eliminated by switching to another formula (like the Kruskal-Szekeres coordinates).
Quote:
Originally Posted by nauthiz
What happens at the Schwarzschild radius when you're using other coordinate systems?
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Quote:
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This region was isolated from the rest of the universe by a place where Schwarzschild's coordinates blew up, though nothing was wrong with the curvature of spacetime there. (This was called the Schwarzschild radius. Later, other coordinate systems were discovered in which the blow-up didn't happen; it was an artifact of the coordinates, a little like the problem of defining the longitude of the North Pole. The physically important thing about the Schwarzschild radius was not the coordinate problem, but the fact that within it the direction into the hole became a direction in time.)
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This quote is from the
Usenet Physics FAQ -- What is a black hole, really? (much more knowledgeable about black holes than I.
) See also the
Black Holes FAQ.