- #36
G. E. Hunter
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is black hole evaporation in the real universe, or is this just theory? Any actual observations of this phenomenon? Actual observed decreases in size or mass over time?
G. E. Hunter said:is black hole evaporation in the real universe, or is this just theory? Any actual observations of this phenomenon? Actual observed decreases in size or mass over time?
I made a slip-up using the term 'coordinate' rather than 'parameter' in that quote, but the gist is the same.pervect said:The entire purpose of the metric is to convert changes in coordinates (r, theta, phi) into changes in length.
right, that's how I basically understood it. But in another thread, the grr factor, even in that context, was explained as something quite different in meaning to a straight metric operator on dr - i.e. one cannot infer a coordinate dr = dL*(grr)-1/2, but rather a relation of differential volume to differential area. While the latter is apparently a standard interpretation, couldn't see where it came from.dr isn't a length, until you multiply it by the appropriate metric coefficient.
Once you have the spatial metric (the induced 3-metric I mentioned earlier), you find the length of any curve by integrating dL along the curve, where dL is given by
[tex]dL^2 = g_{rr} dr^2 + g_{{\theta}{\theta}} d\theta^2 + g_{{\phi}{\phi}} d\phi^2[/tex]
Not being familiar with a lot of GR jargon, loked up http://en.wikipedia.org/wiki/Induced_metric , which gave me just enough clues that 'induced metric' is somewhat akin to the idea behind 'partial derivatives' - in this case holding t constant (mapping on to a spatial hypersurface or somesuch?).Just remember we're using the induced three-metric here, not the space-time metric.
Phew -right about not getting into calculating an infinite number of curves! Can we simplify that even more if one restricts to just measuring along a short tangent surface geodesic r*sinθ*dϕ, or radial displacement dr - assuming exterior Schwarzschild metric where I take it gθθ = gϕϕ = 1 applies?And the distance is just the shortest curve connecting two points (or the greatest lower bound). Since you probably don't want to calculate an infinite number of curves to find the greatest lower bound :-), you use either the calculus of variations, the geodesic equations, or a Lagranian approach to make sure you've chosen the curve that minimizes distance.