
#19
Oct1811, 11:24 AM

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#20
Oct1811, 11:24 AM

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#21
Oct1811, 11:33 AM

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Also, since a test sphere in this spacetime, to an observer next to it, will *not* appear distortedit will be sphericalif it does appear distorted to an observer much further away from the black hole, that distortion would have to be "gravitationally induced optical distortion", would it not? If so, you wouldn't want to "compensate" for that, since it would eliminate the effect you are interested in. 



#22
Oct1811, 11:35 AM

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#23
Oct1811, 11:40 AM

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#24
Oct1811, 11:49 AM

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#25
Oct1811, 11:51 AM

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#26
Oct1811, 12:20 PM

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#27
Oct1811, 12:51 PM

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#28
Oct1811, 12:54 PM

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Now this would all go away if in fact hte SM is *actually* isotropic spatially, so are SC's fooling us? This is why I want that test sphere result so bad! Bed time. 



#29
Oct1811, 01:39 PM

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Perhaps I introduced confusion when I used the word "anisotropy" to refer to the fact that there is more distance between two neighboring 2spheres in the Schwarzschild exterior region than Euclidean geometry would lead one to expect based on the areas of the spheres. The only thing that that shows to be nonisotropic is the "Euclideanness" of the space. I've already described in detail how that "anisotropy" goes away as you descend through the nonvacuum "shell" region; if you want a "physical" explanation of how that works, I would say it's because the nonEuclideanness of the space is due to the mass of the shell being below you, as you descend through the shell, less and less of its mass is below you. "Below" here just means "at a smaller radius", or, if I were to be careful about describing everything in terms of direct observables, "below" means "lying on a 2sphere with a smaller area than the one you are on". It's also worth noting, I think, that even the "nonEuclideanness" of the space, as I described it, is observerdependent; it assumes that the nonEuclideanness is being judged using 2spheres which are at rest relative to the shell. Observers who are freely falling through the exterior vacuum region will *not* see this anisotropy in the space that is "at rest" relative to them; in other words, if a freely falling observer were to measure the areas of two adjacent 2spheres that were falling with him, and measure the distance between the two 2spheres, he would find the relationship between those measurements to be exactly Euclidean. 



#30
Oct1811, 03:30 PM

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#31
Oct1811, 06:07 PM

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The diagram below of a "Flamm's paraboloid" may aid understanding the curvature of space (but not spacetime) outside the event horizon of a black hole. The surface represents a 2D crosssection through 4D spacetime involving the [itex]r[/itex] and [itex]\phi[/itex] coordinates only. In this diagram [itex]r[/itex] is the radius in the horizontal direction. "Ruler" distances in space (e.g. PeterDonis's method of counting small identical objects packed together) are represented by distances along the curved surface. (The vertical direction has no physical meaning at all.)
The diagram goes all the way to the event horizon, where the surface becomes vertical at the bottom of the "trumpet". So, for the example being discussed in this thread, you need to slice off the bottom part of the surface. The interior of the shell could then be represented by a horizontal flat circular disk almost capping the bottom of the remaining trumpet, but there is then a gap to be bridged between the disk and the trumpet to represent the shell. I've no idea whether is possible to construct a curved surface to bridge that gap which would correctly represent the geometry within the shell. (I suspect it might not.) The mathematics of Flamm's paraboloid is discussed on Wikipedia at Schwarzschild metric: Flamm's paraboloid. P.S. Flamm's paraboloid does not represent the gravitational potential. The potential has a somewhat similar shape but it's a completely different formula. 



#32
Oct1811, 06:15 PM

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#33
Oct1811, 07:13 PM

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And even if you can produce a 2D curved surface in 3D space to represent the entire "shell" space 



#34
Oct1811, 09:48 PM

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Oct1811, 09:49 PM

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#36
Oct1811, 10:16 PM

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