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How does GR handle metric transition for a spherical mass shell? 
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#37
Oct1811, 10:28 PM

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#38
Oct1911, 06:52 AM

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But enough of generaized criticism. To answer your specific point Peter, just consider redshift. All but the most uninformed newbe has little trouble appreciating that redshift cannot be locally observed because it is inherently a differential issue  clockrate 'here' vs clockrate out 'there'. We have no conceptual issue with this (well, those insisting it's exclusively a 'tired light' 'energy drain' thing might). So why should spatial measure be any different? What is the apparent fundamental divide? If effect of metric on timerate is properly a relational, nonlocally experienced, justifiably physical thing, what allows that length measure  the spatial component of SM in particular, are *not* likewise a physically meaningful, nonlocally observed relational thing? Seems illogical to me. Want an 'extreme' example? my 'predjudice' is that BH's are, in one important sense, propped up on the basis that, just as SC's indicate, tangent spatial components are unaffected by gravitational potential. If however *all*, spatial components at least, are subject to the J factor, we find that the physical size of a notional BH shrinks to zero before it in fact can qualify as BH  as of course referenced to a distant observer. That leaves out other matters such as whether 'gravity actually gravitates' but indicates that there are drastic consequences as to the proper, physical implications of metric on a relational, 'distant observer' basis. So it's vitally important to know just what that test sphere will, abberation free, actually be *measured* by the distant observer imo. And my assumption is SM via SC's tells us it will be an oblate spheroid with axial ratio J:1. I further think nature has a different idea  it will to first order remain spherical but shrunk. [unavoidably there will be observed second and higher order distortions simply owing to metric necessarily being a function of r in any reasonable metric theory] Hope you all get my drift here. 


#39
Oct1911, 06:54 AM

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"I entirely meant shell matter's contribution to the exterior, SM region, and said so explicitly in #25." You are capable of discerning the fundamental significance of replacing 'in' with 'to' in your above distortion of what I was on about in #25, right? 


#40
Oct1911, 07:01 AM

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Oh, here's a possible fly in the ointment. Add the tiniest puff of fresh, pure mountain air inside the shell. Just a touch. Just enough to reverse the sign of shell hoop stresses and blow the amplitude up by, say, a mere factor of one million. And this is still looking anything but Alicein Wonderland nonsense?! Good luck, genius! 


#41
Oct1911, 07:36 AM

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#42
Oct1911, 08:10 AM

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#43
Oct1911, 09:10 AM

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#44
Oct1911, 09:13 AM

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#45
Oct1911, 10:19 AM

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Second, the redshift is coordinate independent. The "spatial measure" as you are using the term is not, which is why I was so careful in previous posts to describe everything in terms of areas and the "nonEuclideanness" of space, without saying anything definite about "distance measure". See below. It's worth walking through this in some more detail. Go back to the picture I gave in terms of 2spheres with gradually decreasing areas. The areas of these 2spheres, and the volume in between neighboring 2spheres, are physical observables; we can measure them by covering the area or packing the volume with little identical objects and counting them. So the factor K, that I defined, is a coordinateindependent quantity and represents actual physics. However, there are different ways in which this actual physics can be represented in a coordinate system. The different radial coordinate definitions that I described are different ways of *labeling* the 2spheres, and different labelings lead to different conclusions about which "spatial components" are affected by the K factor. If we use the Schwarzschild r coordinate, we label each 2sphere with a coordinate r equal to the square root of its physical area divided by 4 pi. With this labeling, only the radial component of the metric is affected by the K factor; the tangential components are not. However, if we use the isotropic R coordinate, meaning that a 2sphere gets labeled with a "radius" R that does *not* equal the square root of its physical area divided by 4 pi (in the case we're discussing, R will be smaller than that), then all three spatial components *are* affected by the K factor. The answer is that the question is not valid, because the physical definition of the K factor requires that the 2spheres in question are spacelike surfaces. But the 2sphere at the EH, where r = 2M in Schwarzschild coordinates, is not spacelike; it's null. So it is physically impossible to perform the comparison I described using a 2sphere at the EH, and there is therefore no way to physically define the K factor (or the J factor, for that matter) at the EH. 


#46
Oct1911, 12:22 PM

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#47
Oct1911, 12:36 PM

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However, there can't be a static 2sphere "hovering" at the EH, because the EH, as a surface in spacetime, is a null surface, and therefore there can't be a surface of "constant time" that is orthogonal to the EH, in which the 2sphere could be said to lie, and in which the area of the 2sphere could be compared with the volume between it and a neighboring 2sphere. So the main point in what I said above still holds: it's impossible to do the physical measurement at the EH that I was using to define the K factor. I should also note that the above does not entirely apply to the J factor; since there are still timelike worldlines passing through the EH, it is still possible to define a "gravitational redshift" factor there, for an infalling observer. However, this factor cannot apply to a static, "hovering" observer at the EH, since as we've seen there can't be one. So what I said does apply to the J factor for static observers. 


#48
Oct1911, 02:31 PM

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"Oh, here's a possible fly in the ointment. Add the tiniest puff of fresh, pure mountain air inside the shell. Just a touch. Just enough to reverse the sign of shell hoop stresses and blow the amplitude up by, say, a mere factor of one million." If you choose to reject the basic logic of that bit, then recall  you have committed to proving me wrong by calculations I consider doomed to failure  but go ahead and show that I'm the mistaken one. 


#49
Oct1911, 02:35 PM

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So what follows in your comments here are I think perfectly OK only if one already accepts spatial metric permitting finite EH area. Catch 22, seems to me. thanks for your clarification in #47, but same deal. As in my first passage in #19, still see this thing of defining distance in terms of area as another Catch 22. How do you define area divorced from linear length measure? Those 'packing objects' are LxLxL entities, and one must have a clear definition of L, and same goes with the area thing  area A is an LxL object! If A is the primitive, how do you determine it apart from L measure! Could this be a conundrum forced by need to accomodate difficulties with the BH EH issue? Maybe not, but it's my hypothesis. Much later. 


#50
Oct1911, 03:34 PM

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(1) Did you read that previous post where I defined J and K carefully? You'll note that I specified there that the relation J = K^{1} does *not* hold in the nonvacuum region. I was talking about the "shell" scenario there, but the same would apply for the interior of a collapsing body such as a star. (2) The factor J does not apply to the tangential metric components; it applies to the time component, since it's the "redshift factor". So I'm not sure how you're concluding that the tangential metric components would "shrink" by the factor J. 


#51
Oct2011, 07:50 AM

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Interestingly, while SC's were telling me tangent component was invariant, from this directionally dependent c perspective, there is in fact tangent shrinkage of a collapsing objects perimeter by factor J^{1/2}. Still not isotropic, but not as 'bad' as I thought before. 


#52
Oct2011, 08:34 AM

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#53
Oct2011, 09:28 AM

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#54
Oct2011, 12:18 PM

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