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Spherical Mass Shell  What Actually Happens? 
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#1
Oct1511, 06:30 AM

P: 1,115

In another thread I posed basically the folowing problem:
Take the case of a stationary, nonrotating thin spherical shell of uniform area mass density  outer radius r_{b}, inner radius r_{a}, with (r_{b}r_{a})/r_{a} << 1. There is consensus opinion that everywhere exterior and down to r_{b}, spacetime is that of the vacuum SM (Schwarzschild metric), whilst everywhere interior to r_{a}, flat MM (Minkowski metric) applies. Within the shell wall itself, there is a nonzero stressenergy and spacetime is neither vacuum SM or MM, but the particulars of that transition region is of no concern here. Assume then a modest gravitational potential such that r_{s}/r << 1 close to the shell of mass m, with r_{s} = 2Gmc^{2}. To a good first approximation there is a negligible relative drop in potential in going from r_{b} to r_{a}. Of interest is how spacetime affects the spatial and temporal components of a small test object placed in the SM or MM regions  all referenced to a distant stationary observer in asymptotically flat MM  the coordinate reference frame. Let the test object be a small perfect sphere (notionally "perfectly rigid") of diameter D as per coordinate measure in gravityfree space. It also doubles as a clock  emitting a fixed frequency f there. Next the sphere is placed in a stationary relative position: A: Resting just outside the shell at radius r_{b}. It is here subject to SM B: Anywhere inside the shell at radius r<r_{a}. It is here subject to MM. Required is the mathematically correct distortion factors D_{r}'/D, D_{t}'/D, f'/f, now observed for cases A and B, where: D_{r}', D_{t}', are the observed radial, tangential spatial measures in the gravity effected cases, and likewise for f'. Five values altogether are required: Case A: D_{r}'/D_{SM}, D_{t}'/D_{SM}, f'/f_{SM}, Case B: D'/D_{MM}, f'/f_{MM},  given that here flat MM implies D_{r}'= D_{t}' = D'. It should go without saying that only the underlying metric properties are of interest. Assume that any mechanical distortions due to direct or tidal 'g' forces are negligible or corrected for (e.g.; suspension in a flotation tank), and likewise for optical effects (gravitational light bending). Locally, no distortions would be apparent  only as seen 'from infinity' of course. This is an attempt to sort out certain claims that all commonly used coordinate systems will yield identical predictions. 


#2
Oct1511, 06:48 AM

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P: 17,526

I am not quite sure what your setup and desired computations. First, is your sphere an approximately rigid object and you want to know about stresses on it due to gravity or is your sphere a collecition of non interacting particles and you want to see how they move relative to each other? Also, is it free falling or is it at rest wrt the shell or undergoing some other motion? Do you want a set of coordinates which smoothly transitions from interior to exterior?



#3
Oct1511, 07:10 AM

P: 1,115

"It should go without saying that only the underlying metric properties are of interest. Assume that any mechanical distortions due to direct or tidal 'g' forces are negligible or corrected for (e.g.; suspension in a flotation tank), and likewise for optical effects (gravitational light bending)." Just to completely clarify  yes, assume a notionally perfectly rigid test sphere, distorted only via metric measure (as of course referenced 'to infinity'). Locally therefore, nothing would be observed. I have now edited #1 to add "notionally perfectly rigid", plus a bit more about locally unobservable vs coordinate measure! [missed your one about stationary vs orbiting etc. I thought it obvious that stationary was implied unless otherwise stated, but have now further edited in that spec  stationary] 


#4
Oct1511, 08:56 AM

P: 95

Spherical Mass Shell  What Actually Happens?
What's a distortion factor and how is it defined?



#5
Oct1511, 09:11 AM

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#6
Oct1511, 09:17 AM

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It's not clear what you mean by D_{t}'. Do you mean a change in coordinates, or do you mean the distance you would measure using a ruler (or radar equipment)? If you mean coordinates, the Schwartzschild tangential coordinates are angles, not distances, and to convert a coordinate angle to a distance, you use the metric which gives you an answer D_{t}' = D (for small D). The Schwartzschild radial coordinate does not directly measure distance, and to convert the coordinate to a distance, you use the metric which gives you an answer D_{r}' = D (for small D)  if that's what you meant by D_{r}'.
For the interior, it depends what coordinates you use: you have a choice of using Minkowski coordinates directly, or using rescaled Minkowski coordinates that have been multiplied by a constant factor so that they match up with the exterior coordinates. From what you have said in this thread and in the other thread, it sounds like you want to splice together the coordinate systems to form a composite coordinate system to cover both inside and outside without discontinuity, in which case you'll need to use rescaled Minkowski coordinates. Now a similar argument applies to the interior as to the exterior: you use the rescaled Minkowski metric to convert coordinate differences to actual distances, and you'll get D' = D. Now I suspect that isn't really what you meant by D', so you'll have to specify more clearly what you really did mean. 


#7
Oct1511, 09:31 AM

P: 95

Well if you're interested in how the sphere contracts you're going to have to say what it is made of, which will make the analysis complicated. In principle what you should do is specify a matter model (involving elasticity, etc.) and solve the coupled matterEinstein equations, linearized off your background spherical shell spacetime.



#8
Oct1511, 09:52 AM

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#9
Oct1511, 09:53 AM

P: 1,115

"It should go without saying that only the underlying metric properties are of interest. Assume that any mechanical distortions due to direct or tidal 'g' forces are negligible or corrected for (e.g.; suspension in a flotation tank), and likewise for optical effects (gravitational light bending). Locally, no distortions would be apparent  only as seen 'from infinity' of course." Are you saying this cannot be accommodated  one cannot in principle decouple purely metric distortions from the 'mechanical' effects of finite elastic strain? That doesn't seem reasonable to me, but maybe this is simply a misunderstanding. 


#10
Oct1511, 10:38 AM

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#11
Oct1511, 10:56 AM

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#12
Oct1511, 10:56 AM

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P: 17,526

An elastic object will get "squeezed" into an ellipsoidal shape as it falls towards the shell. As it nears the shell the squeezing will reach the maximum. As it falls through the shell it will unsqueeze until as it reaches the interior of the shell it is completely unsqueezed. The proper time between ticks will remain unchanged throughout the fall, and the proper diameter inside the shell will be the same as the proper diameter at infinity. I think what you are really interested in is one continuous coordinate system that covers the whole spacetime. The distant measures you mention will depend on your choice of coordinates. 


#13
Oct1511, 12:03 PM

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#14
Oct1511, 12:05 PM

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#15
Oct1511, 12:08 PM

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#16
Oct1511, 04:06 PM

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P: 17,526

What is the differential comparison technique you have in mind? Perhaps something like sending a sending a light pulse from one side of the sphere to the other, sending a light pulse out to infinity at the start and a second pulse out to infinity at the end, and measuring the time between receiving the two pulses at infinity? 


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