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Spherical Mass Shell - What Actually Happens?

  1. Oct 15, 2011 #1
    In another thread I posed basically the folowing problem:
    Take the case of a stationary, non-rotating thin spherical shell of uniform area mass density - outer radius rb, inner radius ra, with (rb-ra)/ra << 1. There is consensus opinion that everywhere exterior and down to rb, spacetime is that of the vacuum SM (Schwarzschild metric), whilst everywhere interior to ra, flat MM (Minkowski metric) applies. Within the shell wall itself, there is a non-zero stress-energy 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 rs/r << 1 close to the shell of mass m, with rs = 2Gmc-2. To a good first approximation there is a negligible relative drop in potential in going from rb to ra.

    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 gravity-free 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 rb. It is here subject to SM
    B: Anywhere inside the shell at radius r<ra. It is here subject to MM.

    Required is the mathematically correct distortion factors |Dr'/D|, |Dt'/D|, |f'/f|, now observed for cases A and B, where:
    Dr', Dt', are the observed radial, tangential spatial measures in the gravity effected cases, and likewise for f'.

    Five values altogether are required:

    Case A: |Dr'/D|SM, |Dt'/D|SM, |f'/f|SM,
    Case B: |D'/D|MM, |f'/f|MM, - given that here flat MM implies Dr'= Dt' = 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.
    Last edited: Oct 15, 2011
  2. jcsd
  3. Oct 15, 2011 #2


    Staff: Mentor

    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?
  4. Oct 15, 2011 #3
    Umm, well I did try and be very specific - you did read this part?:
    "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]
    Last edited: Oct 15, 2011
  5. Oct 15, 2011 #4
    What's a distortion factor and how is it defined?
  6. Oct 15, 2011 #5
    That is my own terminology, but is it not clear what is meant? I'll briefly repeat. Distant stationary observer determines, using receiver (for frequency f), and telescope (for test sphere diameter components Dr, Dt) values for said test sphere in a gravity-free region. Next, at the same distance from observer, the same measurements are conducted with sphere located variously just outside or inside a spherical mass shell now present. The ratio of readings (after/before) = 'distortion factor(s)'. Perhaps I should have used a different term like 'contraction factor', but since both temporal and spatial components are asked for, seemed 'distortion' was reasonable omnibus term. The definitions in #1 now hopefully make total sense to you.
  7. Oct 15, 2011 #6


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    It's not clear what you mean by Dt'. 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 Dt' = 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 Dr' = D (for small D) -- if that's what you meant by Dr'.

    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.
  8. Oct 15, 2011 #7
    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 matter-Einstein equations, linearized off your background spherical shell spacetime.
  9. Oct 15, 2011 #8
    Had no idea there would be such confusion here and above as to my intent, but there you go. The main aim is to cut through arguments about which coordinate choice is 'better' in describing relational values. I just want to know, given the physical system described, what will a distant observer actually measure, as a before/after exercise. When someone can provide a set of the five values I have asked, we can then discuss the basis for arriving at those values. Hopefully, this will be done using a variety of methods, but that's up to the GR pros here to decide. If I have it correct, you are saying interior to the shell, length scale is as for the case of no shell present?
  10. Oct 15, 2011 #9
    In #1 I wrote: "..Let the test object be a small perfect sphere (notionally "perfectly rigid")..", and further down
    "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.
  11. Oct 15, 2011 #10


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    The problem is that distant observers cannot directly measure distances or times; only a local observer can do that. Distant observers have to set up some sort of convention for assigning cooridinates to distant events, but those coordinates don't usually directly measure distance or time. But the coordinates come with a metric which tells you how to convert coordinate differences into distances and times. Note that even these distances and times are in general "coordinate dependent" because there's more than one way to decompose spacetime into space+time. GR is rather more flexible than SR in this regard. In SR it is generally agreed that there is only one way that we usually decompose spacetime into space+time relative to a given inertial observer, but in GR there are no truly inertial coordinate systems and there's much more choice about decomposition.

    If you are simply looking at the coordinate values and ignoring the metric, interior coordinates are not the same as the "coordinates at infinity" (assuming you splice the different sections together to ensure continuity at the splicing points). In this case the interior metric will be of the form[tex]ds^2 = A^2 \, \, dt^2 - B^2 (dr^2 + r^2(d\theta^2 + \sin^2 \theta \, \, d\phi^2))[/tex] for some constants A, B chosen to get a "smooth splice", or, in Cartesian form[tex]ds^2 = A^2 \, \, dt^2 - B^2 (dx^2 + dy^2 + dz^2)[/tex]If you apply the metric, the "interior" distances will be the same as those "at infinity".
    Last edited: Oct 15, 2011
  12. Oct 15, 2011 #11
    Well, spheres made of different stuff (e.g., with different elasticity properties) will deform differently, and consequently will produce different metrics. It could be that under some assumptions the result will be independent of the matter model--but if you're asking about distortion, I doubt it. Since you seem to be wondering about some established calculation that seems to make preferred use of coordinates, maybe you could link to some reference to see how they define the problem there.
  13. Oct 15, 2011 #12


    Staff: Mentor

    If the sphere is rigid then it is not distorted by definition. There may be tidal forces acting on it, but since it is rigid that doesn't affect its shape. Particularly not if you are interested only in covariant measures from the metric and not in coordinate-dependent measures.

    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.
    Last edited: Oct 15, 2011
  14. Oct 15, 2011 #13
    Agreed that distant observer cannot physically stretch across the light years with calipers. But can directly measure redshifted frequency, and using arbitrarily powerful telescope, determine size on a comparative basis, as I outlined. This ambiguity in slicing up spacetime etc seems troublesome, as there can be no ambiguity in measured results, given the system is specified beforehand, and we have made allowances for all extraneous effects.
    That I believe pins down, according to my list in #1, your finding that for Case B: |D'/D|MM = 1. Now all we need are the other four! I'm pretty sure the redshift values will be as I expect, but will be interested in the exterior spatial values.
  15. Oct 15, 2011 #14
    I'm confused. My understanding is that by metric we understand the spacetime metric of SM or MM, owing to the relatively massive shell, in which the small test body is immersed. And that mechanical distortion of said body does not constitute some added metric, or appreciably alter the embedding metric owing to the shell. Besides, it was stipulated that mechanical distortions are either eliminated or compensated for. And one can simply apply a limiting process, ie let specific stiffness -> 'near infinite', a common enough idealization surely.
  16. Oct 15, 2011 #15
    See my remarks to Sam Gralla on that issue in #14.
    Not really. Just an accurate prediction of what the observer measures, and how obtained. Whatever combo of coordinate systems is used is up to whoever tackles it.
    Please elaborate. Just like the twin cycle clock method I outlined elsewhere, one can use differential comparison techniques to eliminate various 'ambiguities'.
  17. Oct 15, 2011 #16


    Staff: Mentor

    Sure, but then you need to specify the measurement process you are interested in. Otherwise all you have is coordinates which are completely arbitrary.

    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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