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How does GR handle metric transition for a spherical mass shell?

by Q-reeus
Tags: handle, mass, metric, shell, spherical, transition
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George Jones
Feb25-12, 08:59 AM
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P: 6,248
I am looking for a new post, written in succinct scientific style (without a lot commentary), of the form

"For a matter distribution given by ..., calculate ..."
Feb25-12, 10:23 AM
PF Gold
P: 706
Here's an idea for a non-stationary matter distribution:

Assume you have 5000 particles, originating from an event (t=0,x=0) and each is assigned a random x and y rapidity between -3 and 3, traveling away from each other in the xy-plane. Assume that these particles move with constant velocity until time t=1.

At time t=1, the particles spontaneously develop mass. Given one of the particles, calculate the force on this particle resulting from the other 4999 particles.

(The proper-time/coordinate-time of the spontaneous development of mass, and the delay before that mass is detected, would need to be more fully described to answer the question.)
Feb25-12, 02:15 PM
P: 17,530
Quote Quote by George Jones View Post
I'm having trouble tracing this back. What, precisely (and succinctly), is the matter distribution in question.
The details are a little fuzzy, but if I recall correctly Q-reeus was claiming that GR was not self consistent because of the transition from the Schwarzschild metric outside a solid shell to the flat metric inside the solid shell.

Somehow the discussion became focused on the space-space components of the (Ricci) curvature tensor and the space-space components of the stress-energy tensor. Q-reeus though you could neglect the space-space components of the stress-energy tensor simply because the time-time component was larger (which somehow led to a contradiction, though I don't remember how).

So I was going to calculate the metric for a finite-thickness solid shell and show that the space-space components of the stress-energy tensor could not be neglected and that accounting for them resolved the supposed contradictions.
Feb25-12, 06:35 PM
Sci Advisor
P: 7,659
At the risk of hijacking the thread (which seems hopelessly confused anyway), I do have a specific problem along the lines of boundary conditions - one I think I solved correctly, that I presented earlier. I think there was some questions raised about it, but I didn't follow the questions.

If we consider one of the simplest possible forms for the interior metric of a photon star,


eq (1)

[tex]\frac{7}{4}\, dr^2 + r^2 \,d \theta^2 + r^2 sin^2 \theta \, d\phi^2 - \sqrt{\frac{7}{3}}\,r\,dt^2[/tex]

we might ask how do we go about enclosing said interior metric in a thin, massless shell, so we get the exterior Schwarzschild metric. I.e. how do we match up the exterior and interior Schwarzschild soultions at the boundary so that we have a solution for light in a spherical "box" by matching the interior solution given by (1) to some exterioor Schwarzschild solution.

I started with the line element from Wald for the spherically symmetric metric:


-f(r)\,dt^2 + h(r)\,dr^2 + r^2 \left(d \theta ^2 + sin \, \theta \: d\phi^2 \right)

Einsteins' equations give via equations 6.2.3 and 6.2.4 from Wald, General Relativity

[tex]8 \, \pi \rho = \frac{ \left( dh/dr \right) }{r \, h^2} + \frac{1}{r^2} \left( 1 - \frac{1}{h} \right) \; = \; \frac{1}{r^2} \frac{d}{dr} \left[r \, \left(1 - \frac{1}{h} \right) \right]

[tex]8 \, \pi \, P = \frac{ \left( df/dr \right) } {r \, f \, h} - \frac{1}{r^2} \left( 1 - \frac{1}{h} \right) [/tex]

Here [itex]\rho[/itex] and P are the density and pressure in the spherical shell.

Setting [itex]\rho[/itex] to zero and using 6.2.3 immediately tells us that r (1 - 1/h) is constant through the shell. For a thin shell, this means that h is the same inside the shell and outside the shell, because r is the same at the interior of the shell and the exterior of the shell, so h-, h inside the shell, equals h+, h outside the shell.

We can add 6.2.3 and 6.2.4 together to get

[tex]8 \pi \left(\rho + P \right) \; = \; \frac{ \left(dh/dr \right) } {r h^2 }+ \frac{ \left( df/dr \right) } {r \, f \, h} \; = \; \left( \frac{1}{ r \, f \, h^2 } \right) \, \frac{d}{dr} \left[ f \, h \right] [/tex]

So we can see that the product (f * h) can't be constant through the shell. So, we known that the right boundary conditions are that h is constant and f varies. In a shell of finite thickness, f will increase continuously throughout the shell. As we shrink it to zero width, f jumps discontinuously.

Simply put, for a _massless_ shell, we can say that the spatial curvature coefficient, h, is the same inside the shell and outside. This is a consequence of Einstein's equations. While h is constant, f, the time dilation metric coefficient, is NOT constant. This also follows from Einstein's equations.

We can do some more computation and find the exterior metric if we assume that the boundary of the shell is located at r=1. (It turns out we can place it wherever we like).

Then, the metric previously given in (1) is used for r<1, and for r> 1, we use

\frac{dr^2}{1-\frac{3}{7r}}+ r^2 \,d \theta^2 + r^2 sin^2 \theta \, d\phi^2 - \left( 1-\frac{3}{7\,r} \right) dt^2

We can do some more interesting stuff along the lines of comparing the Komar mass to the Schwarzschild mass parameter, but I think it suffices to say that the two agree for the total mass M as judged by the observer in asymptotically flat space-time, but are distrubuted differently in the interior.
Feb26-12, 01:03 AM
P: 1,115
Quote Quote by pervect View Post
At the risk of hijacking the thread (which seems hopelessly confused anyway),...
Feel free to 'hijack' this confused thread, which evidently has kicked back into life, and hopefully end the confusion.
Won't comment on the specifics of your photon gas inside a containing shell model, other than to say that there shell self-gravitation as contribution to it's own stress seems to be, understandably, an entirely absent factor. In my scenario, it is the only contribution. If you go back to #1 hopefully my problem statement is made clear enough, and basically what DaleSpam said in #201 sums it up. Again, the specific model settled on was in #17, but that can be obviously generalized.
I walked away from this thread owing to a general failure to get agreement on being able to apply differential length, radial (dr) vs azimuthal (rdΩ), in coordinate measure, as suggested by the standard Schwarzschild coordinates. As expressed in earlier entries, I accept boundary matching from exterior to interior regions is always possible mathematically. Whether that math is properly based on a physical principle (and I was genuinely shocked when it was claimed shell stresses for an entirely self-gravitating shell would do that trick) is another matter!

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