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How does GR handle metric transition for a spherical mass shell? 
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#199
Feb2512, 08:59 AM

Mentor
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 ..." 


#200
Feb2512, 10:23 AM

PF Gold
P: 706

Here's an idea for a nonstationary 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 xyplane. 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 propertime/coordinatetime 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.) 


#201
Feb2512, 02:15 PM

Mentor
P: 17,530

Somehow the discussion became focused on the spacespace components of the (Ricci) curvature tensor and the spacespace components of the stressenergy tensor. Qreeus though you could neglect the spacespace components of the stressenergy tensor simply because the timetime 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 finitethickness solid shell and show that the spacespace components of the stressenergy tensor could not be neglected and that accounting for them resolved the supposed contradictions. 


#202
Feb2512, 06:35 PM

Emeritus
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, from http://arxiv.org/abs/grqc/9903044 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: eq(2) [tex] f(r)\,dt^2 + h(r)\,dr^2 + r^2 \left(d \theta ^2 + sin \, \theta \: d\phi^2 \right) [/tex] 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] [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 [tex] \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 [/tex] 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 spacetime, but are distrubuted differently in the interior. 


#203
Feb2612, 01:03 AM

P: 1,115

Won't comment on the specifics of your photon gas inside a containing shell model, other than to say that there shell selfgravitation 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 selfgravitating shell would do that trick) is another matter! 


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