waitedavid137
Here's an exact vacuum solution I finally found for the space around a thin infinite sheet of matter in the x-y plane whose pressure terms along x and y are equal and whose values are arbitrary being proportional to k. I've had other solutions corresponding to different pressure states, but this ones new.
User posted image: User posted image
rpenner
Do you think you could demonstrate that it is a vacuum solution with limiting case the Minkowski space-time or is this just going to be a sterile post in a thread that goes nowhere?
waitedavid137
QUOTE (rpenner+Apr 24 2012, 05:43 PM)
Do you think you could demonstrate that it is a vacuum solution with limiting case the Minkowski space-time or is this just going to be a sterile post in a thread that goes nowhere?

Pretty sure no one else really has anything to say about it either way. The way to verify that it is an exact vacuum solution is to get the Einstein tensor and see that its zero and as I've suggested in the past the easiest way to do this is to get grtensorii or other tensor calculus software that can do it and enter the metric. If you're having trouble entering that many terms correctly I'll send you the metric in spreadsheet format so you can copy and paste. This solution HAS spacetime curvature so there is no coordinate transformation that exists that globally reduces it to diag(1,-1,-1,-1). I expressed it in coordinates for which it only immediately reduces to that at the origin in the case k=1, otherwise you would still have to do a local coordinate transformation to get it to reduce to diag(1,-1,-1,-1) there and that you can always do a local coodinate transformation to diag(1,-1,-1,-1) actually because there is spacetime curvature tells you nothing about whether its vacuum there. You have to get the Einstein tensor.

I should mention if you do it with the absolute values for grtensorii it will give you signum functions for the energy density and two pressure terms. The signum functions are delta functions whitch are zero off the wall. If you just drop the absolute values it will return all components zero for the Einstein tensor and you can compute the Riemann tensor to see that there is spacetime curvature.
waitedavid137
Heres a short video I did talking about what led to it and about a couple of other domain walls.