From: goldenfieldquaternions@gmail.com   
      
   On Thursday, June 14, 2018 at 3:37:33 AM UTC-5, Michael Cole wrote:   
   > I have been exploring a fun GR problem.  Consider a uniform planar   
   > mass distribution in the xy plane.  The spacetime metric will only   
   > depend on z.  An elementary argument shows that the metric has the   
   > form g_xx = g_yy = A(z), g_zz = B(z), and g_tt = -C(t).  By a change   
   > of the z coordinate one can simplify by setting B(z) = 1.  It is   
   > not a hard exercise to work out the Ricci tensor, set it equal to   
   > 0, and solve for the functions A and C.  The derivation is similar   
   > to but somewhat simpler than the derivation of the Schwartzchild   
   > metric.  The metric is singular in the xy plane and there is an   
   > analogue of the Schwartzchild radius.   
   >   
   > I am trying to come up with a model for a slab of matter that exists   
   > for -h < z < h.  Does anyone have thoughts for a physically realistic   
   > stress energy tensor that would describe, say, an infinite planar   
   > slab of gravitating fluid.  Perhaps it is absurd to use the phrase   
   > "physically realistic" for this math exercise, but I find the problem   
   > interesting.  There are a couple papers on the idealized massive   
   > plane of thickness 0, but not many.   After all, nature doesn't   
   > present us with infinite massive planes.  But anyways, what would   
   > be a suitable stress energy tensor for a thick slab?   
      
   This appears to be similar to the Kasner spacetime. Look this up in   
   Misner, Thorne and Wheeler book Gravitation.   
      
   LC   
      
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