From: carlip@physics.ucdavis.edu   
      
   On 6/14/18 10:37 AM, 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 is a bit tricky, for a good reason.  One of the nice features   
   of general relativity is that you can't simply stick in sources   
   at arbitrary positions and expect them to stay there.  The Einstein   
   field equations only have solutions if gthe sources are obeying   
   their own equations of motion.   
      
   For a thick slab, gravitational attraction will tend to make the   
   material collapse.  The only way to avoid this is by having very   
   large stresses, comparable in magnitude to the energy density.   
      
   The one reference I know of for a finite plate is Horsky and Novotny,   
   J. Phys. A 2 (1969) 251.   
      
   Steve Carlip   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)