From: carlip@physics.ucdavis.edu   
      
   On 5/1/19 2:25 AM, p.kinsler@ic.ac.uk wrote:   
   > Steven Carlip  wrote:   
   >> Finally, as a slightly more subtle point, the two elements of general   
   >> relativity I listed aren't really independent.  It turns out that the   
   >> Einstein field equations -- the equations that determine the curvature   
   >> of spacetime in terms of its matter content -- are all you need.  If   
   >> you try to write down a solution of the field equations with two   
   >> separate lumps of matter, you find that there is no solution for which   
   >> the lumps are stationary.  Solutions *only* exist if each piece is   
   >> already moving properly in the other's gravitational field.   
      
   > Just a request for clarification - are you saying it can be proven   
   > that no such "both stationary" solutions can exist?   
      
   Yes.   
      
   There are two ways to do this.  The first is simply to solve the   
   field equations for a pair of masses, using a systematic approximation   
   procedure.  You find that the solution automatically has each mass   
   moving as expected in the gravitational field of the other.  This   
   method goes back to Einstein, Infeld, and Hoffmann in 1938, who did   
   the computation to first order in the approximation.  It's been carried   
   out to much higher orders since then.  (In fact, this is the way we   
   compute gravitational radiation from a binary system.)   
      
   Alternatively, you can look for a solution in which you *demand* that   
   both masses be stationary.  You'll find that there is necessarily   
   a singular "strut" -- a line of infinite curvature -- between the   
   masses, holding them apart.  This is not something you put in; it   
   comes out of the calculation. as a consequence of the Einstein field   
   equations.  You can do more: you can use the field equations to   
   calculate the stresses on this strut.  (The curvature is infinite,   
   but along only a single line, so averages are finite.)  If you do,   
   you find that the stresses are just those required to balance the   
   gravitational attraction.  In other words, if you demand a solution   
   with two masses both at rest, the field equations force you to add   
   in an extra strut between them to hold them apart.  I think this result   
   may go back to Weyl; I learned it from Synge's 1960 book _Relativity:   
   The General Theory_.   
      
   Steve Carlip   
      
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