Look at the full set of equations

Here is a table of all of the equations represented by the "Einstein Equation". Note that all of the below equations are automatically satisfied except for a single differential equation, which we must solve to get the function g[r] that determines the spacetime metric.

In[19]:=

  
  Table[eqn[-i,-j],{i,3},{j,3}]

Out[19]=

  {{True, True, True}, {True, True, True}, 
   
                 -(g'[r] + r g''[r])
    {True, True, ------------------- == 
                         g[r]
                      2 E     r
   
      k M DiracDelta[r] DiracDelta[theta]
      -----------------------------------}}
                     g[r]
                    E     r

The fact that all of the equations are satisfied except for the one we need to solve means that our original choice of coordinates based on the rotational and time translation symmetries was consistent with the stress tensor used above. A lot of nasty work in physics can be avoided by understanding the symmetries of the problem before doing the calculations! Same goes in life, I suppose. :-)

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Compute the equations