The stress-energy side of the equations

What is the stress tensor for a single static point mass? The distribution that describes things that exist ONLY at a single point in space is the Dirac delta function, which is defined in Mathematica in an external package that we have to load here:

In[14]:=

  <</LocalLibrary/Mathematica/Packages/Calculus/DiracDelta.m

The correct normalization of this function is important. We have to get back unity when it is integrated over the spacetime volume. But the spacetime volume is measured by the square root of the absolute value of the determinant of the spacetime metric. Therefore the Dirac delta function has to be divided by the square root of the absolute value of the determinant of the spacetime metric. Here is the determinant of the spacetime metric:

In[15]:=


  Detg

Out[15]=

     2 g[r]  2
  -(E       r )

So the stress tensor will have all zero components except for:

In[16]:=


  T[i_,j_] = If[(-i==3&&-j==3),
M DiracDelta[r] DiracDelta[theta] / (r E^g[r]), 0]

Out[16]=

                         M DiracDelta[r] DiracDelta[theta]
  If[-i == 3 && -j == 3, ---------------------------------,
                                         g[r]
                                      r E

    0]

Here is the whole tensor as a list of components in our chosen coordinate system:

In[17]:=


  Table[T[-m,-n],{m,3},{n,3}]

Out[17]=

  {{0, 0, 0}, {0, 0, 0}, {0, 0,

     M DiracDelta[r] DiracDelta[theta]
     ---------------------------------}}
                   g[r]
                  E     r

And here is the component of this tensor that will be relevant for determining the function g[r] in our spacetime metric.

In[18]:=


  T[-3,-3]

Out[18]=

  M DiracDelta[r] DiracDelta[theta]
  ---------------------------------
                g[r]
               E     r

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Compute the equations