The curvature side of the equations

Our chosen spacetime has three dimensions. We will label them with coordinates r, theta and t, where r and theta are like the ordinary ploar coordinates and t is time. We prepared a file for MathTensor call "3dIn.m" where we specified the coordinates and the form we expect the metric to take, accounting for the rotational and time-translation symmtery of our point mass. This input file was fed at an earlier date into the "Components" function of MathTensor, where the components of the curvature tensor were calculated and written into the file being loaded below:

In[10]:=

  
  <<"Library/Mathematica/MathTensor/3dOut.m"

  MetricgFlag has been turned off.

Here is the form of the metric we started with. The function we will need to calculate is the g[r] inside the exponential functions. We are starting with these exponential functions of g[r] because starting this way makes calculating g[r] much faster. (Based on my own experience.)

In[11]:=

  Table[Metricg[-i,-i],{i,1,3}]

Out[11]=

    g[r]   g[r]  2
  {E    , E     r , -1}

Here is the set of expressions for the "curvature side" of the Einstein equation. Notice how many are nonzero:

In[12]:=

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

Out[12]=

                                -(g'[r] + r g''[r])
  {{0, 0, 0}, {0, 0, 0}, {0, 0, -------------------}}
                                        g[r]
                                     2 E     r

Out of the six possible independent terms we are left with just one equation, as we predicted above.

In[13]:=

  
  eqn[-3,-3]

Out[13]=

  -(g'[r] + r g''[r])
  ------------------- == k T
          g[r]              33
       2 E     r

Now we have to decide what the stress tensor is for this system.

Up to

Compute the equations