Solve the Equations

In[20]:=

  
  eq = eqn[-3,-3]

Out[20]=

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

The DiracDelta function has some bugs when it is multiplied so we have to eliminate the denominators by hand.

In[21]:=

  
  eq = - 2 E^g[r] r eq[[1]] == - 2 k M DiracDelta[r] DiracDelta[theta]

Out[21]=

  g'[r] + r g''[r] == -2 k M DiracDelta[r] DiracDelta[theta]

This is the equation we need to solve. We have to integrate each side over both r and theta. The DiracDelta that Mathematica uses isn't normalized properly for an angular DiracDelta[theta], which should be normalized over the interval from 0 to 2 Pi. So while we integrate the left-hand side of the equation from 0 to 2 Pi, we have to trick the right answer out of the right hand side by using the wrong integration limit (i.e. none at all) and then just ignoring the UnitStepFunction[theta] generated thereby.

In[22]:=

  
  lhs = Integrate[eq[[1]], r, {theta,0,2 Pi}]

Out[22]=

  2 Pi r g'[r]

In[23]:=

  
  rhs = Integrate[eq[[2]],r, theta]

Out[23]=

  -2 k M UnitStep[r] UnitStep[theta]

Since the coordinate r has been defined by our choice of coordinates so that the origin of the coordinate system is at r=0, we can set the UnitStep[r] to 1 and proceed with the calculation by dividing lhs and rhs by - 2 Pi r and then integrating both sides to get g[r].

In[24]:=

  
  Integrate[- (k M/Pi) /r, r]

Out[24]=

    k M Log[r]
  -(----------)
        Pi

In[25]:=

  
  k = 8 Pi G

Out[25]=

  8 G Pi

In[26]:=

  
  g[r_] := -(k M/Pi) Log[r]

In[27]:=

  
  g[r]

Out[27]=

  -8 G M Log[r]

In[28]:=

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

Out[28]=

    -8 G M   2 - 8 G M
  {r      , r         , -1}

Remember that M is the mass of the point particle we have placed at the origin of the coordinate system in this spacetime, and G is Newton's gravitational constant, which is a constant of nature that determines the strength of the gravitational force.

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