Integrate the angle as a function of radius

Notice our good fortune above: tdot = 1 (when energy is absorbed into the impact parameter, otherwise we'd have tdot =E, which is really just as good only not as neat looking).

This means that the time coordinate t tells us exactly what the affine parameter v is along photon paths. (Note -- this is not the same as saying that the coordinate time t measures "proper time" along these paths.) So now we can eliminate the rest of our dependence on the affine parameters as follows:

In[106]:=

  
  Simplify[thdot[r]/rdotout[r]]

Out[106]=

            b
  ---------------------
                2
   2           b      2
  c  Sqrt[1 - -----] r
               2  2
              c  r

In[107]:=

  
  Integrate[Simplify[thdot[r]/rdotout[r]],r]

Out[107]=

            b
    ArcSin[---]
           c r
  -(-----------)
         c

In[108]:=

  
  Integrate[Simplify[thdot[r]/rdotin[r]],r]

Out[108]=

          b
  ArcSin[---]
         c r
  -----------
       c

ArcCos[b/cr] only differs by Pi/2 from ArcSin[b/cr]. Note that if we choose ArcCos[b/cr] for the solution, the constant theta0 is then angle at which the test particle reaches its minimum distance r0 = b/c from the point mass.

But we need two functions for theta, one from integrating using rdotin[r] and one for using rdotout[r]:

In[109]:=

  
  theta1[r_,theta0_] = theta0 + ArcCos[b/(c r)]/c

Out[109]=

                   b
           ArcCos[---]
                  c r
  theta0 + -----------
                c

In[110]:=

  
  theta2[r_,theta0_] = theta0 - ArcCos[b/(c r)]/c

Out[110]=

                   b
           ArcCos[---]
                  c r
  theta0 - -----------
                c

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Compute and solve the equations for light propagation