Integrating the time as a function of radius

Now let's calculate the time dependence. The same trick of dividing by the "dot" (dt ~ (tdot/rdot) dr) can be used. We can get away with only using rdotin[r] and not rdotout[r] because we'll be losing that sign difference again later:

In[125]:=

  
  dt[r_] = tdot[r]/rdotin[r]

Out[125]=

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

Now we integrate to find the time, using the parameter t0 for the time at which r = b/c:

In[126]:=

  
  t[r_] = t0 + Integrate[dt[r],r]

Out[126]=

             2    2  2
           -b  + c  r
  -(r Sqrt[-----------]) + t0
               2  2
              c  r

This relationship can be inverted so the radial coordinate r can be expressed in terms of coordinate time t:

In[127]:=

  
  tsq = t[r]^2

Out[127]=

              2    2  2
            -b  + c  r         2
  (-(r Sqrt[-----------]) + t0)
                2  2
               c  r

Now solve for r[t]:

In[128]:=

  
  R = Solve[tsq == t^2, r]

Out[128]=

                2
               b     2              2
  {{r -> -Sqrt[-- + t  - 2 t t0 + t0 ]}, 
                2
               c
   
                2
               b     2              2
    {r -> Sqrt[-- + t  - 2 t t0 + t0 ]}}
                2
               c

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Compute and solve the equations for light propagation