Compute the Killing vectors

Such symmetries in the metric lead to objects called Killing vectors, and there will be one Killing vector for each such symmetry. Finding these Killing vectors is important because they can be used to greatly simplify the computation of the null geodesic paths of light in this spacetime. We'll calculate the Killing vectors for this spacetime using the Killing equation DuKv + Dvku = 0. Here are the vector definitions:

In[56]:=

  
  DefineTensor[K,"K",{{1},1}]

  PermWeight::def: Object K defined

In[57]:=

  
  K[la]

Out[57]=

  K
   a

In[58]:=

  
  K[1] = kr[r]
  K[2] = ktheta[r]
  K[3] = kt[t]

Out[58]=

  kr[r]

Out[59]=

  ktheta[r]

Out[60]=

  kt[t]

In[61]:=

  
  K[-1] = MakeSum[Metricg[-1,la] K[ua]]

Out[61]=

  kr[r]

In[62]:=

  
  K[-2] = MakeSum[Metricg[-2,la] K[ua]]

Out[62]=

             2  2
  (1 - 4 G M)  r  ktheta[r]

In[63]:=

  
  K[-3] = MakeSum[Metricg[-3,la] K[ua]]

Out[63]=

  -kt[t]

In[64]:=

  
  DefineTensor[Kterm,"Kterm",{{2,1},1}]

  PermWeight::sym: Symmetries of Kterm assigned

  PermWeight::def: Object Kterm defined

In[65]:=

  
  DefineTensor[Kterm2,"Kterm2",{{2,1},1}]

  PermWeight::sym: Symmetries of Kterm2 assigned

  PermWeight::def: Object Kterm2 defined

Now we compute the derivative combination Du kv + Dv Ku that appears in the Killing equation:

In[66]:=

  
  Kterm[la_,lb_] = CDtoOD[CD[K[la],lb] + CD[K[lb],la]]

Out[66]=

      p
  -2 G    K  + K    + K
       ab  p    a,b    b,a

In[67]:=

  
  On[EvaluateODFlag]

In[68]:=

  
  Kterm2[la_,lb_] := Simplify[MakeSum[Kterm[la,lb]]]

In[69]:=

  DefineTensor[Keqn,"Keqn",{{2,1},1}]
  

  PermWeight::sym: Symmetries of Keqn assigned

  PermWeight::def: Object Keqn defined

Now we take that derivative term we just calculated and set it to zero to solve for the Killing vectors:

In[70]:=

  
  Keqn[la_,lb_] := Kterm2[la,lb] == 0

Here is the list of differential equations we need to solve to get the Killing vectors:

In[71]:=

  
  Table[Keqn[-i,-j],{i,1,3},{j,1,3}]

Out[71]=

                              2  2
  {{2 kr'[r] == 0, (1 - 4 G M)  r  ktheta'[r] == 0, True}, 
   
                2  2
    {(1 - 4 G M)  r  ktheta'[r] == 0, 
   
                  2
     2 (1 - 4 G M)  r kr[r] == 0, True}, 
   
    {True, True, -2 kt'[t] == 0}}

These equations can be solved by inspection to yield:

In[72]:=

  
  kr[r_] = 0
  kt[r_] = c1
  ktheta[r_] = c2

Out[72]=

  0

Out[73]=

  c1

Out[74]=

  c2

In[75]:=

  
  Table[K[i],{i,3}]

Out[75]=

  {0, c2, c1}

The Killing vectors for this spacetime are any linear combination with constant coefficients of a unit vector in the time direction and a unit vector in the theta direction.

The two symmetries in this spacetime are time translation and rotational invariance. We will use these symmetries and these Killing vectors to speed up the process of solving the geodesic equations and calculate the paths of rays of light around the mass M.

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