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What I'm trying to do is to compute the Killing equations and then solve them using DSolve to determine their solutions. I have a problem understanding where my Mathematica program doesn't work properly as it seems like it doesn't substitute \ [Micro], \ [Nu] and \ [Delta] properly... For simplicity I adapted my program to compute the Killing equations for a 2-sphere where the Killing equations are well known.

The general expression for the Killing equations I decided to work with is of the form:
killing eq.

The equations that I should obtain for the 2-sphere should be of the form:

killing eq.2

    (*coordinates initialisation*)
    Clear[coord, metric, inversemetric, affine, killing, t, r, \[Theta], \
    \[Phi]]



    n = 2


 (*The metric and inverse metric*)
    coord = {\[Theta], \[Phi]}
    metric = {{1, 0}, {0, sin[\[Theta]]*sin[\[Theta]]}}

    inversemetric = Simplify[Inverse[metric]]




 (*Calculating the Christoffel symbols*)
    affine := 
     affine = Simplify[
       Table[(1/2)*
         Sum[(inversemetric[[i, s]])*(D[metric[[s, j]], coord[[k]]] + 
             D[metric[[s, k]], coord[[j]]] - 
             D[metric[[j, k]], coord[[s]]]), {s, 1, n}], {i, 1, n}, {j, 1,
          n}, {k, 1, n}]]



 (*Dysplaying non-null Christoffel symbols*)
    listaffine := 
     Table[If[UnsameQ[affine[[i, j, k]], 
        0], {ToString[\[CapitalGamma][i, j, k]], affine[[i, j, k]]}], {i, 
       1, n}, {j, 1, n}, {k, 1, n}]
    TableForm[Partition[DeleteCases[Flatten[listaffine], Null], 2], 
     TableSpacing -> {2, 2}]








 (*Killing equations*)
    killing := 
     killing = 
      Simplify[Table[
        D[Subscript[\[Xi], coord[[\[Micro]]]][\[Theta], \[Phi]], 
          coord[[\[Nu]]]] + 
         D[Subscript[\[Xi], coord[[\[Nu]]]][\[Theta], \[Phi]], 
          coord[[\[Micro]]]] - 
         2*affine[[\[Delta], \[Micro], \[Nu]]]*
          Subscript[\[Xi], 
           coord[[\[Delta]]]][\[Theta], \[Phi]], {\[Micro], 1, n}, {\[Nu],
          1, n}, {\[Delta], 1, n}]]


(*Dysplaying Killing equations*)
    listkilling := 
     Table[If[UnsameQ[killing[[\[Micro], \[Nu], \[Delta]]], 
        0], {killing[[\[Micro], \[Nu], \[Delta]]], "=  0"}], {\[Micro], 1,
        n}, {\[Nu], 1, n}, {\[Delta], 1, n}]
    TableForm[Partition[DeleteCases[Flatten[listkilling], Null], 2], 
     TableSpacing -> {2, 2}]

The results my code gives are of the form:

killing eq. 3 where I believe there are some obvious differences. First of all there is a suplementary equation (the third one displayer above) and in the first two the derivative si carried for both coordinates instead of only \ [Theta]. Any opinions? Thank you in advance!

PS: How can I get rid of that annoying 'Subscript' and display it properly?

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  • $\begingroup$ I find it somewhat odd that you use \[Micro] instead of \[Mu]... Moreover affine := affine = ... is odd, to. Why don't you just use affine = ...? $\endgroup$ – Henrik Schumacher Dec 27 '18 at 9:39
  • $\begingroup$ @HenrikSchumacher \ [Micro] is equivalent to \ [Mu] it seems so it doesn't bother me. I used := as I previously got some errors. In this way the defined function/equation remains unevaluated until I substitute the subsripts/ other parameters. $\endgroup$ – Vlad G Dec 27 '18 at 14:50
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(*coordinates initialisation*)Clear[coord, metric, inversemetric, \
affine, killing, t, r, \[Theta], \[Phi]]
n = 2
(*The metric and inverse metric*)
coord = {\[Theta], \[Phi]}
metric = {{1, 0}, {0, Sin[\[Theta]]*Sin[\[Theta]]}}

inversemetric = Simplify[Inverse[metric]]
(*Calculating the Christoffel symbols*)
affine := 
 affine = Simplify[
   Table[(1/2)*
     Sum[(inversemetric[[i, s]])*(D[metric[[s, j]], coord[[k]]] + 
         D[metric[[s, k]], coord[[j]]] - 
         D[metric[[j, k]], coord[[s]]]), {s, 1, n}], {i, 1, n}, {j, 1,
      n}, {k, 1, n}]]
(*Dysplaying non-null Christoffel symbols*)listaffine := 
 Table[If[UnsameQ[affine[[i, j, k]], 
    0], {ToString[\[CapitalGamma][i, j, k]], affine[[i, j, k]]}], {i, 
   1, n}, {j, 1, n}, {k, 1, n}]
TableForm[Partition[DeleteCases[Flatten[listaffine], Null], 2], 
 TableSpacing -> {2, 2}]
(*Killing equations*)
killing := 
 killing = 
  Simplify[Table[
    D[Subscript[\[Xi], \[Micro]][\[Theta], \[Phi]], coord[[\[Nu]]]] + 
     D[Subscript[\[Xi], \[Nu]][\[Theta], \[Phi]], 
      coord[[\[Micro]]]] - 
     2*affine[[\[Delta], \[Micro], \[Nu]]]*
      Subscript[\[Xi], \[Delta]][\[Theta], \[Phi]], {\[Micro], 1, 
     n}, {\[Nu], 1, n}, {\[Delta], 1, n}]]


(*Dysplaying Killing equations*)listkilling := 
 Table[If[UnsameQ[killing[[\[Micro], \[Nu], \[Delta]]], 
    0], {killing[[\[Micro], \[Nu], \[Delta]]], "=  0"}], {\[Micro], 1,
    n}, {\[Nu], 1, n}, {\[Delta], 1, n}]
TableForm[Partition[DeleteCases[Flatten[listkilling], Null], 2], 
 TableSpacing -> {2, 2}]

fig1

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  • $\begingroup$ Yes! Thank you sir, I must recognize that I wanted to put that coord[[ ]] in my subscripts but it messed up the things. I am still puzzled by the 3rd and 5th equation. I did the calculations by myself/by hand and substituting the subscripts indeed gives this equation too though non of the referrences I worked with mentioned a 4th equation. Thank you again!!! $\endgroup$ – Vlad G Dec 27 '18 at 14:53
  • $\begingroup$ You're welcome! $\endgroup$ – Alex Trounev Dec 27 '18 at 14:55

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