First cell creates Riemann tensor, Ricci tensor, Ricci scalar in 5d
First cell creates Riemann tensor, Ricci tensor, Ricci scalar in 5d
In[]:=
n=5;X={t,r,ψ,θ,ϕ};gdd={{-v1*r^2,0,0,0,0},{0,v1/r^2,0,0,0},{0,0,v2,0,0},{0,0,0,v2*Power[Sin[ψ],2],0},{0,0,0,0,v2*Power[Sin[ψ],2]*Power[Sin[θ],2]}};gdd//MatrixForm;guu:=guu=Simplify[Inverse[gdd]];guu//MatrixForm;(*Christoffelsymbolshavebeencalculatedasfollows*)Christoffel=FullSimplifyTable*Sum[(guu[[i,s]])*(D[gdd[[s,j]],X[[k]]]+D[gdd[[s,k]],X[[j]]]-D[gdd[[j,k]],X[[s]]]),{s,1,n}],{i,1,n},{j,1,n},{k,1,n};listaffine:=Table[If[UnsameQ[Christoffel[[i,j,k]],0],{ToString[Γ[i,j,k]],Christoffel[[i,j,k]]}],{i,1,n},{k,1,n},{j,1,k}];TableForm[Partition[DeleteCases[Flatten[listaffine],Null],2],TableSpacing{2,2}];(*Riccitensor()havebeencalculatedasfollows*)Riemann=Simplify[Table[D[Christoffel[[i,j,l]],X[[k]]]-D[Christoffel[[i,j,k]],X[[l]]]+Sum[Christoffel[[s,j,l]]*Christoffel[[i,k,s]]-Christoffel[[s,j,k]]*Christoffel[[i,l,s]],{s,1,n}],{i,1,n},{j,1,n},{k,1,n},{l,1,n}]];Ricci=Simplify[Table[Sum[Riemann[[i,j,i,l]],{i,1,n}],{j,1,n},{l,1,n}]];Ricci//MatrixForm;(*Ricciscalariscalculatedbelow*)RR=Table[guu[[i,j]]*Ricci[[i,j]],{i,1,n},{j,1,n}];R=Simplify[Sum[RR[[i,j]],{i,1,n},{j,1,n}]];Riemann//MatrixForm;Remn=Table[Sum[gdd[[i,j]]*Riemann[[j,h]][[k,l]],{j,1,n}],{i,1,n},{h,1,n},{k,1,n},{l,1,n}];Remn//MatrixForm;
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Final answer is given as -
Final answer is given as -
Weyl4=Term1+Term2
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