In[185]:= Clear[coord,metric,inversemetric,christoffel,reimann,ricci,einstein,r,\[Phi],\[Theta],t,x,y,z] In[186]:= n=4 Out[186]= 4 In[187]:= coord={t,r,\[Theta],\[Phi]} Out[187]= {t,r,\[Theta],\[Phi]} In[188]:= metric={{-Exp[2*\[CapitalPhi][r]],0,0,0},{0,Exp[2*\[Lambda][r]],0,0},{0,0,r^2,0},{0,0,0,r^2 Sin[\[Theta]]^2}} Out[188]= {{-E^(2 \[CapitalPhi][r]),0,0,0},{0,E^(2 \[Lambda][r]),0,0},{0,0,r^2,0},{0,0,0,r^2 Sin[\[Theta]]^2}} In[189]:= metric // MatrixForm Out[189]//MatrixForm= (-E^(2 \[CapitalPhi][r]) 0 0 0 0 E^(2 \[Lambda][r]) 0 0 0 0 r^2 0 0 0 0 r^2 Sin[\[Theta]]^2 ) In[190]:= inversemetric =Simplify[Inverse[metric]] Out[190]= {{-E^(-2 \[CapitalPhi][r]),0,0,0},{0,E^(-2 \[Lambda][r]),0,0},{0,0,1/r^2,0},{0,0,0,Csc[\[Theta]]^2/r^2}} In[191]:= inversemetric // MatrixForm Out[191]//MatrixForm= (-E^(-2 \[CapitalPhi][r]) 0 0 0 0 E^(-2 \[Lambda][r]) 0 0 0 0 1/r^2 0 0 0 0 Csc[\[Theta]]^2/r^2 ) In[192]:= christoffel:= Simplify[ Table[(1/2)*Sum[inversemetric[[s,i]]*( D[metric[[j,s]],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} ] ] In[193]:= listchristoffel:= Table[If[UnsameQ[christoffel[[i,j,k]],0],{ToString[\[CapitalGamma][i-1,j-1,k-1]],christoffel[[i,j,k]]}],{i,1,n},{j,1,n},{k,1,j}] In[194]:= TableForm[Partition[DeleteCases[Flatten[listchristoffel],Null],2],TableSpacing ->{2,2}] Out[194]//TableForm= \[CapitalGamma][0, 1, 0] (\[CapitalPhi]^\[Prime])[r] \[CapitalGamma][1, 0, 0] E^(-2 \[Lambda][r]+2 \[CapitalPhi][r]) (\[CapitalPhi]^\[Prime])[r] \[CapitalGamma][1, 1, 1] (\[Lambda]^\[Prime])[r] \[CapitalGamma][1, 2, 2] -E^(-2 \[Lambda][r]) r \[CapitalGamma][1, 3, 3] -E^(-2 \[Lambda][r]) r Sin[\[Theta]]^2 \[CapitalGamma][2, 2, 1] 1/r \[CapitalGamma][2, 3, 3] -Cos[\[Theta]] Sin[\[Theta]] \[CapitalGamma][3, 3, 1] 1/r \[CapitalGamma][3, 3, 2] Cot[\[Theta]] In[195]:= riemann:=riemann=Simplify[Table[D[ christoffel[[i,j,l]], coord[[k]] ] - D[ christoffel[[i,j,k]], coord[[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}]] In[196]:= listriemann:=Table[If[UnsameQ[riemann[[i,j,k,l]],0], {ToString[R[i-1,j-1,k-1,l-1]], riemann[[i,j,k,l]]}],{i,1,n},{j,1,n},{k,1,n},{l,1,k-1}] In[197]:= TableForm[Partition[DeleteCases[Flatten[listriemann],Null],2],TableSpacing -> {2,2}] Out[197]//TableForm= R[0, 1, 1, 0] -(\[Lambda]^\[Prime])[r] (\[CapitalPhi]^\[Prime])[r]+(\[CapitalPhi]^\[Prime])[r]^2+(\[CapitalPhi]^\[Prime]\[Prime])[r] R[0, 2, 2, 0] E^(-2 \[Lambda][r]) r (\[CapitalPhi]^\[Prime])[r] R[0, 3, 3, 0] E^(-2 \[Lambda][r]) r Sin[\[Theta]]^2 (\[CapitalPhi]^\[Prime])[r] R[1, 0, 1, 0] E^(-2 \[Lambda][r]+2 \[CapitalPhi][r]) (-(\[Lambda]^\[Prime])[r] (\[CapitalPhi]^\[Prime])[r]+(\[CapitalPhi]^\[Prime])[r]^2+(\[CapitalPhi]^\[Prime]\[Prime])[r]) R[1, 2, 2, 1] -E^(-2 \[Lambda][r]) r (\[Lambda]^\[Prime])[r] R[1, 3, 3, 1] -E^(-2 \[Lambda][r]) r Sin[\[Theta]]^2 (\[Lambda]^\[Prime])[r] R[2, 0, 2, 0] (E^(-2 \[Lambda][r]+2 \[CapitalPhi][r]) (\[CapitalPhi]^\[Prime])[r])/r R[2, 1, 2, 1] (\[Lambda]^\[Prime])[r]/r R[2, 3, 3, 2] (-1+E^(-2 \[Lambda][r])) Sin[\[Theta]]^2 R[3, 0, 3, 0] (E^(-2 \[Lambda][r]+2 \[CapitalPhi][r]) (\[CapitalPhi]^\[Prime])[r])/r R[3, 1, 3, 1] (\[Lambda]^\[Prime])[r]/r R[3, 2, 3, 2] 1-E^(-2 \[Lambda][r]) In[198]:= ricci:=ricci=Simplify[Table[Sum[riemann[[i,j,i,l]],{i,1,n}],{j,1,n},{l,1,n}]] In[199]:= listricci:=Table[If[UnsameQ[ricci[[j,l]],0],{ToString[R[j-1,l-1]],ricci[[j,l]]}],{j,1,n},{l,1,n}] In[200]:= TableForm[Partition[DeleteCases[Flatten[listricci],Null],2],TableSpacing -> {2,2}] Out[200]//TableForm= R[0, 0] (E^(-2 \[Lambda][r]+2 \[CapitalPhi][r]) ((2-r (\[Lambda]^\[Prime])[r]) (\[CapitalPhi]^\[Prime])[r]+r (\[CapitalPhi]^\[Prime])[r]^2+r (\[CapitalPhi]^\[Prime]\[Prime])[r]))/r R[1, 1] ((\[Lambda]^\[Prime])[r] (2+r (\[CapitalPhi]^\[Prime])[r])-r ((\[CapitalPhi]^\[Prime])[r]^2+(\[CapitalPhi]^\[Prime]\[Prime])[r]))/r R[2, 2] E^(-2 \[Lambda][r]) (-1+E^(2 \[Lambda][r])+r (\[Lambda]^\[Prime])[r]-r (\[CapitalPhi]^\[Prime])[r]) R[3, 3] E^(-2 \[Lambda][r]) Sin[\[Theta]]^2 (-1+E^(2 \[Lambda][r])+r (\[Lambda]^\[Prime])[r]-r (\[CapitalPhi]^\[Prime])[r]) In[201]:= scalar=Simplify[Sum[inversemetric[[i,j]]ricci[[i,j]],{i,1,n},{j,1,n}]] Out[201]= (2 E^(-2 \[Lambda][r]) (-1+E^(2 \[Lambda][r])-2 r (\[CapitalPhi]^\[Prime])[r]-r^2 (\[CapitalPhi]^\[Prime])[r]^2+r (\[Lambda]^\[Prime])[r] (2+r (\[CapitalPhi]^\[Prime])[r])-r^2 (\[CapitalPhi]^\[Prime]\[Prime])[r]))/r^2 In[202]:= einstein:=einstein=Simplify[ricci-(1/2)scalar*metric] In[203]:= listeinstein:=Table[If[UnsameQ[einstein[[j,l]],0],{ToString[G[j-1,l-1]],einstein[[j,l]]}],{j,1,n},{l,1,j}] In[204]:= TableForm[Partition[DeleteCases[Flatten[listeinstein],Null],2],TableSpacing -> {2,2}] Out[204]//TableForm= G[0, 0] (E^(-2 \[Lambda][r]+2 \[CapitalPhi][r]) (-1+E^(2 \[Lambda][r])+2 r (\[Lambda]^\[Prime])[r]))/r^2 G[1, 1] (1-E^(2 \[Lambda][r])+2 r (\[CapitalPhi]^\[Prime])[r])/r^2 G[2, 2] E^(-2 \[Lambda][r]) r ((\[CapitalPhi]^\[Prime])[r]+r (\[CapitalPhi]^\[Prime])[r]^2-(\[Lambda]^\[Prime])[r] (1+r (\[CapitalPhi]^\[Prime])[r])+r (\[CapitalPhi]^\[Prime]\[Prime])[r]) G[3, 3] E^(-2 \[Lambda][r]) r Sin[\[Theta]]^2 ((\[CapitalPhi]^\[Prime])[r]+r (\[CapitalPhi]^\[Prime])[r]^2-(\[Lambda]^\[Prime])[r] (1+r (\[CapitalPhi]^\[Prime])[r])+r (\[CapitalPhi]^\[Prime]\[Prime])[r])