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<< Notation` (*Convenient to add the notation package*)
Symbolize[NotationTemplateTag[Subscript[r, s]]]
Clear[\[GothicCapitalM], Metric, InvMetric,
CHFS, RMT, RT, RS, ET,ETT, r,\[Theta],\[Phi],t];
\[GothicCapitalM] = {t,r,\[Theta],\[Phi]}; n = 4;
f[r_]:= 1- ( 2 M r^2)/(r^3 + 2 M L^2)
\[ScriptM][_u]:= \[ScriptM][u]
SchMetric = {{-(1- (Subscript[r, s] / r)),0,0,0},
               {0,(1 - (Subscript[r, s]  / r) )^(-1),0,0},
               {0,0,r^(2) ,0}, {0,0,0, r^(2) Sin[\[Theta]]^2}};


Metric = FullSimplify[SchMetric];
InvMetric = FullSimplify[Inverse@Metric]; MatrixForm[Metric]
CHFS = FullSimplify[Table[(1/2)*Sum[(InvMetric[[\[Lambda], \[Alpha]]])*
       (D[Metric[[\[Mu], \[Alpha]]], \[GothicCapitalM][[\[Nu]]] ] +
         D[Metric[[\[Nu], \[Alpha]]], \[GothicCapitalM][[\[Mu]]] ] -
         D[Metric[[\[Mu], \[Nu]]], \[GothicCapitalM][[\[Alpha]]] ]), {\[Alpha], 1, n}],
{\[Lambda], 1, n}, {\[Mu], 1, n}, {\[Nu], 1, n}]];
    RMT = FullSimplify[Table[D[CHFS[[\[Kappa], \[Lambda], \[Nu]]], \[GothicCapitalM][[\[Mu]]] ] -
     D[CHFS[[\[Kappa], \[Lambda], \[Mu]]], \[GothicCapitalM][[\[Nu]]] ] +
     Sum[CHFS[[\[Alpha], \[Lambda], \[Nu]]] CHFS[[\[Kappa], \[Mu], \[Alpha]]] -
       CHFS[[\[Alpha], \[Lambda], \[Mu]]] CHFS[[\[Kappa], \[Nu], \[Alpha]]],
      {\[Alpha], 1, n}], {\[Kappa], 1, n}, {\[Lambda], 1, n}, {\[Mu], 1, n}, {\[Nu], 1, n}]];


RT = FullSimplify[Table[Sum[RMT[[\[Alpha], \[Mu], \[Alpha], \[Nu]]], {\[Alpha], 1, n}], {\[Mu], 1, n}, {\[Nu], 1, n}]];
(*Print the Einstein tensor components*)
RS = FullSimplify[Sum[InvMetric[[\[Alpha], \[Beta]]] RT[[\[Alpha], \[Beta]]], {\[Alpha], 1, n}, {\[Beta], 1, n}]]
RMTIndicesDown = Simplify[Table[
Sum[Metric[[\[Kappa], \[Kappa]1]]*RMT[[\[Kappa]1, \[Lambda], \[Mu], \[Nu]]], {\[Kappa]1, 1, n}], {\[Kappa], 1, n}, {\[Lambda], 1, n},
{\[Mu], 1, n}, {\[Nu], 1, n}]];
RMTIndicesUp = Simplify[Table[
Sum[Sum[Sum[InvMetric[[\[Lambda]1, \[Lambda]]]*InvMetric[[\[Mu]1, \[Mu]]]*InvMetric[[\[Nu]1, \[Nu]]]*
        RMT[[\[Kappa], \[Lambda]1, \[Mu]1, \[Nu]1]], {\[Lambda]1, 1, n}], {\[Mu]1, 1, n}], {\[Nu]1, 1, n}], {\[Kappa], 1,  n}, {\[Lambda], 1, n}, {\[Mu], 1, n}, {\[Nu], 1, n}]];


(*We'll print the Ricci tensor components*)

ET = FullSimplify[RT - ((1/2) Metric  RS)];
(*Print the trace of energy momentum tensor*)
EMT = FullSimplify[RT - ((1/2)Metric  RS) / (8 \[Pi] G ) ];
(*Print the trace of Einstein equation*)
ETT = FullSimplify[Sum[ InvMetric[[\[Alpha], \[Beta]]] RT[[\[Alpha], \[Beta]]], {\[Alpha], 1, n}, {\[Beta], 1, n}]- (1/2) Sum[InvMetric[[\[Alpha], \[Beta]]] Metric[[\[Alpha], \[Beta]]],{\[Alpha], 1, n}, {\[Beta], 1, n} ] RS ]

KretschmannScalar = FullSimplify[Sum[
   RMTIndicesDown[[\[Kappa], \[Lambda], \[Mu], \[Nu]]]*RMTIndicesUp[[\[Kappa], \[Lambda], \[Mu], \[Nu]]], {\[Kappa], 1, n}, {\[Lambda], 1,
    n}, {\[Mu], 1, n}, {\[Nu], 1, n}]]


ListCHFS := Table[If[UnsameQ[CHFS[[\[Lambda], \[Mu], \[Nu]]],0],
        {ToString[\[CapitalGamma][\[GothicCapitalM][[\[Lambda]]], \[GothicCapitalM][[\[Mu]]], \[GothicCapitalM][[\[Nu]]]]],
        CHFS[[\[Lambda], \[Mu], \[Nu]]]}], {\[Lambda], 1, n}, {\[Mu], 1, n}, {\[Nu], 1, \[Mu]}]
TableForm[Partition[DeleteCases[Flatten[ListCHFS], Null], 2],
  TableSpacing -> {2, 2}]

ListRMT := Table[If[UnsameQ[RMT[[\[Kappa], \[Lambda], \[Mu], \[Nu]]], 0],
         {ToString[R[\[GothicCapitalM][[\[Kappa]]], \[GothicCapitalM][[\[Lambda]]], \[GothicCapitalM][[\[Mu]]], \[GothicCapitalM][[\[Nu]]]]],
        RMT[[\[Kappa], \[Lambda], \[Mu], \[Nu]]]}] , {\[Kappa], 1, n}, {\[Lambda], 1, n}, {\[Mu], 1,
      n}, {\[Nu], 1, \[Mu] - 1}]
TableForm[Partition[DeleteCases[Flatten[ListRMT], Null], 2],
  TableSpacing -> {2, 2}]


ListRT := Table[If[UnsameQ[RT[[\[Mu], \[Nu]]], 0], {ToString[R[\[GothicCapitalM][[\[Mu]]], \[GothicCapitalM][[\[Nu]]]]],
        RT[[\[Mu], \[Nu]]]}] , {\[Mu], 1, n}, {\[Nu], 1, \[Mu]}]
TableForm[Partition[DeleteCases[Flatten[ListRT], Null], 2],
  TableSpacing -> {2, 2}]

  ListET := FullSimplify[Table[If[UnsameQ[ET[[\[Mu], \[Nu]]], 0], {ToString[G[\[GothicCapitalM][[\[Mu]]], \[GothicCapitalM][[\[Nu]]]]], ET[[\[Mu], \[Nu]]]}], {\[Mu], 1, n}, {\[Nu], 1, \[Mu]}]]
TableForm[Partition[DeleteCases[Flatten[ListET], Null], 2], TableSpacing -> {2, 2}]

ListEMT := FullSimplify[Table[If[UnsameQ[EMT[[\[Mu], \[Nu]]], 0], {ToString[T[\[GothicCapitalM][[\[Mu]]], \[GothicCapitalM][[\[Nu]]]]], EMT[[\[Mu], \[Nu]]]}], {\[Mu], 1, n}, {\[Nu], 1, \[Mu]}]]
TableForm[Partition[DeleteCases[Flatten[ListEMT], Null], 2], TableSpacing -> {2, 2}]