g = ({
       {-Exp[2 α[r]], 0, 0, 0},
       {0, Exp[2 β[r]], 0, 0},
       {0, 0, r^2, 0},
       {0, 0, 0, r^2 Sin[θ]^2}
      }) ; T = ({
       {0, 0, 0, 0},
       {0, 0, 0, 0},
       {0, 0, 0, 0},
       {0, 0, 0, 0}
      }); var = {t, r, θ, ϕ};
    h = Inverse[g]; Print["Inverse Metric=" MatrixForm[h]];

    Print[Style[
      "Christoffel Symbol", {Bold, Underlined, Larger}]]; Γ[
      i_, j_, k_] :=
     Sum[1/2 h[[i,
        l]] (D[g[[k, l]], var[[j]]] + D[g[[j, l]], var[[k]]] -
         D[g[[k, j]], var[[l]]]), {l, 1, 4}]; i := 1; j := 1; k := 1;
    For[i = 1, i <= 4, i++,
      For[j = 1, j <= 4, j++,
       For[If[j < k, k = j], k <= 4, k++,
        If[Γ[i, j, k] =!=  0,
         Print[Subscript[Γ, var[[j]], var[[k]]]^var[[i]],
          "=", FullSimplify[Γ[i, j, k]]]]]]];

    Print[Style["Riemann Tensor", {Bold, Underlined, Larger}]];
    RiemannTensor[a_, b_, c_,
      d_] := (D[Γ[a, d, b],
        var[[c]]]) - (D[Γ[a, c, b],
        var[[d]]]) + Γ[a, c, 1] Γ[1, d,
        b] + Γ[a, c, 2] Γ[2, d,
        b] + Γ[a, c, 3] Γ[3, d,
        b] + Γ[a, c, 4] Γ[4, d,
        b] - Γ[a, d, 1] Γ[1, b,
        c] - Γ[a, d, 2] Γ[2, b,
        c] - Γ[a, d, 3] Γ[3, b,
        c] - Γ[a, d, 4] Γ[4, b,
        c]; a := 1; b := 1; c := 1; d := 1;
    For[a = 1, a <= 4, a++,
      For[b = 1, b <= 4, b++,
       For[c = 1, c <= 4, c++,
        For[If[c < d, d = c], d <= 4, d++,
         If[RiemannTensor[a, b, c, d] =!= 0,
          Print[Subscript[R^var[[a]], var[[b]], var[[c]], var[[d]]], "=",
           FullSimplify[RiemannTensor[a, b, c, d]]]]]]]];

    Print[Style["Ricci Tensor", {Bold, Underlined, Larger}]];
    RicciTensor[m_, n_] :=
     Sum[RiemannTensor[soma, m, soma, n], {soma, 1, 4}]; m := 1; n := 1
    For[m = 1, m <= 4, m++,
      For[If[m < n, n = m], n <= 4, n++,
       If[RicciTensor[m, n] =!= 0,
        Print[Subscript[R, var[[m]], var[[n]]], "=",
         FullSimplify[RicciTensor[m, n]]]]]];

    Clear[RicciTensor];

    Print[Style[
      "Calculating Ricci Tensor directly, without Riemann Tensor", {Bold,
       Underlined, Larger}]]; alpha := 1; beta := 1;
    RicciTensor[alpha_, beta_] :=
      Sum[D[Γ[rho, beta, alpha], var[[rho]]] -
        D[Γ[rho, rho, alpha],
         var[[beta]]] + Γ[rho, rho, lambda] Γ[
          lambda, beta, alpha] - Γ[rho, beta,
          lambda] Γ[lambda, rho, alpha], {lambda, 1, 4,
        1}, {rho, 1, 4, 1}];
    For[alpha = 1, alpha <= 4, alpha++,
      For[If[alpha <= beta, beta = alpha], beta <= 4, beta++,
       If[RicciTensor[alpha, beta] =!= 0,
        Print[Subscript[R, var[[alpha]], var[[beta]]], "=",
         FullSimplify[RicciTensor[alpha, beta]]]]]];

RicciTensor[alpha_, beta_] :=
 Sum[D[Γ[rho, beta, alpha], var[[rho]]] - D[Γ[rho, rho, alpha], var[[beta]]], {rho, 1, 4, 1}]
 + Sum[Γ[rho, rho, lambda] Γ[lambda, beta, alpha]
     - Γ[rho, beta, lambda] Γ[lambda, rho, alpha], {lambda, 1, 4, 1}, {rho, 1, 4, 1}];