A start: entering the metric (textbook equation 13.6):

d[r_]=r^2 + a^2 -2 M r
rho2[r_,t_]=r^2+a^2 Cos[t]^2
gij=DiagonalMatrix[{rho2[r,t]/d[r], rho2[r,t], Sin[t]^2(r^2+a^2+ 2 M r a^2/rho2[r,t] Sin[t]^2),-(1-2 M r/rho2[r,t])}]
gij[[3,4]]= -2 M r a/rho2[r,t] Sin[t]^2
gij[[4,3]]=gij[[3,4]]

In words describe why the (d phi d tau) term in the textbook has a 4, whereas this gij has a 2
The metric tensor's meaning is to define distances: ds^2=gab dxA dxB
In the double sum we will have both (a,b)=(3,4) and (a,b)=(4,3) thus we will get two such terms:
g[[3,4]] dphi dct + g[[4,3]]dct dphi=2 g[[3,4]] dphi dct, since g[[3,4]]=g[[4,3]]
and thus the 2 in g[[3,4]] becomes a 4 in textbook equation 13.6 for ds^2

gijk=ConstantArray[0,{4,4,4}];
Do[ Do[ Do[
gijk[[i,j,k]]=Switch[k,1,D[gij[[i,j]],r],2,D[gij[[i,j]],t],3,0,4,0],{k,4}],{j,4}],{i,4}]

gIJ=Simplify[Inverse[gij]]
Glik=ConstantArray[0,{4,4,4}];
Do[ Do[ Do[
sum=Sum[gIJ[[l,j]] (gijk[[i,j,k]]+gijk[[j,k,i]]-gijk[[k,i,j]]),{j,4}];
Glik[[l,i,k]]=FullSimplify[sum/2],{k,4}],{l,4}],{i,4}]

Glijk=ConstantArray[0,{4,4,4,4}];
Do[ Do[ Do[ Do [
Glijk[[l,i,j,k]]=Switch[k,1,D[Glik[[l,i,j]],r],2,D[Glik[[l,i,j]],t],3,0,4,0],{k,4}],{l,4}],{j,4}],{i,4}]

RDabc=ConstantArray[0,{4,4,4,4}];

Do[ Do[ Do[ Do [
sum1=Sum[Glik[[j,a1,b1]] Glik[[d1,c1,j]]-Glik[[j,a1,c1]] Glik[[d1,b1,j]],{j,4}];
RDabc[[d1,a1,b1,c1]]=-Simplify[sum1+Glijk[[d1,a1,b1,c1]]-Glijk[[d1,a1,c1,b1]] ],{a1,4}],{b1,4}],{c1,4}],{d1,4}]

Rab=ConstantArray[0,{4,4}];
Do[ Do[ 
Rab[[a1,b1]]=FullSimplify[Sum[RDabc[[d1,a1,b1,d1]],{d1,4}]],{a1,4}],{b1,4}]

Rab                                                                    

Out[18]= {{0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}