TensorContract and TensorProduct problem

Question

I am trying the calculate the Kretschmann scalar $K$ of a metric $$K=R^{abcd}R_{abcd}$$ where $R$ is the Riemann tensor. If I do

TensorContract[TensorProduct[R,R],{{1,5},{2,6},{3,7},{4,8}}]

I get a scalar function, but is this code equivalent to the above formula? Should I raise the indices of the first Riemann tensor with the inverse of the metric, or does Mathematica do this for me? Even if the metric is pseudo-Riemannian and with complicated coefficients?

Answer 1

Mathematica contracts indices assuming flat Euclidean metric. You have to lower/raise the indices yourself.

g = DiagonalMatrix[{1, -1, -1, -1}];

RaiseIndex[T_, g_, i_] := 
  Module[{indices = Append[Delete[Range[Length@Dimensions@T], i], i]},
    Transpose[
     TensorContract[TensorProduct[T, g], {i, Length[indices] + 1}], 
     indices]
  ]

Rcontra = Fold[RaiseIndex[#1, g, #2] &, R, {1, 2, 3, 4}];

TensorContract[TensorProduct[R, Rcontra], {{1, 5}, {2, 6}, {3, 7}, {4, 8}}]

Edit: Explanaion of RaiseIndex

Suppose, we want to raise second index $j$ in $R_{ijkl}$: $$ R^{\ j'}_{i\ kl} = R_{ijkl}g^{jj'} $$

However, if we just do R1=TensorContract[R⊗g, {2,5}], then the order of the indices will be $i,k,l,j'$. In other words, we need to transpose the tensor to restore the order $i,j',k,l$. The order of transposition is calculated in variable indices. It says remove index i from sequence {1,2,...n} and put it to the end. In the case where $i=2$ and $n=4$: {1,2,3,4} → {1,3,4,2}. Now when we do Transpose[R1, {1,3,4,2}], with R1 having the order of dimensions $i,k,l,j'$, we basically say put dimension $i$ to the 1st place, put dimension $k$ go to the 3rd place, put dimension $l$ to the 4th place and put dimension $j'$ to the 2nd place. Thus, the order of our dimensions (indices) will be $i,j',k,l$.

Since we need to raise all of 4 indices, Fold is just a smart way to write this:

Rcontra = RaiseIndex[RaiseIndex[RaiseIndex[RaiseIndex[R, g, 1], g, 2], g, 3], g, 4];

Please, read help on Fold to get further insights.