Define tensor made of other tensors in xAct

Question

I'm fairly new to the xAct project for Mathematica, but feel like I understand fairly well how to define an arbitrary tensor using xTensor. My problem is that I want to define a tensor that is composed of other tensors. Specifically, I'm trying to construct an abstract tensor that looks like the following, $\mathcal{H}^{MN}=\left(\begin{array}{cc}g_{ij}-b_{ik}g^{kl}b_{lj}&&b_{ik}g^{kj}\\-g^{ik}b_{kj}&&g^{ij}\end{array}\right)$

This is known as the generalized metric in generalized geometry. Additionally, the indices $M,N$ take values from $1$ to $2D$, while the $i,j$ takes values from 1 to D. The $g_{ij}$ here corresponds to a normal Riemannian metric and the $b_{ij}$ is an antisymmetric matrix. The indices on the generalized metric would be raised and lowered using the O(D,D) invariant metric, $\eta^{MN}=\left(\begin{array}{cc}0&\mathbb{1}\\\mathbb{1}&0\end{array}\right)$.

From the xAct documentation, it isn't clear to me how (or even if) it is possible to abstractly define this using the built-in functions. Any tips or references to useful information would be much appreciated!

Answer 1

In this case, there can be several ways of doing what you want. My way of doing this is to define separately the tensors in different vbundles and then using a MakeRule to define how you write $\cal{H}^{MN}$ in terms of the other tensors. Here is an example of a small code which achieves this:

DefManifold[M, 2 D, IndexRange[a, m]]
DefTensor[H[a, b], M4, Symmetric[{a, b}]]

After defining the main manifold, where the generalized metric lives, you have to define two different vbundles which I called "first" and "second" for lack of better names. Note the AIndex they have, this is important, with that you always know with which one you are dealing.

DefVBundle[first, M4, D, {i1, j1, k1, l1, m1, n1}]
DefVBundle[second, M4, D, {i2, j2, k2, l2, m2, n2}]

Also, the numbers allow you to identify which part of the matrix you are dealing with (eg. if you write H[i1,j2] you are referring to the first "line", second "columm" of $\cal{H}^{MN}$). Now you have to define what you call $g_{ij}$ (with inverse $g^{ij}$). In this case you have to define two different tensors, otherwise xTensor will complain about not having the indices in the right positions. This will become clearer later.

DefTensor[g1[i1, j1], M4, Symmetric[{i1, j1}]]
DefTensor[g2[i2, j2], M4, Symmetric[{i2, j2}]]

Thus the tensor g1[i1,j1] with upper indices is meant to represent the $g_{ij}$ with lower indices. It is complicated I know, but I had not found any better way. After that you define the antisymmetric tensor $b_{ij}$.

DefTensor[B[-a, -b], M4, Antisymmetric[{-a, -b}]]

And define the rules:

MakeRule[{H[i1, j1], g1[i1, j1] - B[i1, -k2] g2[k2, l2] B[-l2, j1]}]
MakeRule[{H[i1, j2], B[i1, -k2] g2[k2, j2]}]
MakeRule[{H[i2, j1], -B[-k2, j1] g2[i2, k2]}]
MakeRule[{H[i2, j2], g2[i2, j2]}]

And you can also define them all together in a big rule. About using the invariant $O(D,D)$ metric, this is a bit harder to implement automatically, but I hope this helps.