Your question is not clear to me. There are four coordinates, could be any symbols, say X = (x1, x2, x3, x4).
IMPORTANT: note that, in Maple, `` is the selection operator. Given a list X := [a, b, c, d], there is no meaning for X. So to make the notation X work in the computer as we do with paper and pencil we need to map X into X[n] for some n that is a positive integer. This is relevant for tensors, say the metric g_[mu,nu], because the order of lines and columns in the matrix representation is according to the ordering of the coordinates, and here again, in the context of a computer, there is no meaning to "the 0th column" of a matrix. Lines and columns are associated to positive integer numbers.
So suppose that the signature is `+ + + -`. The different sign is the last one, in position 4, and that is the timelike position . Thus the mapping we use (as in textbooks or paper and pencil) is that the value 0 for the index always points to the timelike position, and so for this signature X = X, and X[1..3] are the spacelike coordinates, and in the case of the metric g_[0, 0] = g_[4, 4], meaning you get the component in line 4 and column 4 of the matrix representation for g_.
If the signature is `- + + +` then X = X (the timelike coordinate for this signature) and X[2..4] are the three spacelike coordinates and g_[0, 0] = g_[1, 1], you see this in the matrix representation of the metric (enter g_).
To see this mapping at work for the coordinates, use SpacetimeVector(X) with the two different settings of the signature.
If you prefer to avoid having to select the coordinate with a number and forget about this subtlety of what is the timelike position in the signature, you can always Setup(usecoordinatesastensorindices = true). Suppose your coordinates are [t, x, y, z], or [x, y, z, t] it doesn't matter: you can now index directly with the coordinates themselves, as in Christoffel[t,t,t] to get its value.
Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft