Did you manage to solve this problem? If there was a covariant way of writing the `v, t, z`

coordinate part of the metric, the `split_index`

command works great. On the other hand, if you had a fixed dimension, it's easy to specify coordinates and work things out. It would be nice to have an elegant solution to combine these.

There is an ugly solution instead. Which is to use the Kaluza-Klein example (see here) to `split_index`

recursively. And then you can manually write all the equations of motion of interest separately for the t, z, and mixed components. And you would have covariant expressions for the `i, j`

and `\alpha, \beta`

parts.

Here's the ugly potential solution in more detail:

- Start with an expression written in terms of "total" indices
`M, N`

, etc. First use `split_index`

to split the total space-time from these `M, N`

indices to, say, `a, b`

and `A, B`

indices.
- And then apply a subsequent
`split_index`

to split `A, B`

into your `i, j`

and `\alpha, \beta`

indices. Until here, it is currently possible to do using `split_index`

as it stands.
- However, the missing piece is a way to write the
`a, b, c`

part of the metric in terms of the specific coordinates `v, t, z`

with an explicit metric of choice as you have it. If there way to do this last piece of the metric by say specifying `g_{a b}`

to be a the three-metric you have, it would have been great.
- What you can do, instead, is to use
`split_index`

three times for each of the coordinates as is done in the Kaluza-Klein example quoted above. This is admittedly tedious and rather ugly.

But unfortunately, as far as I know, there's no way to carry out the third step together with the first two to achieve what you want to do directly. Perhaps @Kasper has an elegant solution? Please drop a line here if you manage to solve this issue. Otherwise, I hope the tedious solution helps...