need help to write Riemann tensor for a metric of d dimension

[ARUNDHATI GOLDAR]

faild to define a metric which has a arbitary coordinate d. and i want my in terms of d.

[Ilham Prasetyo]

Dear Arundhati Goldar,

I had done this d-dimensional space-times in the past so allow me to reply. As far as I know from other posts and replies in this google group, dimension in xAct must be inputted with a number. May others correct me if I am wrong.

Below are some possible options that I had done in the past. Others can correct me.

You can generate Riemann tensor components from any metric with any dimension, so you can generate for d=4, d=5, and d=6, then see some pattern on these Riemann tensor components. Guess the pattern and test it for d=7 and so on by induction. However, this can be overwhelming because the amount of Riemann tensor components that you check is n^2(n^2 -1)/12 (source: https://en.wikipedia.org/wiki/Riemann_curvature_tensor?useskin=vector#Symmetries_and_identities ).

Another way is manually calculate it by hand. If Torsion tensor is set zero and you use some maximally symmetric space-times, tetrad/vierbein method can be used to deduct the Riemann tensor components, then use some formulas for maximally symmetric space-time (see e.g. https://en.wikipedia.org/wiki/Riemann_curvature_tensor?useskin=vector#Special_cases ). However, this also has some drawbacks, i.e., it can be too long, especially off some off diagonal components of the metric is nonzero, and prone to typos. xAct can be used to check your calculation and guide you to the correct results.

These perhaps are not the only methods, because I find these to be complicated. If you use metric pertubations with spherical harmonic functions like the Regee-Wheeler metric, this is very complicated. A paper (https://arxiv.org/pdf/2010.00593 at section 3.3.1) had done this but I don't know their method yet.

I hope this helps.

Best regards,

Ilham

[Thomas Bäckdahl]

Hi!

For coordinate calculations in xCoba, you do need a number for your dimension. Hence, it is not possible to directly compute curvature etc for a metric with symbolic dimension. At least not at the moment.
However, xTensor itself can handle symbolic dimensions.

For instance
DefConstantSymbol[dim, PrintAs -> "d"]
DefManifold[Man, dim, {\[Mu], \[Nu], \[Rho], \[Sigma], \[Delta], \[Lambda]}];
is OK.

For the particular case in question, the metric seems to have a nice 1+(n-1) split.
One can handle such things abstractly with for instance
DefTensor[normal[\[Mu]], Man, PrintAs -> "N"]
DefMetric[-1, metrich[-\[Mu], -\[Nu]], cd, {"|", "D"}, InducedFrom -> {g, normal}, PrintAs -> "h"]
This defines a normal to the spatial surfaces and an induced metric.
Section 7.8 in xTensorDoc describes some of the tools available.
There are some useful functions like MetricToProjector, ChangeCovD, GaussCodazzi, ExtrinsicKToGradNormal etc.
Using these tools one can express spacetime quantities in terms of intrinsic quantities.
For this to simplify, one would also have to set up some properties of the normal.

For the particular case in question, the spatial metric seems to be conformally flat. One can use this to for instance express the curvature for the spatial metric in terms of derivatives of the conformal factor. This can either be done manually from text book expressions, or one can derive such formulas in xAct.
Unfortunately, I have not found much documentation about the handling of conformal metrics in xAct.
However, one could start by defining a conformal factor and the flat conformal metric.
DefTensor[confactor[], Man, PrintAs -> "\[CapitalOmega]"]
DefMetric[-1, metrichconf[-\[Mu], -\[Nu]], cdconf, {":",  "\!\(\*OverscriptBox[\(D\), \(~\)]\)"}, ConformalTo -> {metrich[-\[Mu], -\[Nu]], confactor[]}, PrintAs -> "\!\(\*OverscriptBox[\(h\), \(~\)]\)", FlatMetric -> True]
Then one would have to play around with ChangeCurvature, ChristoffelToGradConformal, ChangeCovD etc to derive the equations one would like.

Unfortunately, I don't have time to figure out a full procedure on how to do the full calculation at the moment because it is a bit complicated. However, I hope this might give you and/or someone else some ideas on how to deal with metrics with a symbolic dimension.

Regards
Thomas

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[ARUNDHATI GOLDAR]

Dear Ilham Prasetyo,

Thank you very much for your kind response. Regarding the method of guessing, I have tried it and managed to reach a form, but I would like to compare it with a calculated expression for confirmation.

I truly appreciate you sharing the reference paper, and I will take time to study it carefully.

With best regards,
Arundhati

[ARUNDHATI GOLDAR]

Hi Thomas Bäckdahl,

I really appreciate your reply.

I used the following:

DefConstantSymbol[dim, PrintAs -> "d"]

DefManifold[Man, dim, {μ, ν, ρ, σ, δ, λ}];

But in later calculations, I keep getting the error that d is not bound. So, without assigning a specific value to d, it always throws that error.

Thank you again for taking the time to respond.

  Best regards,
Arundhati