The Solution of Einstein's Equation in General Dimensions

Question

In this link, I have learned how to solve Einstein equation in 4 dimensional spacetime, but how to solve it in general dimension? For example, in $(n+1)$ dimensional spacetime, the metric is

$$ \frac{d s^2}{l^2}=\frac{d v^2}{v^2 f(v)}+\frac{1}{v^2}\left(-f(v) d t^2+d z^2+\delta_{i j} d x^i d x^j+h_{\alpha \beta}(v) e^{i k z} d x^\alpha d x^\beta\right) $$

where $i,j=1,\cdots,n-2$, $\alpha,\beta=1,\cdots,n$, Einstein's equation in AdS spacetime is

$$ 0 = R_{a b} - \frac{1}{2}R_{c d} g^{c d} g_{a b} - \frac{n\left(n-1\right) }{2 {l}^2}g_{a b} $$

where $l$ is the AdS radius. How to obtain the components of Einstein's equation?

Comments

Firstly, I don't thing it can be done in a general form... in order to calculate, the dimension of the spacetime has to be fixed (afaik).

Then, Are you interestes in perturbing a Schwarzschild space?

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Not exactly. I'm also interested in other spacetime, e.g. RN-AdS spacetime.

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Hi doxdrum, I have solved it. See more detials in the comment of answer.

Answer 1

The following code should be useful:

{M,N,O,P,Q,R,S}::Indices(full, position=independent);
{\alpha,\beta,\mu,\nu,\sigma,\gamma,\lambda}::Indices(subspace1, position=independent, parent=full);
{\alpha,\beta,\mu,\nu,\sigma,\gamma,\lambda}::Integer(0..d-1).
w::Coordinate;
g_{\mu\nu}::Metric.
g^{\mu\nu}::InverseMetric.
g_{\mu\nu}::Symmetric.
g^{\mu\nu}::Symmetric.
g^{\mu}_{\nu}::KroneckerDelta.
g_{\mu}^{\nu}::KroneckerDelta.

from cdb.utils.indices import *
ex:=g_{M N} g^{M O} P^N;
replace_index(_,r"g",r'O',r'\mu');
split_index(_, $M,\mu,w$,repeat=True);
eliminate_metric(_);
eliminate_kronecker(_);
substitute(_,$P^w->A$);

Answer 2

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:

1. 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.
2. 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.
3. 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.
4. 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...

Comments

Yes, I have solved it. The key point: Combining replace_index and split_index. First, replace explicit indice in higher dimension into indices in lower dimension. Then, split dummy indices into indices in subspaces. Finally, do substitude.

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Hi Eureka. Thank you for your feedback. Could you post an asnwer with a minimal working example of your algorithm? Best wishes.

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Ok, I have updated in another answer. I hope this is useful.