How to calculate Einstein tensor components for this metric?

I try to calculate the Einstein tensor compenents from the eqution:

$$G_{\alpha\beta} = \frac{\nabla_\beta (\partial_\alpha \phi)}{\phi} - \frac{1}{2\phi^2} \left[ \frac{\partial_4 \phi \partial_4 g_{\alpha\beta}}{\phi} - \partial_4 (\partial_4 g_{\alpha\beta} ) + g^{\gamma\delta} \partial_4 g_{\alpha\gamma} \partial_4 g_{\beta\delta} - \frac{g^{\gamma\delta} \partial_4 g_{\gamma\delta} \partial_4 g_{\alpha \beta}}{2}+ \frac{g_{\alpha\beta}}{4}\left( \partial_4 g^{\gamma\delta} \partial_4 g_{\gamma\delta} + ( g^{\gamma\delta} \partial_4 g_{\gamma\delta})^2 \right)\right],$$

The code in the answer in this thread Calculating Einstein tensor components in Kaluza-Klein model works well in case of a sipmle metric like:

$$g_{\alpha\beta} = a^2(t,y) (-1,1,1,1)$$

While now I try to calculate $$G_{\alpha\beta}$$ in case of a perturbed metric 0209156

metric = a[t, y]^2 {{-(1 + 2 n \[CapitalPhi][t, r, \[Theta], \[Phi]]),
n D[S[t, r, \[Theta], \[Phi]], r],
n D[S[t, r, \[Theta], \[Phi]], \[Theta]],
n D[S[t, r, \[Theta], \[Phi]], \[Phi]]},
{n D[S[t, r, \[Theta], \[Phi]], r],
1 - 2 n \[CapitalPsi][t, r, \[Theta], \[Phi]] +
n D[Xi[t, r, \[Theta], \[Phi]], {r, 2}] -
1/3 n D[Xi[t, r, \[Theta], \[Phi]], {\[Theta], 2}] -
1/3 n D[Xi[t, r, \[Theta], \[Phi]], {\[Phi], 2}],
n D[D[Xi[t, r, \[Theta], \[Phi]], r], \[Theta]],
n D[D[Xi[t, r, \[Theta], \[Phi]], r], \[Phi]]},
{n D[S[t, r, \[Theta], \[Phi]], \[Theta]],
n D[D[Xi[t, r, \[Theta], \[Phi]], r], \[Theta]],
1 - 2 n \[CapitalPsi][t, r, \[Theta], \[Phi]] +
n D[Xi[t, r, \[Theta], \[Phi]], {\[Theta], 2}] -
n D[Xi[t, r, \[Theta], \[Phi]], {r, 2}] -
1/3 n D[Xi[t, r, \[Theta], \[Phi]], {\[Phi], 2}],
n D[D[Xi[t, r, \[Theta], \[Phi]], \[Theta]], \[Phi]]},
{n D[S[t, r, \[Theta], \[Phi]], \[Phi]],
n D[D[Xi[t, r, \[Theta], \[Phi]], r], \[Phi]],
n D[D[Xi[t, r, \[Theta], \[Phi]], \[Theta]], \[Phi]],
1 - 2 n \[CapitalPsi][t, r, \[Theta], \[Phi]] +
n D[Xi[t, r, \[Theta], \[Phi]], {\[Phi], 2}] -
1/3 n D[Xi[t, r, \[Theta], \[Phi]], {\[Theta], 2}] -
1/3 n D[Xi[t, r, \[Theta], \[Phi]], {r, 2}]}};


Where: coordList = {t, r, \[Theta], \[Phi]};

And n is a factor of a polynomial series and it’s of the first order {n,0,1}.

For this metric I tried to modify the last step in the answer’s code 1

ein[a_, b_] := Normal[Series[-(term1[a, b] + term2[a, b] + term3[a, b] +
Sum[term4[a, b, g, d] + term5[a, b, g, d] +
term6[a, b, g, d], {g, 4}, {d, 4}])/2/f[t, y]^2, {n,0,1}]]


But the code doesn't work for this new metric. So how to calculate $$G_{\alpha\beta}$$ for the metric?

• If I run the code on the new metric, I get a 5.7MB expression just for ein[1,1]. So one question is, how can you really tell if it works or not with an expression of that size? Since your metric contains off diagonal terms, off-diagonal terms for the other components of the Einstein tensor are present as well, so you've got an enormous output to sift through. I'm not claiming the old code will work, since it was specifically written for a special case, only that it's not easy to tell if it's working or not. So how did you arrive at your conclusion?
– user87932
Nov 16, 2023 at 23:08
• Regarding your use of Series[expr,{n,0,1}], I think this came up in one of your older questions. Series treats 'n' as a continuous variable, not a discrete one. So if you want to expand f[x] to second order around x=0, use Series[f[x],{x,0,2}]; see the first line in the documentation for Series, and examine Series[x^n,{n,0,2}]. You need to compute the perturbed metric first, then use that metric directly in the code. The problem is that Series doesn't handle matrices, so you'll need something like xActxPert to generate the perturbations and substitute terms into it via rules.
– user87932
Nov 17, 2023 at 1:25
• Hello @jdp. Thanks a lot for your reply. I still working on the metric and trying to find a solution around it. Yes, I think the output is huge, and maybe the off-diagonal terms of the metric need to be simplified. Nov 17, 2023 at 6:02
• One thing to keep in mind is that perturbation theory in relativity is complicated by the fact that you have to worry about gauge invariance: (arxiv.org/pdf/0711.0115.pdf for example). I.e. your physical results don't depend on the coordinate system you choose, and this needs to be reflected in your metric and any perturbations of it. As for the code from the other question, each term just implements a single term in your equation, but doesn't place restrictions on whether the metric is diagonal or not, so although not tested, will probably work for other cases. No hiccups for 5.7MB.
– user87932
Nov 17, 2023 at 17:02
• Does this answer your question? How to calculate scalar curvature, Ricci tensor and Christoffel symbols in Mathematica? Nov 20, 2023 at 15:16