computing Einstein Tensor

[M E C]

 Hello, I am trying to calculate the Einstein tensor for the metric.

CTensor{{(-1 + (m r n)/(a^2 Cos[χ]^2 + r^2))/(1 + k Cos[χ] r)^2, 0, 0, -((a m r n Sin[χ]^2)/((1 + k Cos[χ] r)^2 (a^2 Cos[χ]^2 + r^2))), 0}, {0, (a^2 Cos[χ]^2 + r^2)/((1 + k Cos[χ] r)^2 (a^2 + r^2 - m r n)), 0, 0, 0}, {0, 0, (a^2 Cos[χ]^2 + r^2)/(1 + k Cos[χ] r)^2, 0, 0}, {-((a m r n Sin[χ]^2)/((1 + k Cos[χ] r)^2 (a^2 Cos[χ]^2 + r^2))), 0, 0, (Sin[χ]^2 (a^2 + r^2 + (a^2 m r n Sin[χ]^2)/(a^2 Cos[χ]^2 + r^2)))/(1 + k Cos[χ] r)^2, 0}, {0, 0, 0, 0, (Cos[χ]^2 r^2)/(1 + k Cos[χ] r)^2}}, {-B, -B}, 0] 

I have followed the instructions:

<< xAct`xTras`

<< xAct`xTensor`

<< xAct`xCoba`

DefConstantSymbol[{a, m, k, n}]

....

DefManifold[M, 5, {α, β, γ, δ, ι, κ, λ, μ, o, Σ, ρ, σ, τ, Υ, ω, ν}] 

DefChart[B, M, {0, 1, 2, 3, 4}, {t, r, χ, θ, φ}]    

\[Rho]2 = r[]^2 + a^2 Cos[\[Chi][]]^2;

\[CapitalDelta] = r[]^2 - m*r[]^n + a^2;

schw1 = CTensor[....

SetCMetric[schw1, B, SignatureOfMetric -> {3, 1, 0}]

MetricCompute[schw1, B, All]

schw1[-\[Alpha], -\[Beta]]
CD = LC[schw1]

Einstein[CD][\[Alpha], -\[Beta]]

  The system is unable to calculate the Einstein tensor. I also tried using MetricCompute[schw1, B, Einstein, All -> False]. Any recommendations? Thank you very much for reading. Atteched de file

[Leo Stein]

Dear Milko,

xCoba can calculate the Einstein tensor with no problem; the slowest parts for me are computing Riemann with all indices down (about 45 sec), raising an index to make the (1,3) form of Riemann (about 1 minute), and tracing to compute Ricci (about 20 sec). All of these times include applying Simplify with some specific assumptions at each stage, to try to keep the size of the expressions small. If you give Simplify more information, and can sometimes find simpler expressions. This is done with the CVSimplify option to MetricCompute.

Let me note also the following changes to your notebook: for SignatureOfMetric, you probably mea {4,1,0}, not {3,1,0}; and for CD, you probably want to use CovDOfMetric[schw1] instead of LC (they end up giving the same result, but LC recomputes whereas CovDOfMetric uses the cached value that was computed by MetricCompute).

There are some more hopefully useful usage patterns in the attached notebook.

Best

Leo

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[M E C]

  Thank you very much for the help. I will review it in detail.  

[M E C]

  Hello. I have the following question. The following routine is not able to run. Would there be any suggestions to make it run?  

[M E C]

  Apologies for my mistake, in the previous email I attached the wrong file. I am now attaching the correct file. My question was: "I have the following question. The following routine is not able to run. Would there be any suggestions to make it run? " thank you very much for reading