Hi everyone, I'm new to Mathematica so I'm aware I may have committed some atrocious crimes in writing my code.
Basically, I need to study the Einstein tensor for a metric (that i called "improvedmetric") in order to compute the energy-momentum tensor. The problem is that it takes a very very long time for my computer to compute the Einstein tensor, therefore I can never work on it since I'm always stuck at the last line of the code (I'm gonna write it at the end of my message). Have I done something wrong? Is there a clever way to speed up the computation?
I already tried to compute the Ricci tensor and the scalar curvature separately: in this case the computation actually works, but when I try to calculate the Einstein tensor by using its definition, I get a huge monster that I can never get to simplify.
Thank you all in advance!
<< xAct`xCoba`
$Info = False;
$CVVerbose = False;
$PrePrint = ScreenDollarIndices;
indices = Join[{a},
Table[ToExpression[StringJoin["a", ToString[j]]], {j, 1, 20}]
]
DefManifold[M, 4, indices]
DefMetric[-1, g[-a1, -a2], CD, PrintAs -> "g"]
DefChart[chart, M, {0, 1, 2, 3}, {t[], r[], \[Theta][], \[Phi][]}]
DefScalarFunction[F]
DefConstantSymbol[Mass, PrintAs -> "M"]
DefConstantSymbol[Rotation, PrintAs -> "a"]
Delta = r[]^2*F[r[]] + Rotation^2
f = 1/2*r[]^2*(1 - F[r[]])
Psi = r[]^2 + Rotation^2 Cos[\[Theta][]]^2
Sigma = (r[]^2 + Rotation^2)^2 - Rotation^2*Delta*Sin[\[Theta][]]^2
improvedmetric = {
{1 - 2 f/Psi, 0, 0,
4*Rotation*f*Sin[\[Theta][]]^2/Psi}, {0, -Psi/Delta, 0, 0}, {0,
0, -Psi^2, 0}, {4*Rotation*f*Sin[\[Theta][]]^2/Psi, 0,
0, -Sigma*Sin[\[Theta][]]^2/Psi}
}
MetricInBasis[g, -chart, improvedmetric] // TableForm
MetricCompute[g, chart, "Einstein"[-1, -1], CVSimplify -> Simplify]