Time derivative of the metric

[Mattia Villani]

How do I calculate the time derivative of the metric? This is my setup

DefManifold[M6, 6, IndexRange[a, n]]
DefChart[ch, M6, {0, 1, 2, 3, 4, 5}, {t[], r[], x[], xc[], t1[], p1[]}]

DefScalarFunction[{\[Alpha], \[Beta], \[Gamma]}]
met = CTensor[{{1, 0, 0, 0, 0, 0},
    {0, Exp[2 \[Alpha][t[], r[], x[], xc[]]], 0, 0, 0, 0},
    {0, 0, 0, Exp[2 \[Beta][t[], r[], x[], xc[]]], 0, 0},
    {0, 0, Exp[2 \[Beta][t[], r[], x[], xc[]]], 0, 0, 0},
    {0, 0, 0, 0, Exp[2 \[Gamma][t[], r[]]], 0},
    {0, 0, 0, 0, 0,
     Exp[2 \[Gamma][t[], r[]]] Sin[t1[]]^2}}, {-ch, -ch}];
SetCMetric[met, ch, SignatureOfMetric -> {5, 1, 0}];
cd = CovDOfMetric[met];

I know the partial derivative would be something like PDch[-a][met[-b,-c]], but how do I set a=0?

[Jose]

Hi,

I think the derivative you are looking for is PDch[{0, -ch}][met[-b, -c]]. That's the contraction (ContractBasis) of the full derivative PDch[-a][met[-b, -c]] with the (coordinated) basis vector Basis[{0, -ch}, a].

Cheers,

Jose.