I am having trouble calculating exterior derivatives for one forms, maple does not seem to recognize them as one forms.
Sigma2 := (R^2+a^2)^2-Delta*a^2*sin(Theta)^2;
rho2 := R^2+a^2*cos(Theta)^2;
z := 2*M*R/rho2;
interface(typesetting = extended);
with(DifferentialGeometry); with(Tensor);
DGsetup([T, R, Theta, Phi], BlackHole, verbose);
The following coordinates have been protected:
[T, R, Theta, Phi]
The following vector fields have been defined and protected:
[Typesetting:-mcomplete(D_T, Typesetting:-_Hold([
_DG([["vector", BlackHole, []], [[[1], 1]]])])),
Typesetting:-mcomplete(D_R, Typesetting:-_Hold([
_DG([["vector", BlackHole, []], [[[2], 1]]])])),
Typesetting:-mcomplete(`D_Θ`, Typesetting:-_Hold([
_DG([["vector", BlackHole, []], [[[3], 1]]])])),
Typesetting:-mcomplete(`D_Φ`, Typesetting:-_Hold([
_DG([["vector", BlackHole, []], [[[4], 1]]])]))]
The following differential 1-forms have been defined and
protected:
[Typesetting:-mcomplete(dT,
Typesetting:-_Hold([_DG([["form", BlackHole, 1], [[[1], 1]]])])),
Typesetting:-mcomplete(dR,
Typesetting:-_Hold([_DG([["form", BlackHole, 1], [[[2], 1]]])])),
Typesetting:-mcomplete(`dΘ`,
Typesetting:-_Hold([_DG([["form", BlackHole, 1], [[[3], 1]]])])),
Typesetting:-mcomplete(`dΦ`,
Typesetting:-_Hold([_DG([["form", BlackHole, 1], [[[4], 1]]])]))
]
g := evalDG((-1+z)*`&t`(dT, dT)+`&t`(dT, dR)+`&t`(dR, dT)+rho2*`&t`(dTheta, dTheta)-z*a*sin(Theta)^2*(`&t`(dPhi, dT)+`&t`(dT, dPhi))-a*sin(Theta)^2*(`&t`(dR, dPhi)+`&t`(dPhi, dR))+Sigma2*sin(Theta)^2*`&t`(dPhi, dPhi)/rho2);
ON := evalDG(DGGramSchmidt([D_T, D_R, D_Theta, D_Phi], g, signature = [-1, 1, 1, 1]));
e0 := evalDG(ON[1]);
oneform:=RaiseLowerIndices(g,e0,[1]);
ExteriorDerivative(oneform);