On Monday 22 of June 2015 11:06:55 Leo Stein wrote:
> Hi Peter,
>
> Have a look at the file xAct/Documentation/English/xCobaDoc2.nb,
> section 4.1. There is an explicit example of how to define a Basis
> which has a specific Jacobian with respect to another basis. This
> involves giving DefBasis the option BasisChange->CTensor[change,
> {-newbasis, old}] where change is an NxN Jacobian matrix. A lot of
> this is simpler in the new CTensor approach than in the old approach.
Hi Leo,
This is exactly what I needed and with fully developed examples; finally
the calculation went as expected. One problem of the new approach is
that it is undocumented according to ?DefBasis, but the documentation in
xCobaDoc2.nb is excellent, provided that one is aware of its existence!
A couple of minor nitpicks: the matrix of the linear map between a non-
holonomic and a coordinate basis in general cannot be a jacobian,
correct? Also, {-newbasis,old} corresponds to the matrix that transforms
between vectors, correct? But how can this be so, when basis vectors in
Wald's notation have the index up? This has the potential to cause
confusion, because Wald's abstract index notation results in the
opposite placement of indices by the vierbein compared to that of the
form notation.
The next step is to define a 2-form in the chart and then express it in
terms of the Carter orthonormal basis. Is xTerior capable of handling
coordinates this way or should I define the 2-form as a tensor with
antisymmetric indices? Ideally, I would express the orthonormal forms in
terms of the coordinate forms, express the 2-form in terms of the
coordinate basis and then convert it into the orthonormal one.
I wrote the 2-form as a CTensor and soon came to another problem. My
definition is F=cd[-a]@eta[-b] - cd[-b]@eta[-a], where
cd=CovDOfMetric[metric] and eta is a 1-form. The problem is that F
cannot be handled as a CTensor! No index expression works, no basis
transformation etc. I had to work around this by constructing a CTensor
from it by selecting each element in the definition with nested Table.
Shouldn't the result of the covariant derivatives be a CTensor?
> I hope this helps!
> Good luck
> Leo
Very helpful, thanks.
> >>> *in[1]:= <<xAct`xCoba`*
> >>>
> >>> *in[2]:= DefManifold[M, 4, {\[Mu], \[Nu], \[Rho], \[Sigma]}]*
> >>> *in[3]:= DefChart[sphere, M, {0, 1, 2, 3}, {t[], r[], \[Theta][],
> >>> \[Phi][]}, *
> >>> * ExtendedCoordinateDerivatives -> False]*
> >>> *in[4]:= DefMetric[-1, metric[-\[Mu], -\[Nu]], CD, SymbolOfCovD ->
> >>> {";", "D"}, *
> >>> * PrintAs -> "g"]*
> >>> *in[5]:= DefVBundle[tangent, M, 4, {a, b, c, d}]*
> >>> *in[6]:= DefMetric[-1, gflat[-a, -b], PD, SymbolOfCovD -> {",",
> >>> "\[PartialD]"},*
> >>> * FlatMetric -> True]*
> >>> In this step I received an error: *DefMetric::notan: Metrics can
> >>> only
> >>> be defined on a tangent bundle.*