Kerr metric expressed in coordinate chart

[Héloïse]

Dear all,

I am trying to get the Kerr metric expressed in ingoing Kerr coordinates (u, r, \chi, \phi) written as coeff. g_{uu} du^2 + g_{ur} du*dr + [...].

For that, I have written:

DefManifold[M4, 4, {b,c,d,e,f,h,i,j,k,l}]

DefBasis[ingoingK, TangentM4, {0,1,2,3}]

DefChart[ingoingKerr, M4, {0,1,2,3}, {u[], r[], \chi[], \phi[]}, FormatBasis -> {"Partials", "Differentials"}]

DefConstantSymbol[M]

DefConstantSymbol[a]

\rho2 = r[]^2 + a^2 * \chi^2;

\Delta = r[]^2 - 2*M*r[] + a^2;

g = CTensor[{{. . . .}, {. . . .}, {. . . .}, {. . . .}}, {-ingoingKerr, -ingoingKerr}];

g[-a, .b]

BasisExpand[%, {ingoingKerr}].

The problem occurs with the last command, which does not print the metric like g_{uu} du^2 + g_{ur} du*dr + [...].

Could you please tell me why? What shall I do to obtain the form I want?

Thank you in advance!

[Jose]

Hi,

I think we need to see complete code to understand what the problem is. It would be best if you attached a notebook with a few lines of code that show a result you don't expect.

There are several examples with the Kerr metric and frame changes in xCobaDoc.nb. See for example section 3.5.1 on how to work with the Kerr-Newman metric in the rotating frame.

Cheers,

Jose.