[xCoba]&[xTensor] what's the relationship between the Christoffel and Ricci rotation coefficients in a non-coordinate basis (general basis/tetrad) in xCoba.

[Jackson Han]

Hi guys!

I am so confused about the relationship between the Christoffel and rotation coefficients in a non-coordinate basis (general basis/tetrad) in xCoba. Are they the same things in xCoba? For example, when I derive the Christoffel coefficients in a null tetrad for a Schwartz black hole, is it equal to Ricci rotation coefficients after lowering the indices?  I am looking forward to your help or discussion! Thank you so much! Best and take care!

[Jackson Han]

The following is what I found from xCoba.nb--1.2 usagesmessages--1.2.1Bases:

           (* Ricci rotation coefficients *)

           RicciRotation::usage = "RicciRotation[covd, basis] gives the Christoffel tensor of covd from the parallel derivative of basis.";

But I am still a little bit confused...... 

[Jose]

Hi,

Christoffel and RicciRotation are essentially the same thing in xCoba. In fact, the whole code of RicciRotation is

RicciRotation[covd_?CovDQ] := Christoffel[covd];

RicciRotation[covd_?CovDQ, basis_?BasisQ] := Christoffel[covd, PDOfBasis[basis]];

Hence, 1-argument RicciRotation is identical to Christoffel, and 2-arguments RicciRotation takes a basis instead of a covariant derivative in the second argument, so the two arguments of RicciRotation are of different type.

The important thing here is how to express the covariant derivative of the vectors of a basis (or frame). If the basis is coordinated then we usually speak of "Christoffel symbols"; if the basis is orthonormal then we speak of "Ricci rotation coefficients"'; in the general case we call them "connection coefficients". But they are the same thing. The names just stress some property of the basis. In xCoba we generally use a single name: Christoffel. The symbol RicciRotation is there just for code readability.

Cheers,

Jose.