Partial derivatives in xCoba

[Theo]

Hi all,

I am trying to define the components of some timelike vector field u in a coordinate chart (t,x,y,z) (I've already defined the metric in that basis to be in Newtonian gauge). I want the components of u to be related to partial derivatives with respect to t of scalar functions of the coordinates. Ultimately I need this so that I can use the vector field to perform a 3+1 decomposition and do curvature computations in the 3-space.

 I can't work out how to define the vector components in the coordinate chart because I'm not sure how to extract specific components of a parallel derivative (I'd be fine if these partial derivatives e.g. X_{,t} could just be described as scalar functions, but for the purposes of what I need to do this isn't possible). I've attached my notebook. I assume the solution is actually fairly simple, but I'm new to xCoba and getting myself confused by it.

Many thanks!

Theo

[Jose]

Hi,

I think the vector you are looking for is

u = CTensor[{

   1,

   x[] D[X[t[], y[], z[]], t[]] / X[t[], y[], z[]], 

   y[] D[Y[t[], x[], z[]], t[]] / Y[t[], x[], z[]], 

   z[] D[Z[t[], x[], y[]], t[]] / Z[t[], x[], y[]]

}, {expanding}]

I'm not sure what you mean by "to extract specific components of a parallel derivative". The parallel derivative is a covariant derivative like any other one. You can apply it to a CTensor object and it will return another CTensor object with one more index. For example, once you have defined the vector u above, try

PDexpanding[-\[Alpha]][ u[\[Beta] ]]

This will return a CTensor object with two indices.

You have defined two other frames (PN and bianchiI), but you have not yet specified their relations to the coordinated basis. Therefore xCoba cannot yet perform any practical computations with them. All differentiations with respect to the parallel derivatives of those frames will stay formal.

Cheers,

Jose.