Hi,
I was trying to play with cadabra to study Einsteins equations at future null infinity.
I have the coordinates $u,r,\phi,\theta$, for which I use lowercase greek indices ($\mu,\nu,...$) and I want to use the coordinate $\phi, \theta$ just to label points on the sphere, which I would like to label with uppercase latin indices (A,B,C,..)
Now I would like to define a metric just on the sphere, so I used in its definition only the indices A,B, hoping that it makes it an object only of those indices. Apparently this is not the case, since if I just evaluate the metric, I still get components written out for the u and r directions and if I try to complete the rule ssid, I get the error that the metric has zero determinant. Is there a way to make \gamma_{A B} an object only of the intrinsic coordinates \phi \theta such that e.g. complete works?
Here my minimal example:
{u,r,\phi,\theta}::Coordinate;
{\mu,\nu,\rho,\sigma}::Indices(higher,values={u,r,\phi,\theta}, position=fixed);
{A,B}::Indices(lower,values={\phi,\theta}, position=fixed,parent=higher);
\gamma_{A B}::Metric(lower, signature = 2);
\gamma^{A B}::InverseMetric(lower, signature = 2);
#\gamma_{A B}::Depends(r,\phi,\theta);
ssid:= {\gamma_{\phi \phi} = r**2*\sin(\theta)**2,
\gamma_{\theta \theta} = r**2
};
ex1 := \gamma_{B A};
evaluate(ex1);
complete(ssid, $\gamma^{A B}$);