I am trying to use TensoriaCalc to calculate the components of the Ricci and the Riemann tensor of the following metric: $R^{2} \left(d\theta^{2} + \sin^{2}\left(\theta \right)d\phi^{2} \right)$;
According to the web guide, I should use this code:
Clear[i, j, R, \[Theta], g];
g["2Sphere"] = Metric[
SubMinus[i], SubMinus[j],
R^2 ({{1, 0},{0, Sin[\[Theta]]^2}}),
CoordinateSystem -> {\[Theta], \[Phi]},
TensorName -> "h",
StartIndex -> 1,
ChristoffelOperator -> FullSimplify,
RiemannOperator -> FullSimplify,
RicciOperator -> FullSimplify,
RicciScalarOperator -> FullSimplify
]
and the output should be:
However, using the same code in my notebook, in Mathematica version 12.3, I have the following output:
which is not the result I was expecting.
Does anyone have any idea why this happens?
in this answer
how you can do this kind of computations without any packages. It is done for an $\text{AdS}_5$ spacetime, but you should be able to easily adjust to the case of $\text{S}^2$. I have added as many explanatory comments, as possible. If you want you can have a look. If you are having difficulties adjusting to your metric, let me know and I can help. $\endgroup$