I am currently reading some parts of "Rotating Relativistic Stars" by Friedman and Stergioulas and I have to say mathematics should NOT be taught by astrophysicists... Anyway, I've encountered the term "rotational scalar" and it raised a lot of questions. It all boils down to one big question in the end...
Consider a 2-sphere in a spherically symmetric 4D manifold. SO(3) acts on the manifold via some diffeomorphisms. I've tried to come up with a diffeomorphism that preserves the spherical symmetry but is not in SO(3). A candidate is a mapping that rotates the sphere around an axis, but with varying angular velocity (twisting the sphere). Is it possible to construct such a mapping so that it is a diffeomorphism and does it preserve the spherical symmetry? If it is possible, is such a diffeomorphism a part of SO(3)?
I am not really sure I myself understand my question, but let's start a discussion and see... Trying to ask this helps anyway :)