We may be able to learn if a red rag has any effect ("taurus" means "bull") - perhaps you mean a "torus"? Sort of a doughnut shape?
The second moment of mass is just the moment of inertia.
So we know what that is.
E=mc^2 would be the rest-mass energy of something.
Just blindly substituting a rest-mass energy density for mass in the moment of inertia calculation will still get you the (non-relativistic) moment of inertia as before - just in different units.