  We develop the Hamiltonian theory of axial perturbations around a general
time-dependent spherical background spacetime. Using the fact that the
linearized constraints are gauge generators, we isolate the physical and
unconstrained axial gravitational wave in a Hamiltonian pair of variables.
Then, switching to a more geometrical description of the system, we construct
the only scalar combination of them. We obtain the well-known Gerlach and
Sengupta scalar for axial perturbations, with no known equivalent for polar
perturbations. The strategy suggested and tested here will be applied to the
polar case in a separate article.
