Gravitational dynamics in a 2 +1 +1 decomposed spacetime along nonorthogonal double foliations: Hamiltonian evolution and gauge fixing
Motivated by situations with temporal evolution and spatial symmetries both singled out, we develop a new 2 +1 +1 decomposition of spacetime, based on a nonorthogonal double foliation. Time evolution proceeds along the leaves of the spatial foliation. We identify the gravitational variables in the velocity phase-space as the 2-metric (induced on the intersection Σt χ of the hypersurfaces of the foliations), the 2 +1 components of the spatial shift vector, together with the extrinsic curvature, normal fundamental form and normal fundamental scalar of Σt χ, all constructed with the normal to the temporal foliation. This work generalizes a previous decomposition based on orthogonal foliations, a formalism lacking one metric variable, now reintroduced. The new metric variable is related to (i) the angle of a Lorentz-rotation between the nonorthogonal bases adapted to the foliations, and (ii) to the vorticity of these basis vectors. As a first application of the formalism, we work out the Hamiltonian dynamics of general relativity in terms of the variables identified as canonical, generalizing previous work. As a second application we present the unambiguous gauge-fixing suitable to discuss the even sector scalar-type perturbations of spherically symmetric and static spacetimes in generic scalar-tensor gravitational theories, which has been obstructed in the formalism of orthogonal double foliation.