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Motivated by situations with temporal evolution and spatial symmetries bothsingled out, we develop a new 2+1+1 decomposition of spacetime, based on anonorthogonal double foliation. Time evolution proceeds along the leaves of thespatial foliation. We identify the gravitational variables in the velocityphase-space as the 2-metric (induced on the intersection $\Sigma_{t\chi }$ ofthe hypersurfaces of the foliations), the 2+1 components of the spatial shiftvector, together with the extrinsic curvature, normal fundamental form andnormal fundamental scalar of $\Sigma _{t\chi }$, all constructed with thenormal to the temporal foliation. This work generalizes a previousdecomposition based on orthogonal foliations, a formalism lacking one metricvariable, now reintroduced. The new metric variable is related to (i) the angleof a Lorentz-rotation between the nonorthogonal bases adapted to thefoliations, and (ii) to the vorticity of these basis vectors. As a firstapplication of the formalism, we work out the Hamiltonian dynamics of generalrelativity in terms of the variables identified as canonical, generalizingprevious work. As a second application we present the unambiguous gauge-fixingsuitable to discuss the even sector scalar-type perturbations of sphericallysymmetric and static spacetimes in generic scalar-tensor gravitationaltheories, which has been obstructed in the formalism of orthogonal doublefoliation.

Gravitational dynamics in a 2+1+1 decomposed spacetime along nonorthogonal double foliations: Hamiltonian evolution and gauge fixing