Abstract null geometry, energy-momentum map and applications to the constraint tensor

Abstract

We introduce and study the notion of null manifold. This is a smooth manifold ${\mathcal N}$ endowed with a degenerate metric $\gamma$ with one-dimensional radical at every point. We also define the notion of ruled null manifold, which is a special case of null manifolds. We prove that ruled null manifolds are in one-to-one correspondence with equivalence classes of null metric hypersurface data. This correspondence is used to endow any null manifold $({\mathcal N},\gamma)$ with a family of torsion-free connections related to each other by a well-defined gauge group. The whole construction allows one to define and use geometric notions on arbitrary null manifolds. The paper has a second part where we introduce a canonical map on any null metric hypersurface data and use its algebraic properties to define a canonical decomposition of any symmetric (0,2)-covariant tensor. This decomposition, together with two new differential operators compatible with this splitting, are used to decompose the constraint tensor in full generality and at the purely abstract level. This leads to a hierarchical structure of the (detached) Einstein vacuum null constraint equations without the need of introducing special coordinates or special foliations. The results are applied to study null shells arising from the matching of two spacetimes across null boundaries. The equations governing such objects are obtained in hierarchical form without imposing any topological, gauge or coordinate conditions on the shell.