Narrow Results

The study of gravitational waves in the presence of a cosmological constant has led to interesting forms of the de Sitter and anti-de Sitter line elements based on families of null hypersurfaces. The forms are interesting because they focus attention on the geometry of null hypersurfaces in spacetimes of constant curvature. Two examples are worked out in some detail. The first originated in the study of collisions of impulsive gravitational waves in which the post-collision spacetime is a solution of Einstein’s field equations with a cosmological constant, and the second originated in the generalization of plane fronted gravitational waves with parallel rays to include a cosmological constant.

Horizons of black holes or cosmologies are peculiar loci of spacetime, where interesting physical effects take place, some of which are probed by recent (EHT and LIGO) and future experiments (ET and LISA). We discuss that there are boundary degrees of freedom residing at the horizon. We describe their symmetries and their interactions with gravitational waves. This fits into a larger picture of boundary plus bulk degrees of freedom and their interactions in gauge theories. Existence and dynamics of the near horizon degrees of freedom could be crucial to address fundamental questions and apparent paradoxes in black holes physics.

Using the principle of least action, the motion equations for a singular hypersurface of arbitrary type in quadratic gravity are derived. Equations containing the “external pressure” and the “external flow” components of the surface energy–momentum tensor together with the Lichnerowicz conditions serve to find the hypersurface itself, while the remaining ones define arbitrary functions that arise due to the implicit presence of the delta function derivative. It turns out that neither double layers nor thin shells exist for the quadratic Gauss–Bonnet term. It is shown that there is no “external pressure” for null singular hypersurfaces. The Lichnerowicz conditions imply the continuity of the scalar curvature in the case of spherically symmetric null singular hypersurfaces. These hypersurfaces must be thin shells if the Lichnerowicz conditions are necessary. It is shown that for this particular case the Lichnerowicz conditions can be completely removed therefore a spherically symmetric null double layer exists. Spherically symmetric null singular hypersurfaces in conformal gravity are explored as application.

We study codimension two trapped submanifolds contained into one of the two following null hypersurfaces of de Sitter spacetime: (i) the future component of the light cone, and (ii) the past infinite of the steady state space. For codimension two compact spacelike submanifolds in the light cone we show that they are conformally diffeomorphic to the round sphere. This fact enables us to deduce that the problem of characterizing compact marginally trapped submanifolds into the light cone is equivalent to solving the Yamabe problem on the round sphere, allowing us to obtain our main classification result for such submanifolds. We also fully describe the codimension two compact marginally trapped submanifolds contained into the past infinite of the steady state space and characterize those having parallel mean curvature field. Finally, we consider the more general case of codimension two complete, non-compact, weakly trapped spacelike submanifolds contained into the light cone.

Normal Gaussian prescription breaks down when treating null boundaries since the normal to a null boundary declines into tangency to the boundary. The normal extrinsic curvature is therefore disabled as a carrier of transverse geometrical information for a null boundary. In this paper we will construct a distributional approach based on admissible coordinates to find junction conditions and dynamics of null boundaries. Since a general null boundary can consist of matter and stress-energy distribution, we consider the general case of a null "shell" or "boundary layer". As a simple example, junction conditions and dynamics of a spherical null shell with non-vanishing energy–momentum content are derived.

We study null hypersurfaces of a Lorentzian manifold with a closed rigging for the hypersurface. We derive inequalities involving Ricci tensors, scalar curvature, squared mean curvatures for a null hypersurface with a closed rigging of a Lorentzian space form and for a screen homothetic null hypersurface of a Lorentzian manifold. We also establish a generalized Chen–Ricci inequality for a screen homothetic null hypersurface of a Lorentzian manifold with a closed rigging for the hypersurface.

We prove the existence of a "canonical foliation" on a null incoming hypersurface. Here "canonical" denotes a foliation whose level function is a solution of a given nonlinear system of equations. The existence of this foliation is an important ingredient for the construction of asymptotically flat global Einstein spacetimes, with appropriate initial conditions.