The intersection of a hyperplane with a lightcone in the Minkowski spacetime

TL;DR

This paper explores the geometric properties of hyperplane-lightcone intersections in Minkowski spacetime, providing insights into trapped surface formation and extending previous anisotropic criteria in vacuum.

Contribution

It analyzes the intrinsic and extrinsic geometry of hyperplane-lightcone intersections, offering a geometric interpretation of Green's function and clarifying conditions for trapped surface formation.

Findings

Intersection geometry depends on hyperplane type: spacelike, null, or timelike.

Null hyperplane intersection yields a noncompact marginal trapped surface.

Provides a geometric interpretation of Green's function on the sphere.

Abstract

Klainerman, Luk and Rodnianski derived an anisotropic criterion for formation of trapped surfaces in vacuum, extending the original trapped surface formation theorem of Christodoulou. The effort to understand their result led us to study the intersection of a hyperplane with a lightcone in the Minkowski spacetime. For the intrinsic geometry of the intersection, depending on the hyperplane being spacelike, null or timelike, it has the constant positive, zero or negative Gaussian curvature. For the extrinsic geometry of the intersection, we find that it is a noncompact marginal trapped surface when the hyperplane is null. In this case, we find a geometric interpretation of the Green's function of the Laplacian on the standard sphere. In the end, we contribute a clearer understanding of the anisotropic criterion for formation of trapped surfaces in vacuum.