The Boundary of the Future

TL;DR

This paper characterizes the boundary of the future of a surface in spacetime using null geodesics, with implications for holographic screens and area laws in theoretical physics.

Contribution

It provides a precise geometric description of the future boundary of a surface, linking null geodesics and conjugate points, with applications to holography.

Findings

Boundary of the future is formed by null geodesics orthogonal to the surface without conjugate points.

The theorem aids in proving holographic screens satisfy an area law.

Applications include insights into light sheets and holographic principles.

Abstract

We prove that the boundary of the future of a surface $K$ consists precisely of the points $p$ that lie on a null geodesic orthogonal to $K$ such that between $K$ and $p$ there are no points conjugate to $K$ nor intersections with another such geodesic. Our theorem has applications to holographic screens and their associated light sheets and in particular enters the proof that holographic screens satisfy an area law.