Energy estimate for initial data on a characteristic cone

TL;DR

This paper investigates energy estimates for initial data on a characteristic cone in Einstein's equations, aiming to establish solution existence using adapted null coordinates and building on Rendall's work.

Contribution

It introduces a new approach to energy estimates on characteristic cones for Einstein equations using cone-adapted coordinates and explores their implications for initial data analysis.

Findings

Developed energy estimates in cone-adapted coordinates.

Linked initial data constraints to energy estimate frameworks.

Extended methods to Einstein equations with characteristic initial data.

Abstract

The Einstein equations in wave map gauge are a geometric second order system for a Lorentzian metric. To study existence of solutions of this hyperbolic quasi diagonal system with initial data on a characteristic cone which are not zero in a neighbourhood of the vertex one can appeal to theorems due to Cagnac and Dossa, proved for a scalar wave equation, for initial data in functional spaces relevant for their proofs. It is difficult to check that the initial data that we have constructed as solutions of the Einstein wave-map gauge constraints satisfy the more general of the Cagnac-Dossa hypotheses which uses weighted energy estimates. In this paper we start a new study of energy estimates using on the cone coordinates adapted to its null structure which are precisely the coordinates used to solve the constraints, following work of Rendall who considered the Cauchy problem for Einstein equations with data on two intersecting characteristic surfaces.