Local existence in retarded time under a weak decay on complete null cones

TL;DR

This paper extends the local existence results for solutions to the vacuum Einstein equations on complete null cones to cases with slower decay of initial data, ensuring the Hawking mass limit exists at infinity.

Contribution

It proves local existence in retarded time under weaker decay assumptions on initial data, broadening previous results and confirming the Hawking mass limit's existence.

Findings

Local existence in retarded time holds with slower decay of initial data.

Hawking mass limit exists at infinity under these conditions.

Results extend previous work to more general decay rates.

Abstract

In the previous paper \cite{L-Z}, for a characteristic problem with not necessarily small initial data given on a complete null cone decaying like that in the work \cite{Ch-K} of the stability of Minkowski spacetime by Christodoulou and Klainerman, we proved the local existence in retarded time, which means the solution to the vacuum Einstein equations exists in a uniform future neighborhood, while the global existence in retarded time is the weak cosmic censorship conjecture. In this paper, we prove that the local existence in retarded time still holds when the data is assumed to decay slower, like that in Bieri's work \cite{Bie} on the extension to the stability of Minkowski spacetime. Such decay guarantees the existence of the limit of the Hawking mass on the initial null cone, when approaching to infinity, in an optimal way.