Gluing for the constraints for higher spin fields

Abstract

This short note is a follow-up on the paper by Beig and Chru\'sciel regarding the use of potentials to perform a gluing and shielding of initial data for Maxwell fields and linearised gravity. Based on a work in collaboration with Andersson and B\"ackdahl, this gluing and shielding procedure is generalised to higher spin fields. The approach is based on a generalisation of the de Rham complex to higher spin fields providing a parametrization of the set of constraints, as well as standard elliptic theory to prove the existence of a potential for solutions to the constraints for higher spin fields

On linearised vacuum constraint equations on Einstein manifolds*

“…which is a special case of the "conformal complex" derived in [2] (compare [6,8]), which applies to any locally conformally flat metric. Inspection of the identities above leads to the following elliptic complex: 11) where the composition of the fourth and fifth arrow is div 2 H = 0, and the composition of the third and fourth arrow vanishes by taking formal adjoints:…”

“…which is a special case of the 'conformal complex' derived in [2] (compare [6,8]), which applies to any locally conformally flat metric. Inspection of the identities above leads to the following elliptic complex:…”

“…Recall that in [3] (compare [8]) we provided a simple way of shielding gravity linearised at Minkowski space in transverse-traceless gauge (in a sense made precise by theorem 1.1 below), based on third-order unconstrained potentials for transverse-traceless tensors introduced in [2]. One of the motivations for the current work was to provide a shielding construction for linearised vacuum gravity which applies to initial data with a non-zero cosmological constant Λ, regardless of its sign and of the gauge.…”