The characteristic gluing problem for the Einstein vacuum equations. Linear and non-linear analysis

Abstract

This is the second paper in a series of papers adressing the characteristic gluing problem for the Einstein vacuum equations. We solve the codimension-10 characteristic gluing problem for characteristic data which are close to the Minkowski data. We derive an infinite-dimensional space of gauge-dependent charges and a 10-dimensional space of gauge-invariant charges that are conserved by the linearized null constraint equations and act as obstructions to the gluing problem. The gauge-dependent charges can be matched by applying angular and transversal gauge transformations of the characteristic data. By making use of a special hierarchy of radial weights of the null constraint equations, we construct the null lapse function and the conformal geometry of the characteristic hypersurface, and we show that the aforementioned charges are in fact the only obstructions to the gluing problem. Modulo the gauge-invariant charges, the resulting solution of the null constraint equations is C m+2 for any specified integer m ≥ 0 in the tangential directions and C 2 in the transversal directions to the characteristic hypersurface. We also show that higherorder (in all directions) gluing is possible along bifurcated characteristic hypersurfaces (modulo the gauge-invariant charges).

Azimuthal instabilities on extremal Kerr

“…In view of the existence of a bijection between the behaviour of φ m=0 along H + and rφ m=0 in I + (valid only in axisymmetry) [CT84,BF13], the conservation law of Aretakis along H + may be interpreted as a manifestation of the Newman-Penrose conservation law along I + that is present in general asymptotically flat settings [NP68]. The relation between conservation laws and characteristic initial data gluing problems for wave equations was investigated further in [Are17] and has recently been analyzed in the context of the Einstein equations in [ACR21a,ACR21c,ACR21b,CR22].…”

Gluing variations

“…In a recent series of pioneering papers, Aretakis, Czimek and Rodnianski [1][2][3] presented a gluing construction for characteristic initial data for four-dimensional vacuum Einstein equations. The purpose of this paper is to show that related gluing constructions can be done using a spacelike gluing à la Corvino [18].…”

“…The purpose of this paper is to show that related gluing constructions can be done using a spacelike gluing à la Corvino [18]. While the construction in [1][2][3] uses the structure of the four-dimensional Einstein equations in a substantial way, our approach applies to any dimensions. As a bonus, we allow a non-vanishing cosmological constant.…”

“…The aim of this work is to show that spacelike gluings can be used to construct spacetimes with properties similar to those resulting from the construction of [3]. In the approach described here the hypersurface N 2 of Question 1.1 is again obtained by moving slightly N 2 within (M 2 , g 2 ), but we give up the requirement that N 1 and N 2 are subsets of the same smooth null hypersurface in the final spacetime.…”

“…In Section 8 we show how to carry out a variation of the characteristic gluing using spacelike gluing. In Appendix A we show how the sphere data of [3] relate to our codimension-two data in Bondi parameterisation. In Appendix B we show existence of a preferred, unique set of Bondi coordinates associated with a null hypersurface N with a cross-section S. In Appendix C we show, by quite general considerations, that spacetime Killing vectors provide an obstruction to carry-out certain gluing constructions.…”