[Paper Review] The characteristic gluing problem for the Einstein equations and applications

Stefanos Aretakis, Stefan Czimek+1 more|arXiv (Cornell University)|Jul 6, 2021

Black Holes and Theoretical Physics33 references4 citations

TL;DR

This paper introduces a codimension-10 characteristic gluing construction for the Einstein vacuum equations, using conserved gauge-invariant charges at null infinity to glue asymptotically flat initial data to Kerr spacetime data. The method achieves no loss of decay in the transition region and provides alternative proofs of the Corvino–Schoen and Carlotto–Schoen spacelike gluing theorems, resolving an open problem on decay preservation.

ABSTRACT

In this paper we introduce the characteristic gluing problem for the Einstein vacuum equations. We present a codimension-$10$ gluing construction for characteristic initial data which are close to the Minkowski data and we show that the $10$-dimensional obstruction space consists of gauge-invariant charges which are conserved by the linearized null constraint equations. By relating these $10$ charges to the ADM energy, linear momentum, angular momentum and the center-of-mass we prove that asymptotically flat data can be characteristically glued (including the $10$ charges) to the data of a suitably chosen Kerr spacetime, obtaining as a corollary an alternative proof of the Corvino--Schoen spacelike gluing construction. Moreover, we derive a localized version of our construction where the given data restricted on an angular sector is characteristically glued to the Minkowski data restricted on another angular sector. As a corollary we obtain an alternative proof of the Carlotto-Schoen localized spacelike gluing construction. Our method yields no loss of decay in the transition region, resolving an open problem. We also discuss a number of other applications.

Motivation & Objective

• To formulate and solve the characteristic gluing problem for the Einstein vacuum equations in a neighborhood of null infinity.
• To identify the 10-dimensional obstruction space as consisting of gauge-invariant charges conserved by the linearized null constraint equations.
• To demonstrate that asymptotically flat initial data can be characteristically glued to Kerr data, preserving ADM energy, momentum, angular momentum, and center-of-mass.
• To develop a localized gluing construction where data on an angular sector are glued to Minkowski data on another sector, preserving decay properties.
• To provide a new, decay-preserving method that resolves an open problem in localized spacelike gluing.

Proposed method

• The authors use the double null foliation framework to analyze the null structure equations and Ricci coefficients on a bifurcate null hypersurface.
• They identify conserved charges (E, P, L, G) from the linearized constraint equations at Minkowski spacetime, which correspond to ADM energy, momentum, angular momentum, and center-of-mass.
• A perturbation scheme is constructed using sphere data and diffeomorphisms to solve the linearized gluing problem with prescribed charges.
• A recursive iteration scheme is employed on a sequence of spheres to control the decay of the perturbation and the associated charges.
• The method incorporates a localization lemma that allows gluing on angular sectors by controlling the transition region without loss of decay.
• Well-posedness of the characteristic initial value problem is applied in high regularity to propagate the constructed data to spacelike hypersurfaces with controlled decay.

Experimental results

Research questions

• RQ1 What is the codimension of the obstruction space in the characteristic gluing problem for the Einstein vacuum equations?
• RQ2 How can conserved charges on null infinity be related to ADM quantities such as energy, momentum, and angular momentum?
• RQ3 Can asymptotically flat initial data be characteristically glued to Kerr data while preserving all 10 conserved charges?
• RQ4 Is it possible to perform localized gluing on angular sectors without losing decay in the transition region?
• RQ5 Can the characteristic gluing method yield an alternative proof of the Corvino–Schoen and Carlotto–Schoen spacelike gluing theorems?

Key findings

• The characteristic gluing problem admits a codimension-10 obstruction space, corresponding to the 10 conserved gauge-invariant charges (E, P, L, G) at null infinity.
• These 10 charges are geometrically interpreted as ADM energy, linear momentum, angular momentum, and center-of-mass, enabling exact matching to Kerr data.
• The construction achieves no loss of decay in the transition region, resolving a long-standing open problem in localized gluing.
• The method yields an alternative proof of the Corvino–Schoen spacelike gluing theorem via characteristic gluing to Kerr spacetime.
• A localized version of the gluing is constructed where data on an angular sector are glued to Minkowski data on another sector, with decay preserved at all scales.
• The induced spacelike initial data on a large annulus are shown to be O((1+2c)^{-n} R^{-1})-close to the original data in scale-invariant norms, confirming asymptotic flatness and energy-momentum control.