Multi-localized time-symmetric initial data for the Einstein vacuum equations

. We construct a class of time-symmetric initial data sets for the Einstein vacuum equation modeling elementary configurations of multiple “al- most isolated” systems. Each such initial data set consists of a collection of several localized sources of gravitational radiation, and lies in a family of data sets which is closed under scaling out the distances between the systems by arbitrarily large amounts. This class contains data sets which are not asymp- totically flat, but to which nonetheless a finite ADM mass can be ascribed. The construction proceeds by a gluing scheme using the Brill–Lindquist metric as a template. Such initial data are motivated in part by a desire to understand the dynamical interaction of distant systems in the context of general rela- tivity. As a by-product of the construction, we produce complete, scalar-flat initial data with trivial topology and infinitely many minimal spheres, as well as initial data with infinitely many Einstein–Rosen bridges.

[1]  Yuchen Mao,et al.  Localized initial data for Einstein equations , 2022, 2210.09437.

[2]  Stefanos Aretakis,et al.  The characteristic gluing problem for the Einstein equations and applications , 2021, 2107.02441.

[3]  Federico Pasqualotto,et al.  Global Stability for Nonlinear Wave Equations with Multi-Localized Initial Data , 2019, Annals of PDE.

[4]  Iva Stavrov Allen,et al.  Geometrostatic Manifolds of Small ADM Mass , 2017, Communications on Pure and Applied Mathematics.

[5]  Lan-Hsuan Huang,et al.  Localized deformation for initial data sets with the dominant energy condition , 2016, Calculus of Variations and Partial Differential Equations.

[6]  R. Schoen,et al.  Localizing solutions of the Einstein constraint equations , 2014, 1407.4766.

[7]  M. Eichmair,et al.  Deformation of scalar curvature and volume , 2012, 1211.6168.

[8]  P. Yu,et al.  Construction of Cauchy Data of Vacuum Einstein field equations Evolving to Black Holes , 2012, 1207.3164.

[9]  P. Chruściel,et al.  Construction of N-Body Initial Data Sets in General Relativity , 2010, 1004.1355.

[10]  J. Isenberg,et al.  Construction of N-body time-symmetric initial data sets in general relativity , 2009, 0909.1101.

[11]  Justin Corvino.  On the Existence and Stability of the Penrose Compactification , 2007 .

[12]  Justin Corvino.  A note on asymptotically flat metrics on ℝ³ which are scalar-flat and admit minimal spheres , 2005 .

[13]  R. Bartnik,et al.  The Constraint equations , 2004, gr-qc/0405092.

[14]  P. Miao.  Asymptotically flat and scalar flat metrics on R^3 admitting a horizon , 2003 .

[15]  R. Schoen,et al.  On the Asymptotics for the Vacuum Einstein Constraint Equations , 2003, gr-qc/0301071.

[16]  P. Chruściel,et al.  On mapping properties of the general relativistic constraints operator in weighted function spaces , 2003, gr-qc/0301073.

[17]  P. Chruściel,et al.  On 'many-black-hole' vacuum spacetimes , 2002, gr-qc/0210103.

[18]  Justin Corvino.  Scalar Curvature Deformation and a Gluing Construction for the Einstein Constraint Equations , 2000 .

[19]  J. Lohkamp.  Scalar curvature and hammocks , 1999 .

[20]  S. Kuksin.  On long-time behavior solutions of nonlinear wave equations , 1994 .

[21]  Ó Murchadha N,et al.  Trapped surfaces due to concentration of gravitational radiation. , 1991, Physical review letters.

[22]  R. Bartnik.  The mass of an asymptotically flat manifold , 1986 .

[23]  Shing-Tung Yau,et al.  On the proof of the positive mass conjecture in general relativity , 1979 .

[24]  J. Marsden,et al.  Deformations of the scalar curvature , 1975 .

[25]  Charles W. Misner,et al.  THE METHOD OF IMAGES IN GEOMETROSTATICS , 1963 .

[26]  R. W. Lindquist,et al.  INTERACTION ENERGY IN GEOMETROSTATICS , 1963 .

[27]  Y. Fourès-Bruhat,et al.  Théorème d'existence pour certains systèmes d'équations aux dérivées partielles non linéaires , 1952 .

[28]  D.,et al.  The global nonlinear stability of the Minkowski space , 2018 .

[29]  R. Schoen,et al.  Initial data and the Einstein constraint equations , 2015 .

[30]  I. Rodnianski.  The Cauchy problem in General Relativity , 2006 .

[31]  Piotr T Chruściel,et al.  Existence of non-trivial, vacuum, asymptotically simple spacetimes , 2002 .

[32]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[33]  R. Geroch,et al.  Global aspects of the Cauchy problem in general relativity , 1969 .