Asymptotic Flatness at Spatial Infinity  

In General > s.a. ADM formulation; canonical general relativity; initial-value formulation; multipole moments \ solutions of general relativity.
* Idea: One gets information on conserved quantities of spacetime, but there are no equations, the dynamics is not recorded.
* History: It took about ten years more to develop than the structure at \(\cal I\), mainly because of the intricate differentiable structure; 2016, Supertranslations are used to prove that black holes can have soft hair.
@ Early work: in Lichnerowicz 39 [for stationary spacetimes].
@ General references: Geroch JMP(72), in(77); Sommers JMP(78); Ashtekar in(80), FP(85); Beig & Schmidt CMP(82); Ashtekar & Magnon JMP(84); Beig PRS(84); Winicour FP(85); Chruściel JMP(89), JMP(89); Petrov IJMPD(95), IJMPD(97); Hayward PRD(03)gq; Compère & Dehouck CQG(11)-a1106 [without imposing parity conditions].
@ With symmetries: Beig GRG(80) [static]; Beig & Simon GRG(80), Kennefick & Ó Murchadha CQG(95)gq/93 [stationary]; Beig & Chruściel JMP(96)gq/95; > s.a. initial-value formulation.
@ Existence: Reula CMP(89); Lindblad & Rodnianski CMP(05)m.AP/03 [wave coordinates].
@ Numerical: Husa AIP(00)gq/01; Zenginoğlu JPCS(07)gq/06.
@ Other: Friedrich CMP(88) [radiativity condition]; Ashtekar & Romano CQG(92) [i0 as boundary]; Thiemann CQG(95)gq/93 [Ashtekar variables]; Herberthson CQG(98)gq/97 [diff]; Finster & Kraus CJM(05)m.DG/03 [curvature estimates]; Shiromizu & Tomizawa PRD(04)gq, Tanabe et al JMP(09)-a0902 [higher dimensions].

Spi Formalism > s.a. Penrose Diagram; Ripple.
$ Def: A spacetime (M, g) is said to be asymptotically flat at spatial infinity if there exists a conformally related spacetime (M', g'), which is C everywhere except at a point i0, where M' is C>1 and g' is C>0, together with an imbedding i: MM, such that
(1) \(\bar J\)(i0) = M' \ M, i.e., i0 is at spatial infinity;
(2) There exists a conformal factor Ω: M' → \(\mathbb R\), C2 at i0, C elsewhere, such that g'ab|M = Ω2 gab, Ω\(|_{i^0}\) = ∇'a Ω\(|_{i^0}\) = 0, ∇'a'b Ω\(|_{i^0}\) = 2 g'ab (Ω ~ ρ–2);
(3) Rab admits a regular direction-dependent limit at i0 (matter sources fall off like ρ–4).
* Re boundary conditions: If the metric were C1 at i0, the mass would vanish; If the metric were C0, the mass would not be defined; R'abcd blows up at i0, but Ω1/2 R'abcdRabcd(η), a regular direction-dependent limit; Its Weyl part is coded in the electric and magnetic parts, with potentials provided by the Ricci part.
* Relationships: Implies asymptotic flatness at spatial infinity in the ADM sense.
* Hyperboloid \(\cal D\): The most useful construction at spi is the hyperboloid \(\cal D\) of unit timelike vectors, the "space of directions of approach to i0"; It has a natural metric hab = gabηaηb.
* First-order structure: Universal, the C>1 manifold with well-defined tangent space and metric.
* Connections: They correspond to equivalence classes of connections on spacetime; They need not always be the same.
* Higher-order structure: Not meaningful, reflected in the fact that the curvature blows up at spi.
@ References: Bergmann & Smith PRD(93) [structure]; Valiente a0808 [regularity conditions].

Ambiguities > s.a. asymptotic flatness.
* Non-uniqueness: Equivalent conformal completions may be obtained by supertranslations, inequivalent ones by a 4-parameter family of logarithmic transformations; If we use Ω' = ω Ω, where ω = 1 and C>0 at i0, i.e., ω = 1 + Ω1/2 α, where α...; If ω is C0 at i0, we get an inequivalent completion (e.g., related by a log translation).
* Logarithmic transformations: An ambiguity in the choice of flat metric η (in addition to supertranslations); If η is one such metric, with Cartesian chart xm, then η' with x'm:= xm + Cm ln ρ, for all Cm, will also do; In some cases there is a preferred or asymptotic frame; In the spi framework, a 4-parameter family of inequivalent, logarithmically related completions which give the same physical answers, and can be considered as gauge.
@ Logarithmic transformations: Bergmann PR(61); Beig & Schmidt CMP(82); Ashtekar FP(85).

Symmetries / The Spi Group
* Idea: The set of all diffeomorphisms that leave the spi structure invariant, modulo those which generate the identity at i0 and leave each ripple fixed, \(\cal G\) = \(\cal D\)/I.
* Structure: Similar to the BMS group, a semidirect product of the Lorentz group and the supertranslation group; It has a preferred translation subgroup, but not a preferred Lorentz (and hence Poincaré) subgroup.
* Generators: Vector fields Xa such that at i0, Xa ∈ C>0 and X a = 0 (not to move i0), ∇'(aXb) = 0 (not to change the metric, asymptotic Killing vector fields), and ∇'a'(b Xc) = ∇'a ω g'bc, for some ω (= 1 at i0) (so X can be associated with some change in conformal factor); If ∇a ω = 0, then X generates the identity at i0; It belongs to I.
* Remark: X defines a vector field on \(\cal D\) by X' a:= lim Ω–1/2 X a (tangential to \(\cal D\) since ∇(a Xb) = 0).
@ References: Goldberg PRD(90); Perng JMP(99)gq/98; Lusanna & de Pietri gq/99; Szabados CQG(03)gq; Dehouck PhD-a1112 [electric and magnetic aspects]; Troessaert a1704 [BMS4 algebra as the asymptotic symmetry algebra].

Energy-Momentum > s.a. ADM formalism; energy in general relativity [and generalization].
$ Def: The energy-momentum component along a tangent vector Va at i0 is

Va pa = \(1\over8\pi G\)C Eab Vb dSa ,

where Eab is the electric part of the Weyl tensor on the hyperboloid \(\cal D\) at i0, and C is any cross section of \(\cal D\).
@ Relationship with ADM: Śniatycki RPMP(89); Huang & Zhang gq/05-proc, ScCh(07)gq/06.


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