Asymptotically Flat Spacetimes |

**In General** > s.a. initial-value formulation.

* __Idea__: To define a convenient
notion (if there is one) of isolated system in general relativity, without
gauge ambiguities or awkward limiting procedures; There may be several
inequivalent definitions; At a quantum level, one expects gravitons to be
surrounded by clouds of soft gravitons, so infrared problems may affect the
notion of asymptotic flatness; It turns out that the effect is to enlarge
the Poincaré group to the BMS group.

* __Motivation__: The theory of isolated systems
is used in the proof of positive energy theorems, the definition of gravitational radiation
in exact general relativity, the setup of computational simulations to extract information
on the energy-momentum emitted during binary mergers and the analysis of the evaporation
of black holes, through the use of appropriate Hilbert spaces of asymptotic states.

* __Questions__: (i) Is there a general
algorithm for finding out whether a given spacetime is asymptotically flat?
(ii) Is the property of being asymptotically flat stable under small perturbations?
(iii) Structure of the space of solutions; (iv) Structure near spi; (v) Include matter.

* __Idea, in 4D general relativity__:

(1) Complete the spacetime by an appropriate conformal rescaling of the metric (there is gauge ambiguity);

(2) Detach the extra points attached and get an abstract manifold representing infinity;

(3) Divide the fields here into universal, geometrical structure, and physical or dynamical structure;

(4) Combine tensors of the two types to get physical information and interpret it.

> __Online resources__: see
Wikipedia page.

**Unification of Null and Spatial Infinity**
> s.a. Penrose Diagram.

* __1982__: It is not clear yet whether
the framework is compatible with the presence of radiation and non-zero mass.

* __Remark__: In an asymptotically empty and flat at
null and spatial infinity (AEFANSI) spacetime, \(\cal I\)^{±}
arises as *∂J*^{ ±}(i^{0})
\ i^{0}.

@ __References__: Ashtekar & Hansen JMP(78);
Ashtekar in(80);
Newman CMP(89);
Schmidt CQG(91);
Friedrich JGP(98), gq/98-GR15;
Friedrich & Kánnár AdP(00)gq/99-conf;
Hayward PRD(03)gq,
JKPS(04)gq/03-conf [advanced/retarded conformal factors].

**Related Concepts**
> s.a. *H*-Space; Peeling Property;
types of spacetimes [strongly asymptotically predictable].

* __Supertranslations__: Angle-dependent
translations at infinity, an Abelian subgroup of the BMS group corresponding to

*x*^{a}
\(\mapsto\) *x*^{a}
+ *f*^{ a}(*θ*,
*φ*, *χ*) ,

where *θ*, *φ*, *χ* are hyperbolic angles on the
surfaces *r* = constant; __At scri__: They form an infinite-dimensional Lie
ideal ST of the BMS group, such that BMS/ST is the Lie algebra of the Lorentz group;
__At spi__: They correspond to the additive group of functions on the hyperboloid
\(\cal D\); More physically, they are a global aspect of the gravitational memory effect;
> s.a. Gravitational Memory.

@ __Supertranslations__:
Carlip a1608 [in 2+1 dimensions].

**References**
> s.a. at null infinity and spatial infinity;
canonical general relativity; DSR.

@ __General__: Bergmann in(64);
Penrose in(68) [asymptotic simplicity];
Geroch in(77) [intro];
Geroch & Xanthopoulos JMP(78) [stability];
Beig in(88);
Frauendiener LRR(00),
LRR(04);
Krtouš & Podolský CQG(06)gq [asymptotic structure in higher dimensions];
Müller a1411
[spacetimes admitting conformal compactifications];
Grant & Tod GRG(15)-a1502 [obstructions to smoothness];
Bousso et al PRD(18)-a1709 [asymptotic charges cannot be observed in finite time];
focus issue CQG(18) [BMS asymptotic symmetries];
Neves FS(18)-a1803 [conformal infinities as natural places];
Schöller a2003 [symmetries and conservation laws];
Beyer et al IJMPD(20)-a2005 [rev];
Fernández-Álvarez & Senovilla a2105 [new approach].

@ __Solutions__: Friedrich CMP(86),
JDG(91) [with Yang-Mills fields];
Roberts gq/98 [against asymptotic flatness];
Dain & Friedrich CMP(01)gq [data with prescribed regularity];
Anderson CQG(01) [vacuum 4+1, non-vacuum 3+1];
Anderson & Chruściel CMP(05)gq/04 [vacuum, all even *D*];
Tod a0902-wd,
Bičák et al CQG(10)-a1003 [periodic in time];
Lübbe & Valiente Kroon CQG(09)-a0903 [existence];
Reiris CQG(14),
CQG(14)
[stationary isolated systems are asymptotically flat].

@ __Timelike infinity__: Moreschi CQG(87);
Friedrich CMP(88) [radiativity condition];
Cutler CQG(89);
Gen & Shiromizu JMP(98)gq/97;
Friedrich CMP(13)-a1306 [initial-value problem around past timelike infinity].

@ __Quantum asymptotic flatness__:
summary CQG+(14) [in lqg].

**Modified Types of Spacetimes**
> see de sitter spacetime; hořava-lifshitz gravity.

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