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 relativityr, without gauge ambiguities or awkward limiting
procedures; There may be several inequivalent definitions; At a quantum level,
one expects
gravitons to be surrounded by a cloud 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 a1709 [asymptotic charges cannot be observed in finite time]; focus issue CQG(18) [BMS asymptotic symmetries].

@ __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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feb 2018