Asymptotically
Flat Spacetimes| Asymptotically
Flat Spacetimes |
In General > s.a. initial-value
formulation.
* Motivation: 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.
* 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 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.
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
nonzero
mass.
* Remark: In an asymptotically
empty and flat at null and spatial infty (AEFANSI) spacetime, 
+/– arises
as 
J+/–(i0)
\ i0.
@ References: Ashtekar & Hansen JMP(78);
Ashtekar in(80); Newman CMP(89);
Schmidt CQG(91);
Friedrich JGP(98), gq/98-GR15;
Friedrich & Kannar AdP(00)gq/99-in;
Hayward PRD(03)gq,
JKPS(04)gq/03-in
[advanced/retarded conformal factors].
Related Concepts > s.a. H-Space; types
of spacetimes [strongly asymptotically
predictable].
* Supertranslations: Angle-dependent translations at infinity, an
Abelian subgroup of the BMS group corresponding to
xa 
xa
+ f a(
,
, 
)
,
where , , 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
.
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);
Krtous & Podolsky CQG(06)gq [asymptotic
structure in higher dimensions].
@ 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 & Chrusciel CMP(05)gq/04 [vacuum,
all even D]; Tod a0902 [periodic
in time no-go result]; Lübbe & Valiente Kroon CQG(09)-a0903 [existence].
@ Timelike infinity: Moreschi CQG(87);
Friedrich CMP(88) [radiativity condition]; Cutler CQG(89);
Gen & Shiromizu
JMP(98)gq/97.
Modified Types of Spacetimes > see de sitter spacetime; horava-lifshitz gravity.
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sep 2009