Asymptotic
Flatness at Null Infinity| Asymptotic
Flatness at Null Infinity |
In General > s.a. [asymptotic flatness]; Peeling; Penrose
Diagram.
* Idea: The description
here is slightly more involved than that at spatial infty, but it is more interesting,
partly because one can study dynamics;
it is
more "differential", as opposed to algebraic.
$ Def: A vacuum spacetime
(M, g)
is said to be asymptotically flat at null infinity if there exists an asymptote,
a spacetime (M', g'),
with boundary 
("scri"),
such that
(1) can be written as –
+,
with + 
J–[int(M)]
= – J+[int(M)]
= Ø;
(2) There is a neighborhood of which
is strongly causal;
(3) g'ab = 
2
gab, |scri =
0, 
a |scri 
0
(gives 
r–1
near ), g'ab ab |scri =
0.
* Metric: The metric
one gets is degenerate – it is better to work with concomitants than
covariant derivatives.
* Curvature: A convenient field to define, substituting the Ricci
tensor, is Sab:= Rab – 
R gab.
* Coordinate: In practice, use some advanced or retarded null coordinate u,
and x, whose inverse measures affine length along integral curves
of du. Then x = 0 is past/future .
@ General references: Bondi et al PRS(62);
Penrose PRL(63), PRS(65);
Couch & Torrence JMP(72);
Geroch in(77); Geroch & Horowitz PRL(78);
Hayward gq/03-in
[refined def]; Marolf & Ross CQG(03)
[new definition of causal completion]; Newman & Nurowski CQG(06)
[CR structure]; Kozameh et al a0802 [Bondi 4-momentum and physical content].
@ Cauchy data at scri: Geroch JMP(78);
Schmidt & Stewart PRS(88).
@ Smoothness of scri: Andersson & Chrusciel CMP(94); Valiente-Kroon
CQG(01) [detectability], CMP(04)gq/02,
CMP(04)gq/03 [and
rad].
@ Solutions: Ashtekar & Dray pr(80), CMP(81);
Friedrich PRS(81);
Cutler & Wald CQG(89);
Moreschi JMP(90)
[FRW spacetimes]; Ashtekar et al PRD(97)gq [Einstein-Rosen
waves]; Hübner CQG(98)gq/97 [Gowdy
spacetimes, toroidal null infinity]; Valiente-Kroon JMP(00)gq/99 [one
Killing vector field]; Chrusciel & Delay
CQG(02)gq [non-trivial];
Chrusciel gq/02-in.
@ Higher-dimensional: Hollands & Ishibashi JMP(05)gq/03,
ht/03-in;
Hollands & Wald CQG(04)gq [non-existence
of in odd dimensions].
@ Related topics: Ashtekar & Sen JMP(82)
[with singularities, and NUT charge]; Ashtekar et al PRD(97)gq/96 [Killing
vector field reduced]; Garfinkle gq/99 [electromagnetic
field]; Moreschi & Dain JMP(98)gq/02 [center
of mass]; Tafel & Pukas
CQG(00)
[comparison of approaches]; Tomizawa & Siino gq/02 [topology
at
upper end]; Dappiaggi RVMP(08) [free scalar field theory].
Symmetries: The BMS Group (Bondi-Metzner-Sachs)
* Idea: The symmetry
group at null infinity; It consists, in some sense, of the Lorentz group and
the infinite-dimensional supertranslation group
(name by Sachs), but the former is not uniquely defined; there is a unique
4D
subgroup of the supertranslation group that can be identified with
asymptotic
translations, but the notion of pure rotation or pure boost,
without supertranslation ambiguities, in general does not exist.
* Structure: It is the
semidirect product of the group of conformal mappings on the Riemann sphere S and
the vector space (abelian group!) of smooth real-valued
functions on S [@ Geroch & Newman JMP(71)].
* Re boundary conditions:
One does not know of simple conditions to impose to reduce it to the Poincaré group,
like the vanishing of the magnetic part of the Weyl
tensor in the case of spatial infinity.
* Generators: A vector
field X on is
a generator of the BMS group if, for some scalar field k on satisfying 
n k =
0,
X qab =
2 k qab , X na
= –k na .
@ General references: Trautman PPAS(58);
Bondi et al PRS(62);
Sachs PR(62),
PRS(62);
McCarthy JMP(72);
Ashtekar & Xanthopoulos JMP(78);
Ashtekar & Schmidt
JMP(80);
McCarthy
PRS(92)
[generalization]; Barnich & Troessaert a0909 [and conformal transformations].
@ Representations: Melas JMP(04)
[and gravitational instantons]; Dappiaggi PLB(05)ht/04 [uirr's
and particle classification].
@ Related topics: Bicak & Pravdova JMP(98)gq [electrovac];
Longhi & Materassi
JMP(99)ht/98 [realization];
Barnich & Compère CQG(07)
[3D, central extension].
Dynamics: Conserved Quantities and News > s.a. [energy
in general relativity]; conservation laws.
* Idea: The infinitesimal generators of the BMS group are associated
with
fluxes of conserved quantities at .
* Conserved quantities: One can define the Bondi 4-momentum.
* Bondi news tensor:
The tensor Nab
at , which measures
the amount of energy-momentum carried to infinity by gravitational waves.
@ General references: Sachs PR(62);
Ashtekar & Magnon-Ashtekar JMP(79);
Ashtekar& Streubel PRS(81);
Dray & Streubel CQG(84);
Dray CQG(85);
Goldberg PRD(90);
Valiente
GRG(99)gq/98 [BS
vs NP]; Jezierski CQG(02)gq.
@ Bondi energy-momentum, mass: Bondi et al PRS(62);
Ashtekar & Horowitz PLA(82)
[cannot be null]; Jezierski APPB(98)gq/97 [relationship
with quasilocal]; Katz & Lerer CQG(97);
Zhang ATMP(06)gq/05 [relationship
with ADM]; Kozameh et al CQG(08)
[physical content]; Ishibashi CQG(08)
[higher-dimensional]; > s.a. gravitational energy-momentum.
@ Mass positivity: Israel & Nester PLA(81); Schoen & Yau PRL(82).
@ Angular momentum: Ashtekar in(80);
Cresswell & Zimmerman CQG(86); Moreschi CQG(86);
Nahmad-Achar CQG(87).
@ Bondi news:
Bishop et al PRD(97)gq,
Bishop & Deshingkar PRD(03)gq,
Deadman & Stewart CQG(09)-a0902 [numerical].
@ Related topics: Friedrich CQG(03)gq/02 [spin-2
fields]; Kozameh et al gq/06 [universal
cut functions, type II].
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send feedback and suggestions to bombelli at olemiss.edu – modified 18
sep
2009