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 infinity, 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 \(\cal I\)^{} ("scri"), such that

(1) \(\cal I\) can be written as \(\cal I\)^{–}
∪ \(\cal I\)^{}^{+},
with \(\cal I\)^{}^{+} ∩ *J*^{–}[int(*M*)]
= \(\cal I\)^{}^{–} ∩ *J*^{+}[int(*M*)]
= Ø;

(2) There is a neighborhood of \(\cal I\)^{} which is strongly causal;

(3) The metrics are conformally related, *g'*_{ab} = Ω^{2}
*g*_{ab}, \(\Omega|_{\cal I}\)_{} =
0, ∇_{a} \(\Omega|_{\cal I}\)_{} ≠ 0
(gives Ω ~ *r*^{–1}
near \(\cal I\)), *g'*_{ab} ∇_{a}∇_{b} \(\Omega|_{\cal I}\)_{} =
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 *S*_{ab}:= *R*_{ab} – \(1\over6\)*R* *g*_{ab}.

* __Coordinate__: In practice, use some advanced or retarded null coordinate *u*,
and *x*, whose inverse measures affine length along integral curves
of d*u*; Then *x* = 0 is past/future \(\cal I\).

@ __General references__: Bondi et al PRS(62);
Penrose PRL(63), PRS(65);
Couch & Torrence JMP(72);
Geroch in(77); Geroch & Horowitz PRL(78);
Hayward KJPS-gq/03-conf
[refined def]; Marolf & Ross CQG(03)
[new definition of causal completion]; Newman & Nurowski CQG(06)
[CR structure]; Kozameh et al CQG(08)-a0802 [Bondi
4-momentum and physical content]; Jadczyk RPMP(12) [geometry and shape of Minkowski space conformal infinity]; Helfer PRD(14)-a1407 [radiative spacetimes are not asymptotically flat]; Ashtekar a1409-in [rev]; Harris a1611 [connections at \(\cal I\)^{+}].

@ __Cauchy data at scri__: Geroch JMP(78);
Schmidt & Stewart PRS(88); Chruściel & Paetz CQG(13)-a1307 [vacuum, existence].

@ __Smoothness of scri__: Andersson & Chruściel CMP(94); Valiente-Kroon
CQG(01) [detectability], CMP(04)gq/02,
CMP(04)gq/03 [and
radiation]; Chruściel & Paetz AHP(15)-a1403 [characteristic initial data]; Li & Zhu a1406 [local extension of future null infinity].

@ __Solutions__: Ashtekar & Dray pr(80), CMP(81);
Friedrich PRS(81);
Cutler & Wald CQG(89);
Moreschi JMP(90)
[FLRW 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]; Chruściel & Delay
CQG(02)gq [non-trivial];
Chruściel gq/02-proc.

@ __Higher-dimensional__: Hollands & Ishibashi JMP(05)gq/03,
ht/03-conf;
Hollands & Wald CQG(04)gq [non-existence
of \(\cal I\)^{} in odd dimensions]; Tanabe et al PRD(11)-a1104; > s.a. weyl tensor.

@ __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-wd [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]; Newman a1701 [structures at future null infinity]; > s.a. de sitter spacetime.

**Symmetries: The BMS Group** (Bondi-Metzner-Sachs) > s.a. vacuum states.

* __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)];
While in higher dimensions the symmetry algebra realizes the Poincaré algebra, in 3 and 4 spacetime dimensions
it contains infinitesimal supertranslations, that have been known since the 1960s, and infinitesimal superrotations.

* __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 \(\cal I\)^{} is
a generator of the BMS group if, for some scalar field *k* on \(\cal I\)^{} satisfying \(\cal L\)^{}_{n }*k* =
0,

\(\cal L\)^{}_{X} *q*_{ab} =
2 *k* *q*_{ab} , \(\cal L\)^{}_{X} *n*^{a}
= –*k* *n*^{a} .

@ __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 PRL(10)-a0909 [and conformal transformations]; Barnich & Lambert AMP(12)-a1102 [Newman-Unti approach]; Barnich & Troessaert JHEP(11)-a1106 [BMS charge algebra]; Barnich & Troessaert JHEP(16)-a1601 [finite]; Hollands et al a1612 [gravitational memory and supertranslation, in four and higher dimensions].

@ __Representations__: Melas JMP(04)
[and gravitational instantons]; Dappiaggi PLB(05)ht/04 [uirr's
and particle classification].

@ __Central extensions__:
Barnich & Compère CQG(07)
[3D]; Barnich & Troessaert PoS-a1102 [in arbitrary dimensions, classification].

@ __3D, representations__:
Barnich & Oblak JHEP(14)-a1403, JHEP(15)-a1502, Oblak CMP(15)-a1502 [induced and coadjoint]; Melas a1703.

@ __Related topics__: Bičák & Pravdová JMP(98)gq [electrovac];
Longhi & Materassi
JMP(99)ht/98 [realization]; Flanagan & Nichols PRD(17)-a1510 [extended BMS group and conserved charges].

**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 \(\cal I\)^{}.

* __Conserved quantities__: One can define the Bondi 4-momentum.

* __Bondi news tensor__:
The tensor *N*_{ab}
at \(\cal I\)^{}, 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 quantities]; Kozameh et al CQG(08)
[physical content]; Ishibashi CQG(08)
[higher-dimensional]; Chmielowiec & Kijowski RPMP(09)-a0812 [Hamiltonian
description]; Alexakis & Shao JEMS(16)-a1308 [bounds];
Chruściel & Paetz CQG(14)-a1401 [elementary proof of positivity]; > s.a. gravitational
energy-momentum.

@ __Mass positivity__: Israel & Nester PLA(81); Schoen & Yau PRL(82); Hollands & Thorne CMP(15)-a1307 [in higher dimensions].

@ __Angular momentum__: Ashtekar in(80);
Cresswell & Zimmerman CQG(86);
Moreschi CQG(86);
Nahmad-Achar CQG(87);
Helfer PRD(10);
Tanabe et al PRD(12)-a1203 [in higher dimensions].

@ __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 CQG(07)gq/06 [universal
cut functions, type II].

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