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 \(\cal I\)^{+}];
Costa e Silva et al a1807 [new definitions]; Ashtekar et al a1808 [BMS group, soft charges and infrared issues].

@ __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(16)-a1601 [finite];
Hollands et al a1612
[gravitational memory and supertranslation, in four and higher dimensions];
Alessio & Esposito IJGMP(17)-a1709 [pedagogical review];
Calò a1805 [relation to the Carroll group].

@ __Charges at infinity__: Barnich & Troessaert JHEP(11)-a1106 [BMS algebra];
Flanagan & Nichols PRD(17)-a1510 [extended BMS group and conserved charges];
Campoleoni et al a1712 [massless fields in any dimension].

@ __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].

**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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