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 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
= Ω2
gab,
\(\Omega|_{\cal I}\) = 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 Sab:=
Rab − \(1\over6\)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 \(\cal I\).
@ 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 GRG(18)-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];
Kehrberger a2105,
a2105 [non-smoothness of 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. symplectic structures; 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 qab = 2 k qab , \(\cal L\)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);
Barnich & Troessaert PRL(10)-a0909 [and conformal transformations];
Hollands et al CQG(17)-a1612
[gravitational memory and supertranslation, in four and higher dimensions];
Alessio & Esposito IJGMP(17)-a1709 [pedagogical review];
Calò a1805 [relation to the Carroll group].
@ Generalizations, approaches: McCarthy PRS(92);
Barnich & Lambert AMP(12)-a1102 [Newman-Unti approach];
Barnich & Troessaert JHEP(16)-a1601 [finite];
Ananth et al a2012 [without reference to asymptotic limits].
@ Charges at infinity: Barnich & Troessaert JHEP(11)-a1106 [BMS algebra];
Flanagan & Nichols PRD(17)-a1510 [extended BMS group and conserved charges];
Campoleoni et al Univ-a1712 [massless fields in any dimension];
Grant et al a2105 [Wald-Zoupas formalism].
@ 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 JMP(17)-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 Nab
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];
Wieland JHEP(21)-a2012 [as a time-dependent Hamiltonian];
Frauendiener & Stevens a2104;
> 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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