Asymptotic Flatness at Spatial Infinity| Asymptotic Flatness at Spatial Infinity |
In General
> s.a. ADM formulation; canonical general relativity;
initial-value formulation; multipole moments
\ solutions of general relativity.
* Idea: One gets information
on conserved quantities of spacetime, but there are no equations, the dynamics
is not recorded.
* History: It took about ten
years more to develop than the structure at \(\cal I\), mainly because of the
intricate differentiable structure; 2016, Supertranslations are used to prove
that black holes can have soft hair.
@ Early work: in Lichnerowicz 39 [for stationary spacetimes].
@ General references: Geroch JMP(72),
in(77);
Sommers JMP(78);
Ashtekar in(80),
FP(85);
Beig & Schmidt CMP(82);
Ashtekar & Magnon JMP(84);
Beig PRS(84);
Winicour FP(85);
Chruściel JMP(89),
JMP(89);
Petrov IJMPD(95),
IJMPD(97);
Hayward PRD(03)gq;
Compère & Dehouck CQG(11)-a1106 [without imposing parity conditions];
Henneaux & Troessaert a1904 [rev];
Ali Mohamed & Valiente Kroon a2103 [comparison of Ashtekar's and Friedrich's formalisms].
@ With symmetries: Beig GRG(80) [static];
Beig & Simon GRG(80),
Kennefick & Ó Murchadha CQG(95)gq/93 [stationary];
Beig & Chruściel JMP(96)gq/95;
Prabhu & Shehzad CQG(20)-a1912 [asymptotic symmetries and charges];
> s.a. initial-value formulation.
@ Existence: Reula CMP(89);
Lindblad & Rodnianski CMP(05)m.AP/03 [wave coordinates].
@ Numerical: Husa AIP(00)gq/01;
Zenginoğlu JPCS(07)gq/06.
@ Other:
Friedrich CMP(88) [radiativity condition];
Ashtekar & Romano CQG(92) [i0 as boundary];
Thiemann CQG(95)gq/93 [Ashtekar variables];
Herberthson CQG(98)gq/97 [diff];
Finster & Kraus CJM(05)m.DG/03 [curvature estimates];
Shiromizu & Tomizawa PRD(04)gq,
Tanabe et al JMP(09)-a0902 [higher dimensions];
Henneaux & Troessaert JHEP(18)-a1803 [asymptotic symmetries of Maxwell theory],
JHEP(18)-a1805 [Einstein-Maxwell system];
Gibbons a1902.
Spi Formalism
> s.a. Penrose Diagram; Ripple.
$ Def: A spacetime (M, g)
is said to be asymptotically flat at spatial infinity if there exists a conformally related
spacetime (M', g'), which is C∞
everywhere except at a point i0,
where M' is C>1 and
g' is C>0, together with
an embedding i: M → M, such that
(1) \(\bar J\)(i0) = M' \ M,
i.e., i0 is at spatial infinity;
(2) There exists a conformal factor Ω: M' → \(\mathbb R\),
C2 at i0,
C∞ elsewhere, such that
g'ab|M
= Ω2 gab,
Ω\(|_{i^0}\) = ∇'a Ω\(|_{i^0}\) = 0,
∇'a∇'b
Ω\(|_{i^0}\) = 2 g'ab
(Ω ~ ρ−2);
(3) Rab admits a regular
direction-dependent limit at i0 (matter sources
fall off like ρ−4).
* Re boundary conditions: If the
metric were C1 at i0,
the mass would vanish; If the metric were C0, the mass
would not be defined; R'abcd blows up
at i0, but Ω1/2
R'abcd →
Rabcd(η),
a regular direction-dependent limit; Its Weyl part is coded in the electric
and magnetic parts, with potentials provided by the Ricci part.
* Relationships: It
implies asymptotic flatness at spatial infinity in the ADM sense.
* Hyperboloid \(\cal D\): The most
useful construction at spi is the hyperboloid \(\cal D\) of unit timelike vectors,
the "space of directions of approach to i0";
It has a natural metric hab
= gab −
ηaηb.
* First-order structure: Universal,
the C>1 manifold with well-defined tangent
space and metric.
* Connections: They correspond to
equivalence classes of connections on spacetime; They need not always be the same.
* Higher-order structure: Not
meaningful, reflected in the fact that the curvature blows up at spi.
@ References:
Bergmann & Smith PRD(93) [structure];
Valiente a0808 [regularity conditions].
Ambiguities > s.a. asymptotic flatness.
* Non-uniqueness: Equivalent conformal
completions may be obtained by supertranslations, inequivalent ones by a 4-parameter
family of logarithmic transformations; If we use Ω' = ω Ω,
where ω = 1 and C>0 at
i0, i.e., ω =
1 + Ω1/2 α, where
α...; If ω is C0
at i0, we get an inequivalent completion
(e.g., related by a log translation).
* Logarithmic transformations:
An ambiguity in the choice of flat metric η (in addition to
supertranslations); If η is one such metric, with Cartesian
chart xm, then η'
with x'm:=
xm
+ Cm ln ρ,
for all Cm, will also do; In some
cases there is a preferred or asymptotic frame; In the spi framework, a 4-parameter
family of inequivalent, logarithmically related completions which give the same
physical answers, and can be considered as gauge.
@ Logarithmic transformations:
Bergmann PR(61);
Beig & Schmidt CMP(82);
Ashtekar FP(85).
Symmetries / The Spi Group
* Idea: The set of all
diffeomorphisms that leave the spi structure invariant, modulo those
which generate the identity at i0
and leave each ripple fixed, \(\cal G\) = \(\cal D\)/I.
* Structure: Similar
to the BMS group, a semidirect product of the Lorentz group and the
supertranslation group; It has a preferred translation subgroup,
but not a preferred Lorentz (and hence Poincaré) subgroup.
* Generators: Vector
fields Xa
such that at i0,
Xa
∈ C>0 and
X a
= 0 (not to move i0),
∇'(a
Xb) = 0
(not to change the metric, asymptotic Killing vector fields),
and ∇'a
∇'(b
Xc)
= ∇'a
ω g'bc,
for some ω (= 1 at i0)
(so X can be associated with some change in conformal factor);
If ∇a ω = 0,
then X generates the identity at i0;
It belongs to I.
* Remark: X defines a vector
field on \(\cal D\) by X' a:= lim
Ω−1/2 X a
(tangential to \(\cal D\) since ∇(a
Xb) = 0).
@ References: Goldberg PRD(90);
Perng JMP(99)gq/98;
Lusanna & de Pietri gq/99;
Szabados CQG(03)gq;
Dehouck PhD-a1112 [electric and magnetic aspects];
Troessaert CQG(18)-a1704 [BMS4 algebra as the asymptotic symmetry algebra];
Henneaux & Troessaert JHEP(18)-a1801 [new boundary conditions and the BMS group].
Energy-Momentum > s.a. ADM formalism;
energy in general relativity [and generalization].
$ Def: The energy-momentum component
along a tangent vector Va at
i0 is
Va pa = \(1\over8\pi G\)∫C Eab Vb dSa ,
where Eab
is the electric part of the Weyl tensor on the hyperboloid
\(\cal D\) at i0,
and C is any cross section of \(\cal D\).
@ Relationship with ADM:
Śniatycki RPMP(89);
Huang & Zhang gq/05-proc,
ScCh(07)gq/06.
main page
– abbreviations
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– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 6 mar 2021