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].

@ __With symmetries__: Beig GRG(80) [static];
Beig & Simon GRG(80),
Kennefick & Ó Murchadha CQG(95)gq/93 [stationary];
Beig & Chruściel JMP(96)gq/95;
> 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) [i^{0} 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 i^{0},
where *M*' is C^{>1}
and *g'* is C^{>0}, together
with an imbedding *i*: *M* → *M*, such that

(1) \(\bar J\)(i^{0}) = *M'* \ *M*,
i.e., i^{0} is at spatial infinity;

(2) There exists a conformal factor Ω: *M*' → \(\mathbb R\),
C^{2} at i^{0},
C^{∞} elsewhere, such that
*g'*_{ab}|_{M}
= Ω^{2} *g*_{ab},
Ω\(|_{i^0}\) = ∇*'*_{a} Ω\(|_{i^0}\) = 0,
∇*'*_{a}∇*'*_{b}
Ω\(|_{i^0}\) = 2 *g'*_{ab}
(Ω ~ *ρ*^{−2});

(3) *R*_{ab} admits a regular
direction-dependent limit at i^{0} (matter
sources fall off like *ρ*^{−4}).

* __Re boundary conditions__: If the metric
were C^{1} at i^{0}, the mass would vanish;
If the metric were C^{0}, the mass would not be defined;
*R'*_{abcd} blows up at i^{0},
but Ω^{1/2} *R*'_{abcd}
→ *R*_{abcd}(*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 i^{0}";
It has a natural metric *h*_{ab}
= *g*_{ab} −
*η*_{a}*η*_{b}.

* __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
i^{0}, i.e., *ω* =
1 + Ω^{1/2} *α*, where
*α*...; If *ω* is C^{0}
at i^{0}, 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 *x*^{m}, then *η*'
with *x*'^{m}:=
*x*^{m}
+ *C*^{m} ln *ρ*,
for all C^{m}, 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 i^{0}
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 *X*^{a}
such that at i^{0},
*X*^{a}
∈ C^{>0} and
*X*^{ a}
= 0 (not to move i^{0}),
∇*'*_{(a}
*X*_{b)} = 0
(not to change the metric, asymptotic Killing vector fields),
and ∇*'*_{a}
∇*'*_{(b}
*X*_{c)}
= ∇*'*_{a}
*ω* *g'*_{bc},
for some *ω* (= 1 at i^{0})
(so *X* can be associated with some change in conformal factor);
If ∇_{a} *ω* = 0,
then *X* generates the identity at i^{0};
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}
*X*_{b)} = 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 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 *V*^{a} at
i^{0} is

*V*^{a} *p*_{a}
= \(1\over8\pi G\)∫_{C}
*E*_{ab}* V*^{b}
d*S*^{a} ,

where *E*_{ab}
is the electric part of the Weyl tensor on the hyperboloid
\(\cal D\) at i^{0},
and *C* is any cross section of \(\cal D\).

@ __Relationship with ADM__:
Śniatycki RPMP(89);
Huang & Zhang gq/05-proc,
ScCh(07)gq/06.

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