Asymptotic
Flatness at Spatial Infinity| Asymptotic
Flatness at Spatial Infinity |
In General > s.a. [solutions
of general relativity]; ADM;
canonical general relativity; initial
value formulation; multipole
moments.
* 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 
,
mainly because of the intricate differentiable structure.
@ Early work: in Lichnerowicz 39 [for stationary spacetimes].
@ General: 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);
Chrusciel JMP(89),
JMP(89);
Petrov IJMPD(95),
IJMPD(97);
Hayward PRD(03)gq.
@ With symmetries: Beig GRG(80) [static]; Beig & Simon GRG(80),
Kennefick & Ó Murchadha
CQG(95)gq/93 [stationary];
Beig & Chrusciel
JMP(96)gq/95; > s.a. initial
value formulation.
@ Existence: Reula CMP(89); Lindblad & Rodnianski CMP(05)m.AP/03
[wave coordinates].
@ Symmetries, conservation laws: Goldberg PRD(90); Perng JMP(99)gq/98;
Lusanna & de
Pietri gq/99;
Szabados CQG(03)gq.
@ 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];
Husa gq/01-in
[numerical]; Finster & Kraus CJM(05)m.DG/03 [curvature
estimates]; Shiromizu & Tomizawa PRD(04)gq [D > 4];
Zenginoglu gq/06-in
[numerical].
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 spacetime
(M', g'),
which is Cinfty everywhere
except at a point
i0,
where M' is C>1 and g' is
C>0,
together
with an imbedding i: M → M, such that
(1) J-bar(i0) = M' \ M,
i.e., i0 is
at spatial infinity;
(2) 
:
M' → R,
C2 at
i0, Cinfty elsewhere,
g'ab|M = 2
gab,
|i^0 = 
'a |i^0 =
0,'a'b |i^0 =
2 g'ab ( 
–2);
(3) Rab admits a regular direction-dependent limit at i0 (matter
sources fall off like –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 1/2 R'abcd →
Rabcd(
),
a regular
direction-dependent limit; Its Weyl part is coded in the electric and
magnetic
parts, with potentials provided by the Ricci part.
* Relationships: Implies
asymptotic flatness at spatial infinity in the ADM sense.
* Hyperboloid 
:
The
most useful construction at spi is the hyperboloid
of unit timelike vectors,
the "space of directions of approach
to i0"; It has a natural metric hab = gab – ab.
* 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].
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 ' = 
,
where = 1 and C>0 at
i0, i.e., =
1 + 1/2 
,
where ...;
If
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 (in
addition to supertranslations); If is
one such metric, with Cartesian
chart xm,
then '
with x'm:= xm + Cm ln ,
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, 
= /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), '(aXb) =
0 (not to change
the
metric, asymptotic Killing vector fields), and 'a '(b Xc) = 'a g'bc,
for some (=
1
at i0) (so X can
be
associated with some change in conformal factor); If a =
0, then
X generates the identity at i0; It
belongs
to I.
* Remark: X defines
a vector field
on by X' a:=
lim –1/2 X a
(tangential to since (a Xb)
= 0).
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
= (1/8
G) 
C Eab Vb dSa ,
where Eab is the electric
part of the Weyl tensor on the hyperboloid
at i0,
and C is
any cross section of .
@ Relationship with ADM: Sniatycki RPMP(89);
Huang & Zhang gq/05, gq/06.
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
22 jun 2008