Quasi-Asimptotically Flat Spacetimes and Their ADM Mass 1 Introduction

25 Pages·1996·0.28 MB·English

Quasi-Asimptotically Flat Spacetimes and Their
ADM Mass
(cid:3) y
Ulises Nucamendi and Daniel Sudarsky
y
Instituto de Ciencias Nucleares
Universidad Nacional Aut(cid:19)onoma de M(cid:19)exico
A. P. 70-543, M(cid:19)exico, D. F. 04510, M(cid:19)exico.
(cid:3)
Departamento de F(cid:19)(cid:16)sica
Centro de Investigaci(cid:19)on y de Estudios Avanzados del I.P.N.
A. P. 14-741, M(cid:19)exico, D. F. 07000, M(cid:19)exico.
Abstract
We de(cid:12)ne spacetimes that are asymptotically (cid:13)at, except for a de(cid:12)cit solid
angle(cid:11), andpresentade(cid:12)nitionoftheir\ADM" mass, whichis(cid:12)niteforthisclass
of spacetimes, and, in particular, coincides with the value of the parameter M of
the global monopole spacetime studied by Vilenkin and Barriola [1]. Moreover,
we show that the de(cid:12)nition is coordinate independent, and explain why it can, in
some cases, be negative.
1 Introduction
Spacetimes that are asymptotically (cid:13)at (A.F.), and the properties of their associated
ADM mass (or, more generally, four momentum) have been studied exhaustively [2], [3],
[4], [5], [6], as these are taken as the natural idealizations of those spacetimes that repre-
sent isolated objects in General Relativity. However, these are not the only idealizations
that are of interest, since, in fact, our own universe, having a nonzero average density,
is not asymptotically (cid:13)at, and, moreover, localized solutions that can not be naturally
accommodated within the asymptotically (cid:13)at framework, can naturally be considered as
describing regions of our universe. Such is the case for, the so called global monopoles
2
studied by Vilenkin and Barriola [1], for which the energy density drops o(cid:11) only as 1=r .
Thus, these monopole spacetimes are not asymptotically (cid:13)at in the standard sense. In
particular they have a divergent value for the expression that would have de(cid:12)ned their
(cid:3)
email: ulises@(cid:12)s.cinvestav.mx
y
email: [email protected]
1
ADM mass. However, it is clear that for r su(cid:14)ciently large, that the monopole density
becomes smaller than the mean matter density of the universe, the particular form of its
subsequent rate of decay is of noconsequence whatsoever. Thus, it should make physical
sense to seek a notion of the asymptotic behavior of a spacetime that is appropriate for
the description of such type of \quasi-isolated objects" and to study the properties that
can be de(cid:12)ned for them.
In this paper, we exhibit the (cid:12)rst steps of such a program for the class of spacetimes
that are asymptotically (cid:13)at, but for a de(cid:12)cit solid angle (cid:11) (A.F.D.A.(cid:11)), which we de(cid:12)ne
morespeci(cid:12)cally below. This classincludes theglobalmonopoleofVilenkin andBarriola
[1]andsome perturbationsthereof. Wegiveade(cid:12)nitionofmassthatis(cid:12)nitefortheclass
of (A.F.D.A.(cid:11)) spacetimes, and, moreover, coincides with the value of the parameter
M of the global monopole solution. Finally, we brie(cid:13)y discuss some aspects of the
asymptotic symmetry group of these spacetimes.
Previous works along these general lines include the analysis of Abbott and Deser [7]
of the canonical mass for asymptotically De Sitter and Anti De Sitter spacetimes (See
also [8]). In that study, the authors encountered problems related to the fact that in
the De Sitter case the \mass" can only be associated with an horizon sized region of a
Cauchy hypersurface, and in the Anti De Sitter case there is no Cauchy hypersurface at
all. In the present work, we do not encounter those problems.
We shall adhere to the following conventions on index notation in this paper: Greek
indices ((cid:11), (cid:12), (cid:22), (cid:23),...) range from 0 to 3, and denote tensors on (four-dimensional)
spacetime. Latin indices, alphabetically located after the letter i (i,j,k,...), denote inter-
nal indices in the space of scalar (cid:12)elds, and range from 1 to 3; whereas Latin indices
from the beginning of the alphabet (a,b,c,d,...) range from 1 to 3, and denote tensors
on a spatial hypersurface (cid:6).
Metrictensorsareemployed throughoutthepaper: gdenotesthespacetime metric, h
denotes the metric on a spatial hypersurface (cid:6). The corresponding covariant derivatives
are denoted , for the metric g, and D for the metric h.
r
The signature of the metric g is ( ;+;+;+). Geometrized units, for which GN =
(cid:0)
c = 1 are used in this paper.
2 The Global Monopole Spacetime
The theory of a scalar (cid:12)eld with spontaneously broken internal O(3) symmetry, mini-
mally coupled to gravitation, is described by the action:
(cid:22) 4
S = ( g)[(1=16(cid:25))R (1=2) (cid:30)i (cid:22)(cid:30)i V((cid:30))]d x: (1)
Z q (cid:0) (cid:0) r r (cid:0)
where R is the scalar curvature of the spacetime metric, (cid:30)i is a triplet of scalar (cid:12)elds,
2 1=2
and V((cid:30)), is potential depending only on the magnitude (cid:30) = ( i(cid:30)i) , which we will
2 2 2 P
usually take to be the \Mexican Hat" V((cid:30)) = ((cid:21)=4)((cid:30) v ) .
(cid:0)
We are interested in spacetimes with topology (cid:6) R, where (cid:6) has the topology of
3 (cid:2) 2
(R B) C, with B a 3-ball, and C a compact manifold with S boundary. The
(cid:0) [
2
requirement that (cid:30) v in the asymptotic regions separates the con(cid:12)guration space into
!
topological sectors according to the winding number of the asymptotic behavior of (cid:30)i.
We will focus on the sector with winding number one, corresponding to the asymptotic
behavior characteristic of the Hedgehog ansatz:
(cid:30)i vxi=r: (2)
(cid:25)
i
where the x are asymptotic cartesian coordinates. Within this sector, there is a static,
spherically symmetric solution [1] with metric given by:
2 2 2 2 2 2 2
ds = B(r)dt +A(r)dr +r (d(cid:18) +sin ((cid:18))d’ ); (3)
(cid:0)
and scalar (cid:12)eld
(cid:30)i = vf(r)xi=r (4)
and with the following asymptotic behavior of,
(cid:0)1 2 2
B = A = 1 (cid:11) 2M=r +O(1=r ); f 1+O(1=r ) (5)
(cid:0) (cid:0) (cid:25)
2
where (cid:11) = 8(cid:25)v .
1=2 (cid:0)1=2
Rede(cid:12)ning the r and t coordinates as r (1 (cid:11)) r and t (1 (cid:11)) t, respectively,
(cid:0)3=2 ! (cid:0) ! (cid:0)
and de(cid:12)ning M = M(1 (cid:11)) , we obtain the asymptotic form for the metric:
(cid:0)
f
2 2 (cid:0)1 2 2 2 2 2
ds = (1 2M=r)dt +(1 2M=r) dr +(1 (cid:11))r (d(cid:18) +sin ((cid:18))d’ ): (6)
(cid:0) (cid:0) (cid:0) (cid:0)
f f
The parameter M has previously been associated with the mass of the con(cid:12)guration (
despite the fact fthat the ADM mass formally diverges) because it can be seen that the
proper acceleration of the ((cid:18);’;r) = constant world lines is a = M=(r(r 2M)). Thus
(cid:0) (cid:0)
M plays a role of a Newtonian mass. However, let us emphasize thfat the ADfM mass of
tfhe solution is not de(cid:12)ned and that the formal application of the ADM formula actually
diverges. It is also rather unexpected that in the speci(cid:12)c solution M turns out to be
negative [9]. f
Another intriguing question is posed by the fact that in [10] and [11] a very close
connection between staticity and extrema of mass was found, and thus it is somehow
surprising that here we (cid:12)nd static solutions that do not seem to be extrema of anything.
We will see that these points seem to be completely resolved by the introduction
of the notion of A.F.D.A (cid:11) spacetime and the new de(cid:12)nition of ADM mass that is
appropriate for them.
3
3 The New Class of Spacetimes and Their ADM
Mass
We start with the de(cid:12)nition of a standard spacetime which is going to play the role that
Minkowski spacetime plays for the case of asymptotically (cid:13)at spacetimes; namely, it is
going to be used in the speci(cid:12)cation of the asymptotic behavior de(cid:12)ning the the new
class of spacetimes, and as a benchmark for the de(cid:12)nition of the \ADM mass".
We take the spacetime to be (R3 ~0) R (~0 is the origin of R3) with the metric :
(cid:0) (cid:2)
2 0 (cid:22) (cid:23) 2 2 2 2 2 2
ds = g(cid:22)(cid:23)dx dx = dt +dr +r (1 (cid:11))(d(cid:18) +sin ((cid:18))d’ ): (7)
(cid:0) (cid:0)
We will call this spacetime the standard asymptotically-(cid:13)at-but-for-a-de(cid:12)cit-angle (cid:11)
spacetime or (S.A.F.D.A (cid:11)).
It can be considered as the global monopole solution in the limit when (cid:21) , or,
! 1
more precisely, as the solution for the model in which the potential has been replaced
3 2
by the constraint i=1(cid:30)i(cid:30)i = v .
P
Theasymptotic featuresofthisspacetimecanbeanalyzedbycarryingoutacompact-
i(cid:12)cation analogous to the standard conformal compacti(cid:12)cation of Minkowski spacetime
[3],[5], [6], [12],[13]. In fact, we can introduce new coordinates u,v according to,
u = t+r; v = t r: (8)
(cid:0)
In these coordinates, the S.A.F.D.A (cid:11) metric takes the form,
2 1 2 2 2 2
ds = dudv+ (v u) (1 (cid:11))(d(cid:18) +sin ((cid:18))d’ ); (9)
(cid:0) 4 (cid:0) (cid:0)
which is seen to be conformally related to the metric
~2 2 2 2 2 2 2
ds = dT +dR +(1 (cid:11))(sin R)(d(cid:18) +sin ((cid:18))d’ ); (10)
(cid:0) (cid:0)
2
i.e., d~s = (cid:10)2ds2 with conformal factor (cid:10)2 = 24 2 , where
(1+v )(1+u )
T = arctanv +arctanu; R = arctanv arctanu; (11)
(cid:0)
and where T;R have the following ranges,
(cid:25) < T +R < (cid:25); (cid:25) < T R < (cid:25); 0 < R < (cid:25): (12)
(cid:0) (cid:0) (cid:0)
This spacetime can be extended to T = (cid:25) + R and T = (cid:25) R for R (0;(cid:25)),
(cid:0)(cid:0) (cid:0) + 2
which correspond to past null in(cid:12)nity , and future null in(cid:12)nity , respectively.
J J
Unfortunately, this spacetime can not be extended to (T = 0, R = (cid:25)), which would
4
0
have corresponded to spatial in(cid:12)nity { , because here there is a true singularity that is
evidenced by the fact that the scalar curvature diverges at R = (cid:25) (in fact, the curvature
2
scalar is (6 4(cid:11)+2(cid:11)cot R)=(1 (cid:11))). There is a similar singularity at r = 0 present in
(cid:0) (cid:0)
the real spacetime, but we will not be concerned with it because we will be interested in
spacetimeswhicharesimilartotheS.A.F.D.A(cid:11)spacetimeonlyintheasymptoticregion,
and these will include many regular spacetimes in which this conical singularity will be
\smoothed out". In the S.A.F.D.A (cid:11) spacetime, the presence of the singularity is the
price that we pay in order to have a very simple spacetime to take as the standard one.
We could have equally chosen any of the \smoothed out" spacetimes as the standard,
but there seems to be no canonical choice.
(cid:0) + 0
The fact that we can introduce the concepts of and , but not { , suggests that
J J
in the terminology of [14] these spacetimes would be \asymptotically simple", but not
\asymptotically empty".
De(cid:12)nition: We will say that a spacetime (or a spacetime region) (M;g(cid:22)(cid:23)), with topol-
3
ogy (R B) R, is asymptotically (cid:13)at, but for a de(cid:12)cit angle (cid:11) (A.F.D.A (cid:11)), if there
(cid:0) (cid:2)
exist coordinates (t;r;(cid:18);’), for which the metric can be written as
0
g(cid:22)(cid:23) = g(cid:22)(cid:23) +g~(cid:22)(cid:23); (13)
where g~(cid:22)(cid:23) has the form:
(cid:22) (cid:23) 2 2
g~(cid:22)(cid:23)dx dx = attdt +arrdr +2artdrdt
2 2 2 2
+r [a(cid:18)(cid:18)d(cid:18) +a’’sin ((cid:18))d’ +2a(cid:18)’sin((cid:18))d(cid:18)d’] (14)
+2r[at(cid:18)dtd(cid:18) +ar(cid:18)drd(cid:18)]+2r[at’sin((cid:18))dtd’+ar’sin((cid:18))drd’];
with the functions a(cid:22)(cid:23) O(1=r)(notethatthe a(cid:22)(cid:23) depend onthechoice ofbackground).
(cid:25)
We review now the \3 + 1" hamiltonian formulation of the Einstein-Scalar (E-S) theory,
analogous to that given in [11], and proceed to specialize considerations to the phase
space of regular, asymptotically-(cid:13)at-but-for-a-de(cid:12)cit-angle (cid:11) initial data.
ab
Initial data in E-S theory consists of the speci(cid:12)cation of the (cid:12)elds (hab;(cid:25) ;(cid:30)i;Pi) on
th
a three-dimensional manifold, (cid:6). Here hab is a Riemannian metric on (cid:6), (cid:30)i is the i
ab
scalar (cid:12)eld component , (cid:25) is the canonically conjugate momentum to hab, and Pi is
the momentum canonically conjugate to (cid:30)i.
Einstein-Higgs theory is a theory with constraints. On a hypersurface, (cid:6), the allowed
initial data are restricted to those that at each point x (cid:6) satisfy
2
(3) ab 2
0 = 0 = ph[ R+1=h((cid:25)ab(cid:25) (1=2)(cid:25) )]
C (cid:0) (cid:0)
ab
+(PiPi=2ph)+ph((1=2)h Da(cid:30)iDb(cid:30)i +V((cid:30))); (15)
b
0 = a = 2phDb((cid:25)a =ph)+PiDa(cid:30)i: (16)
C (cid:0)
5
A (cid:12)xed volume element (cid:17)abc for the manifold (cid:6) is assumed to be given, and h relates the
volume element (cid:15)abc corresponding to the metric hab tothe former through(cid:15)abc = ph(cid:17)abc.
The equations of motion of E-S theory can be derived from a Hamiltonian H. (See also
[15]).
(cid:22)
H = (N (cid:22)): (17)
Z(cid:6) C
where the (cid:12)xed volume element (cid:17)abc is understood in all volume integrals over (cid:6).
(cid:22) 0 a
N = (N ;N ) corresponds to the \lapse" function and \shift" vector, respectively, of
(cid:22)
the foliation of the \evolved spacetime". Recall that N are not dynamical variables,
and can be chosen arbitrarily .
A general variation of the initial data will produce a variation in the Hamiltonian
that can be written as:
ab ab i i
(cid:14)H = ( (cid:14)hab + ab(cid:14)(cid:25) + (cid:14)(cid:30)i + (cid:14)Pi)+Surface terms: (18)
Z(cid:6) P Q R S
The evolution equations can be obtained from Hamilton’s principle, if we restrict con-
sideration to variations of compact support,
(cid:25)_ab = ab; h_ab = ab; P_i = i; (cid:30)_i = i; (19)
(cid:0)P Q (cid:0)R S
ab
where speci(cid:12)c expressions for ; ab; i, and i, and the surface terms are given by:
P Q R S
ab 0 (3) ab (3) ab 0 ab cd 2
= phN [ R (1=2) Rh ] (N =2ph)h [(cid:25)cd(cid:25) (1=2)(cid:25) ]
P (cid:0) (cid:0) (cid:0)
0 ac b ab a b 0 ab c 0
+2(N =ph)[(cid:25) (cid:25)c (1=2)(cid:25)(cid:25) ] ph[D D N h D DcN ]
(cid:0) (cid:0) (cid:0)
c ab c(a b) 0 ab
phDc(N (cid:25) =ph)+2(cid:25) DcN N h (PiPi=2ph)
(cid:0) (cid:0)
0 ab cd 0 a b
+(1=2)phN h [(1=2)h Dc(cid:30)iDd(cid:30)i +V((cid:30))] phN D (cid:30)iD (cid:30)i; (20)
(cid:0)
0
ab = (2N =ph)[(cid:25)ab (1=2)hab(cid:25)]+2D(aNb); (21)
Q (cid:0)
0 @V((cid:30)) a 0 b a
i = phN [ DaD (cid:30)i] phDb(N )D ((cid:30)i) phDa(N Pi=ph); (22)
R @(cid:30)i (cid:0) (cid:0) (cid:0)
a 0
i = N Da(cid:30)i +N Pi=ph; (23)
S
0 ac bd ab cd
surfaces terms = dSa[(DbN )(h h h h )(cid:14)hcd
Z@(cid:6) (cid:0)
6
0 ac bd ab cd b a
N (h h h h )Db((cid:14)hcd) 2(N =ph)(cid:14)(cid:25)b
(cid:0) (cid:0) (cid:0)
a bc 0 a a
+(N =ph)(cid:25) (cid:14)hbc +N (D (cid:30)i)(cid:14)(cid:30)i+(N =ph)Pi(cid:14)(cid:30)i]: (24)
These eqs. are known to be equivalent to the four-dimensional E-S equations. However,
as pointed out by Teitelboim[15], this is not a satisfactory application of Hamilton‘s
principle, which must consider unrestricted variations within the phase space. If we
specify the phase space to be that of asymptotically (cid:13)at regular initial data, and con-
sider evolution that corresponds asymptotically to a \time translation", the problem is
resolved by adding asurface term to the Hamiltonian. The surface term is just the ADM
mass. If we now specify phase space tobe thatof(A.F.D.A.(cid:11))regular initialdata, which
we will de(cid:12)ne below, it will turn out that, again, a surface term can be added to the
Hamiltonian that results in a satisfactory Hamiltonian for the application of Hamilton‘s
principle, for evolution corresponding asymptotically to a \time translation", this is:
0
N 1+O(1=r); (25)
(cid:25)
a
N O(1=r): (26)
(cid:25)
De(cid:12)nition: We will call A.F.D.A.(cid:11) regular initial data to a speci(cid:12)cation of the (cid:12)elds
ab
(hab;(cid:25) ;(cid:30)i;Pi)onathree-dimensionalmanifold,(cid:6),whichful(cid:12)llsthefollowingconditions
at in(cid:12)nity,
0
hab hab +(cid:14)hab; (cid:14)hab O(1=r); (27)
(cid:25) (cid:25)
2
hab;c O(1=r ); (28)
(cid:25)
ab 2
(cid:25) O(1=r ); (29)
(cid:25)
2
(cid:30)i vxi=r+O(1=r ); (30)
(cid:25)
Pi O(1=r); (31)
(cid:25)
0
where hac is the S.A.F.D.A (cid:11) spatial metric de(cid:12)ned for the hypersurface t = constant
0
(7), and Db the covariant derivative associated with it.
The surface term, whose variation will cancel the nonvanishing surface term in (24),
when asymptotic conditions are imposed, is:
0ac 0bd 0ab 0cd 0
16(cid:25)(1(cid:0)(cid:11))MADM(cid:11) = Z@(cid:6)dSa(h h (cid:0)h h )Db(hcd): (32)
7
This is clearly the natural generalization of the ADM mass, (in fact, it looks just like
the usual ADM formula, but with the quantities associated with the (cid:13)at metric replaced
by the S.A.F.D.A.(cid:11) metric), and just like this, it is the numerical value of the true
Hamiltonian ( a true generator of \time translations"); so, it is natural to interpret this
as the mass (or energy) of the A.F.D.A.(cid:11) spacetimes.
Thisinterpretationisreinforcedbythefactthat,whenappliedtotheGlobalMonopole
solution, it yields the value:
MADM(cid:11) = M~: (33)
4 The new mass formula is well de(cid:12)ned
The problem, in principle, with the above de(cid:12)nition stems from the fact that the for-
mula (32)involves geometries in two di(cid:11)erent spaces. Speci(cid:12)cally, we are using covariant
derivatives associated with one metric, and applying it to a second one. Thus, what
0 0
we have in fact is the following: A standard-setting Riemmanian Manifold ((cid:6) ;h ), a
test Riemmanian Manifold ((cid:6);h), and a mapping (we assume it is a di(cid:11)eomorphism)
0 (cid:3)
(cid:8) : (cid:6) (cid:6). The metric appearing in eq. (32) is actually (cid:8) (h). So, it is not clear, in
!
principle, that the de(cid:12)nition of MADM(cid:11) does not depend on (cid:8). Therefore, we need to
0
considertwosuchdi(cid:11)eomorphisms(cid:8)1,(cid:8)2 : (cid:6) (cid:6)(whichmustpreservetheasymptotic
!
form of the metric h, written in the coordinates (r;(cid:18);’), associated with the S.A.F.D.A
(cid:11)), and see that the value of ADM(cid:11) mass is the same for (cid:8)1 and (cid:8)2. This amounts to
a 0a a
considering a change of variables (x x = y ), which preserves the asymptotic form
!
of h, and then dropping the primes and substituting directly into the expression (32)
0 0
(without making any change in the variables there, i.e., without change hab and Db). It
0 3 0 a a
will be convenient to write everything in a chart : (cid:6) R on (cid:6) with x = (p)
!
Cartesian coordinates de(cid:12)ned by means oftheir usual relation with spherical coordinates
0 0
(r;(cid:18);’), p (cid:6) . In these coordinates, the S.A.F.D.A (cid:11) metric h is written as,
2
0 a b 2
hab = [(1 (cid:11))(cid:14)ab +(cid:11)x x =r ]: (34)
(cid:0)
We will use the following relation in the paper,
0 b a
xa = habx = x ; (35)
then we can write,
0 2
hab = [(1 (cid:11))(cid:14)ab +(cid:11)xaxb=r ]: (36)
(cid:0)
(cid:0)1 3 (cid:0)1 3
Using the charts (cid:8)1 : (cid:6) R and (cid:8)2 : (cid:6) R (strictly speaking, these maps
(cid:14) ! (cid:14) ! (cid:3) (cid:3)
will be de(cid:12)ned only in the asymptotic region), we can write (cid:8)1(h), and (cid:8)2(h) in the
8
a a (cid:0)1 a a (cid:0)1
"cartesian" coordinates x = (cid:8)1 (q), and y = (cid:8)2 (q), for q (cid:6), which we
(cid:14) (cid:14) 2
will refer to as h(1)ab and h(2)ab, respectively, thus we have:
2
h(1)ab = [(1 (cid:11))(cid:14)ab +(cid:11)xaxb=r +Aab]; (37)
(cid:0)
02
h(2)ab = [(1 (cid:11))(cid:14)ab +(cid:11)yayb=r +Bab]; (38)
(cid:0)
here we have,
0 @Aab 2 @Bab 02
Aab O(1=r); Bab O(1=r); c O(1=r ); c O(1=r ); (39)
(cid:25) (cid:25) @x (cid:25) @y (cid:25)
2 3 a a 02 3 a a a b
where r = a=1x x , r = a=1y y . The coordinates x and y are related by the
P a a P(cid:0)1 (cid:0)1 b
di(cid:11)eomorphism x = (cid:8)1 (cid:8)2 y .
f g (cid:14) (cid:14) (cid:14) f g
The proof that the value of ADM(cid:11) mass is independent of di(cid:11)eomorphisms (cid:8) that
preserve the asymptotic form of the metric h is basically a repetition (with some modi-
(cid:12)cations) of the proof of the analogous statement for the ADM mass of asymptotically
(cid:13)at spacetimes which has been given in [16].
Lemma 1: The conditions (39) imply that h(1)ab and h(2)ab are uniformly elliptic, this
0 a
is, forr;r su(cid:14)ciently large, there exist positive constants C1;C2, such that, vector X ,
8
3 3
(cid:0)1 a a a b a a
Ci X X h(i)abX X Ci X X ; (40)
aX=1 (cid:20) (cid:20) aX=1
where i = 1;2.
Proof: We will prove it, say, for h(1)ab. First we note that,
C
C > 0; r0 > 1 so that r > r0; Aab : (41)
9 9 8 j j (cid:20) r
From Schwartz’s inequality, we obtain,
3 a b 3 3 3
xaX xbX c c 1=2 b b 1=2 c c
( X X ) ( X X ) = (X X ): (42)
2
aX;b=1 r (cid:20) Xc=1 Xb=1 Xc=1
From the above inequalities, using (1 (cid:11)) > 0, (cid:11) > 0, and choosing D > (1 (cid:11)), we
(cid:0) (cid:0)
conclude,
3
a b a a
S1 = (D+(cid:11)+9C) > 0 so that r > r0 > 1; h(1)abX X S1 X X : (43)
9 8 (cid:20) aX=1
9
From inequality (41), we obtain,
3 3 3
9C c c 9C c c a b
r > r1 > r0; X X X X X X Aab: (44)
8 (cid:0) r1 cX=1 (cid:20) (cid:0) r cX=1 (cid:20) aX;b=1
Using the above inequalities, and choosing r1 9C=(1 (cid:11)), and r1 > r0, we have,
(cid:21) (cid:0)
3
9C a a a b
S2 = (1 (cid:11) ) > 0; so that r > r1; S2 X X h(i)abX X : (45)
9 (cid:0) (cid:0) r1 8 aX=1 (cid:20)
We take r2 = max(r0;r1), S = max(S1;S2), and then eqs. (43), (45) prove the lemma.
a a (cid:3)
Lemma 2: Let x and y be coordinate systems on (cid:6) for which the metrics (cid:8)1(h),
(cid:3) f g f g
(cid:8)2(h), respectively, preserve the asymptotic form (eqs. (37), (38)), and such that the
a a (cid:0)1 (cid:0)1 b
di(cid:11)eomorphism x = (cid:8)1 (cid:8)2 y is at least twice di(cid:11)erentiable. Then,
f g (cid:14) (cid:14) (cid:14) f g
this di(cid:11)eomorphism, and its inverse have the form,
3
a a b a
xa(y) = x (y) = Wby +(cid:17) (y); (46)
Xb=1
3
a (cid:0)1 a b a
ya(x) = y (x) = (W )bx +(cid:16) (x); (47)
Xb=1
where,
a a
a a @(cid:17) (y) @(cid:16) (x) a
(cid:17) O(1); (cid:16) O(1); b O(1=r); b O(1=r); Wb SO(3):
(cid:25) (cid:25) @y (cid:25) @x (cid:25) 2
(48)
Proof: We can regard the di(cid:11)eomorphism as de(cid:12)ning a change of coordinates,
c d
@x @x
h(2)ab(y) = h(1)cd(x(y)) a b; (49)
@y @y
c d
@y @y
h(1)ab(y) = h(2)cd(y(x)) a b: (50)
@x @x
th th
Introducing (cid:14)ab = ea(k)eb(k) (where ea(k) is the a -component of the k unitary cartesian
vector), contracting (49), (50) with (cid:14)ab and using the property (40), we obtain,
3 c c 3 c d
1 @x @x @x @x
a a h(1)cd(x(y)) a a = (cid:14)abh(2)ab(y) = h(2)abea(k)eb(k)
C aX;c=1 @y @y (cid:20) aX=1 @y @y
3
C ec(k)ec(k) = 3C; (51)
(cid:20) Xc=1
10