As has been noted before, we have the conserved quantities ,
and
given by
Equations(25
) – (27
). We shall also use the notation
,
,
and
In practice, it is often convenient to apply an isometry to a solution so that the conserved quantities
become as simple as possible. This is achieved in [75, Lemma 8.2, p. 681]:
Lemma 1 Consider a solution to Equations (11)–(12
). If
, there is an
isometry such that if
and
are the constants of the transformed solution, then
and
. If
, there is an isometry such that the
constants of the transformed solution are
and
or
.
Analyzing the asymptotic behavior of the transformed solution and then transforming back is often more convenient than analyzing the original solution.
Returning to the question of the asymptotics, we wish to interpret the conserved quantities as ODEs for
and
. Due to [75
, Lemma 8.1, p. 680], we have the following result:
Naively, it would seem natural to consider the integral expressions to be error terms and to interpret what remains as ODEs for the averages. On the other hand, it would then seem that we have too many equations. In the end, the situation turns out to be somewhat more complicated; see below.
Among other things, Equations (61) – (63
) imply the existence of a constant
such that
is that it decays as . In other words, first taking the average and then taking the square leads to
decay of the form
. First taking the square and then taking the average leads to decay of
the form
. This behavior reflects the same sort of asymptotics as those characterized by
Equation (52
).
Naively estimating the integral on the right-hand side of Equation (61) leads to the conclusion that it is
bounded but no more. Consequently, it seems unreasonable to think of this term as an error term. On the
other hand, integrating with respect to time might lead to an improvement. In fact, since
decays very quickly, see Equation (64
), replacing
with
in Equation (61
) leads to a
term, which tends to zero. Consequently, we can replace
by
in (61
) with a
small error. Integrating Equation (61
) and using such ideas leads to [75
, Lemma 8.9, p. 685]:
Note that in the case of , this result gives detailed information concerning the asymptotics of
;
see [75
, Corollary 8.10, p. 685]:
Corollary 1 Consider a solution to Equations (11)-(12
). If
there is a constant
and
a
such that
for all .
Clearly, Lemma 3 yields important information concerning the asymptotics. Is it possible to apply similar ideas
to Equations (62) and (63
)? It turns out to be necessary to combine both equations in order to obtain a
single equation for
. The problem is the last term in Equation (62
) and the second to last term in
Equation (63
). However, combining partial integrations, Taylor expansions in the last term
in Equation (62
) with various estimates, such as Equation (64
), leads to [75
, Lemma 8.8,
p. 684]:
In the case of it is convenient to apply this result with
. This leads to [75
,
Proposition 8.11, p. 685]:
Proposition 5 Consider a solution to Equations (11)–(12
). If
, then there is a constant
and a
such that for
,
In case of , we consequently have detailed information concerning the asymptotic behavior of
as
well. Furthermore, as was observed earlier, given a solution with the property that
, there
is an isometry of hyperbolic space such that the transformed solution is such that the corresponding
equals zero. Thus, the only case that remains to be analyzed is
. This
case is more complicated (but more interesting). We shall therefore omit a description of the
analysis.
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Living Rev. Relativity 13, (2010), 2
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