Proposition 2 Let be a solution to Equations (16
)–(17
) and assume
.
If
converges to
, then there is an open interval
containing
,
,
, polynomials
for all
and a
such that for all
It is worth noting that the above proposition proves that if , then
is smooth in the
neighborhood of
. In other words, knowledge concerning
at one point can sometimes yield
conclusions in the neighborhood of that point; see [79
, Remark 1.6, p. 985].
Equation (48) essentially has the same content as Equation (38
). In order to see this, define the object
inside the norm on the left-hand side of Equation (48
) to be
. Then
Using the above expansions and equations, expressions for the higher-order time derivatives can be derived.
http://www.livingreviews.org/lrr-2010-2 |
Living Rev. Relativity 13, (2010), 2
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