Theorem 4 Consider a solution to Equations (11)–(12
). Then there is a
and a
such that
for all
, the energy
defined by Equation (53
) satisfies
In fact, this result can be improved somewhat to [75, Theorem 1.6, p. 664]:
Theorem 5 Consider a solution to Equations (11)–(12
). Then if
is given by Equation (53
), there is
a
, a
and a constant
such that
Note that, in some respects, this result leads to conclusions that, on an intuitive level, are somewhat
contradictory. First, since converges to zero, it is clear that the spatial variation of the solution, i.e.,
, converges to zero. Consequently, it seems natural to expect the solution to behave as a spatially
homogeneous solution to the equations. Thus, consider a non–spatially-homogeneous solution and a
spatially homogeneous solution, which is supposed to approximate it, and let
and
denote the
corresponding energies. Then, due to Theorem 5,
even though both limits should be zero if the solution is well approximated by a spatially homogeneous solution.
http://www.livingreviews.org/lrr-2010-2 |
Living Rev. Relativity 13, (2010), 2
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