

These are other types of approaches which take more into account
the longitudinal modes of the theory, namely those which are
related to the energy or matter content of the space time, and
those which do not propagate as waves. The nature of these modes
implies that these approaches seem to need a global knowledge of
the solution, which in practice appears by imposing either
elliptic or parabolic equations, the latter as a way to drive the
solution close to satisfying an elliptic equation for larger
times. Systems of that sort have already been used in
applications: In [16
], a hyperbolic system with lapse given by an elliptic equation
is used in the proof of global existence of small data. The
elliptic equation is used to impose the maximal slice condition
during evolution, that is
. In that work, the
first order system
is for the electric and magnetic parts of the Weyl tensor, while
the metric, connection, and extrinsic curvature tensor are
obtained by
solving
elliptic equations on each slice. For their aims, obtaining a
priori estimates, this suffices. For numerical simulations of
evolution, it is better to solve, as much as possible, evolution
equations, and not elliptic ones. Thus for this aim, equations
-hopefully hyperbolic or at least parabolic- should be added to
evolve the above mentioned (lower order in derivatives)
variables. This has improved recently in [55
], and [56] with a slight generalization to [16
] in admitting arbitrarily prescribed
P
's. In particular, in [55] a complete proof of well posedness of mixed symmetric
hyperbolic-elliptic systems is given. Such a proof must be
implicit somewhere in [16], and a general argument has been given in [48
]. Surprisingly, such a result, which has a rather simple
argument based on the standard elliptic and hyperbolic estimates,
has not before had the clean proof it deserves. This gauge has
been used to show existence of near Newtonian solutions by [57].
In [48] a different elliptic condition is imposed in order to study
near Newtonian solutions. An elliptic system is considered for
both lapse and shift. It is similar, but not equal, to the above
gauge, for in this work a much stronger condition is required on
the order of approach of relativistic solutions to the Newtonian
limit. This implies globally controlling not only the lapse, but
also the shift.
The last two works mentioned hint at some interplay between
the problems of finding well behaving gauges for near Newtonian
solutions and for long term evolution. The argument has been
that, since in this gauge the principal part of the equations is
well behaved near the -singular- Newtonian limit, and since the
rest of the terms of the hyperbolic system go to zero on that
limit, one expects for the time the solution exists to go to
infinity as one approaches the Newtonian limit. Thus the gauge
should be well behaved until then. I cite [59] for recent work on this and [60] for a well behaved system in asymptotically null slices
amenable to study slow solutions near null infinity.
In the frame and in Ashtekar's representations one could even
consider first order elliptic (spinorial) equations to fix the
gauge variables. In the frame representation one can even fix
gauge variables via an algebraic condition.


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Hyperbolic methods for Einstein's Equations
Oscar A. Reula
http://www.livingreviews.org/lrr-1998-3
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to
livrev@aei-potsdam.mpg.de
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