The more important application of the frame representation has
been the conformal system obtained by Friedrich, [13,
14], (see §
1), where in a fixed gauge he got a symmetric hyperbolic system
which allowed him to study global solutions. Later, using similar
techniques and spinors, he found a symmetric hyperbolic system
with the remarkable property that lapse and shift appear in an
undifferentiated form, allowing for greater freedom in relating
them to the geometry without hampering hyperbolicity [34]. In [35] he introduces new symmetric systems for frame components where
one can arbitrarily prescribe the gauge functions, which in this
case does not only include the equivalent to the lapse-shift
pair, but also a three by three matrix fixing the rotation of the
frame. In this case, these gauge functions enter up to first
derivatives. This compares very favorably with the ADM
representation schema where the lapse-shift entered up to second
order derivatives. Friedrich also finds a symmetric hyperbolic
system with the generalized harmonic time condition. Contrary to
the systems in the ADM formalism, where the issue is rather
trivial, these systems do not seem to allow for a writing in flux
conservative form. We do not consider that a serious drawback.
The structure of Einstein's equations is very different than
those of fluids, where the unavoidable presence of shocks makes
it important to write them that way. Indeed the reason fluids
have shocks can be attributed to their genuinely non-linear
character, [28], a property not shared by Einstein's theory. (More about this
in the next section, §
4
.)
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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 |