Another approach to deal with the diffeomorphism freedom of Einstein's equations is by first removing the diffeomorphism invariance. This is done by prescribing the time foliation along evolution, that is, by prescribing a lapse-shift pair along evolution. This removes the diffeomorphism invariance up to three-dimensional diffeomorphisms at the initial surface. Sometimes four dimensional covariance is also broken by splitting Einstein's equations, and possibly other supplementary equations, with respect to that foliation, and then recombining the split pieces in a suitable way. The resulting equations are equivalent to the original ones -for it is just a linear combination of them- so they have the same solutions. Notice that here we are doing more than just a 3+1 decomposition, since in general one is recombining space-space components of the Einstein tensor with time-time, and time-space components in a non-covariant way, and taking as equations this combination, or even transforming the equations to first order in derivatives of the variables by defining new variables and equations and modifying them.
After this procedure is done, one obtains a system which is
symmetric hyperbolic for most choices of
given
lapse-shift functions, once they are suitably re-scaled.
Subsequent arguments go very much on adding equations for the
lapse-shift vector in order to make the whole system well posed,
and presumably useful for some application. It is instructive to
think of these modifications of the evolution equations from the
point of view of the initial value formulation. There one starts
by solving the constraint equations, the time-time and time-space
components of the Einstein tensor, at the initial surface. With
the initial data thus obtained, one finds the solution to the
evolution equations which are taken to be the space-space
components of the Einstein tensor. Since that evolution preserves
the constraints, (The vector field generating the flow in
phase-space is tangent to the constraint sub-manifold.), one can
forget about the constraint equations and think of the evolution
equations as providing an evolution for the whole phase-space. In
this sense, the modification one is making affects the evolution
vector outside the constraint sub-manifold, leaving the vector
intact at it. Uniqueness of solutions, which follows from the
well posedness of the system, then implies that the solutions
stay on the sub-manifold. Nevertheless, and we shall return to
this point, as shown in [58], there is no guarantee that the sub-manifold of constraint
solutions is stable with respect to the evolution vector field as
extended on the whole phase-space. This is an important point for
numerical simulations.
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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 |