Elliptic extensions

5.3 Elliptic extensions

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 [20

], 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 $P ≡ 0$ . 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 [19

] and [1 ] with a slight generalization to [20

] in admitting arbitrarily prescribed $P$ ’s. In particular, in [19 ] a complete proof of well posedness of mixed symmetric hyperbolic-elliptic systems is given. Such a proof must be implicit somewhere in [20 ], and a general argument has been given in [30

]. 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 [52 ].

In [30 ] 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 [42 ] for recent work on this and [41 ] 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.