5 Concluding Remarks

24 bounded regions of spacetime. Gravitational physics and its mathematical expression have been smoothly melded. 23 ^ @ 0 g ij = ?2NK ij ; (82) that has no slicing term analogous to S ij. It has characteristics along the light cone and @ 0. It is not quasi-diagonal because of third order derivatives of g ij arising in (81) from r 2 N. This system is hyperbolic non-strict 21]. On the other hand, the fourth order equation for g ij deduced from (81) and (82) is also not strictly hyperbolic because of the appearance of @ 2 0 in its principal operator (@ 2 0 2). It seems that the system, in its present form, cannot be written as a rst order symmetrizable system; but it is potentially very useful, especially in perturbation theory, because the lapse and the shift need not be perturbed or satisfy any particular gauge conditions. The system (81) and (82) can be proven to be equivalent to the full Einstein equations using the twice-contracted Bianchi identities and the appropriate Cauchy data: G = 0 and ij = 0 on 0. This system can be directly adapted for the study of the non-linear perturbative regime of high-frequency wave propagation 22, 23]. One of the numerous attractive reasons for writing general relativity in ux-conservative FOSH form, as emphasized recently in 9], is to facilitate emulating the successful strategy applied in computational uid dynamics, rather than tailoring the form of Einstein's equations to suit each particular application. However, in uid dynamics in ux-conservative FOSH form, the only characteristics that can appear are the physical ones associated with the \sound cone" or \\ow lines," because there are no analogs of the spatial coordinate and slicing \gauges" of general relativity that give rise to constraints and can lead to unphysical characteristics. In this paper, by modifying the equations of motion of Einstein's theory in a speciic way (R ij ! ij), and by choosing time slicings in which the lapse function obeys a causal wave equation, we have incorporated full spatial coordinate freedom while assuring that only physical characteristics are present. Among other things, this means that causal boundary conditions can be imposed naturally on 22 which the event lies. By writing (74) and (75) in the equivalent forms 1 p 2