The Generalized Thin-sandwich Problem and Its Local Solvability

We consider Einstein Gravity coupled to matter consisting of a gauge field with any compact gauge group and coupled scalar fields. We investigate under what conditions a free specification of a spatial field configuration and its time derivative determines a solution to the field equations (thin-sandwich problem). We establish sufficient conditions under which the thin-sandwich problem can be solved locally in field space. Introduction In this paper we consider the initial value problem for Einstein gravity plus matter in spacetimes Σ× R, where Σ is a closed orientable 3-manifold. We are interested in the question of how to find initial data which satisfy the constraints. The most popular approach here is a powerful method devised by Lichnerowicz, ChoquetBruhat, York and others, henceforth referred to as the “conformal method”. (See [I] for a brief review and [CY] for more details.) Of the gravitational variables it allows to freely specify the conformal class of the initial 3-metric, the conformally rescaled transverse-traceless components of the extrinsic curvature and a constant trace thereof (i.e. Σ must have constant mean curvature). Given these data, the constraints turn into a quasilinear elliptic system of second order for the conformal factor (scalar function) and the transverse momentum (vector field), which decouples due to the constant mean-curvature condition. The disadvantages of this 1 method are that it does not easily generalize to data of variable mean curvature and that it does not allow to control the local scales of the physical quantities initially, since the freely specifiable data (gravitational and non-gravitational) are related to the actual physical quantities by some rescalings with suitable powers of the conformal factor. In particular, one has no control over the conformal part of the initial 3-geometry. In this paper we are concerned with the so called “thin-sandwich method”, which differs from the one just mentioned insofar as it aims to define solutions to the Einstein equations by a free specification of the initial field configuration and its coordinate-time-derivative. The constraints are now read as equations for the gauge parameters (lapse, shift, ..). From the conformal point of view this means that one tries to trade in the freedom to specify the gauge parameters for the freedom to specify the conformal part of the metric and the longitudinal part of the momentum. The disadvantages mentioned above would then be overcome, but unfortunately the equations (for the gauge parameters) turn out to be nonelliptic in general [BO][CF]. However, for certain open subsets of initial data they are elliptic and can be locally solved. This was first shown in [BF] and will be shown in a wider context with dynamical matter here. We note that historically this approach arose from the question (then formulated as a conjecture) of whether the specification of two 3-geometries uniquely determine an interpolating Einstein space-time (thick-sandwich problem). For nearby geometries infinitesimally close in time this turns into the thin-sandwich problem (see [W] (chapter 4) and [BSW]) which we now describe in more detail. In a space-time neighborhood Σ × R of the Cauchy surface Σ, we use the standard parametrization of the space-time metric g, g = −α dt⊗ dt+ gab(dx + βdt)⊗ (dx + βdt), (1.1) where g is the (t-dependent) Riemannian metric of Σ and α, β are (t-dependent) scalarand vector fields on Σ, known as lapse and shift. The extrinsic curvature reads K = 1 2α(∂t − Lβ)g, (1.2) {1} By ellipticity of non-linear differential operators one means the ellipticity of its linearization, which depends on the point (in field space) about which one linearizes. The usual statement that the thin-sandwich equations are not elliptic merely asserts the existence of points where the linearization is not elliptic, but not that the domain of ellipticity is empty.