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Hyperbolic reductions for Einstein's equations

Helmut Friedrich

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We consider the problem of reducing initial value problems for Einstein's field equations to initial value problems for hyperbolic systems, a problem of importance for numerical as well as analytical investigations of gravitational fields. The main steps and the most important objectives in designing hyperbolic reductions are discussed. Various reductions which have already been studied in the literature or which can easily be derived from previous discussions of the field equations are pointed out and some of their specific features are indicated. We propose new reductions based on the use of the Bianchi equation for the conformal Weyl tensor. These reductions involve symmetric hyperbolic systems of propagation equations and allow a number of different gauge conditions. They use unknowns in a most economic way, supplying direct and non-redundant information about the geometry of the time slicing and the four-dimensional spacetime. Some of this information is directly related to concepts of gravitational radiation. All these reductions can be extended to include the conformal field equations. Those which are based on the ADM representation of the metric can be rewritten in flux conserving form.


PACS

04.20.Cv Fundamental problems and general formalism

04.30.Nk Wave propagation and interactions

04.20.Gz Spacetime topology, causal structure, spinor structure

04.20.Ex Initial value problem, existence and uniqueness of solutions

MSC

83C75 Space-time singularities, cosmic censorship, etc.

65L05 Initial value problems

83C35 Gravitational waves

83C05 Einstein's equations (general structure, canonical formalism, Cauchy problems)

35L45 Initial value problems for hyperbolic systems of first-order PDE

35E15 Initial value problems

Subjects

Gravitation and cosmology

Dates

Issue 6 (June 1996)

Received 26 January 1996, in final form 2 April 1996



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