A Phase Space Approach to the Conformal Construction of Non-vacuum Initial Data Sets in General Relativity

Abstract

We present a uniform (and unambiguous) procedure for scaling the matter fields in implementing the conformal method to parameterize and construct solutions of Einstein constraint equations with coupled matter sources. The approach is based on a phase space representation of the spacetime matter fields after a careful \(n+1\) decomposition into spatial fields B and conjugate momenta \(\Pi _B\), which are specified directly and are conformally invariant quantities. We show that if the Einstein-matter field theory is specified by a Lagrangian which is diffeomorphism invariant and involves no dependence on derivatives of the spacetime metric in the matter portion of the Lagrangian, then fixing B and \(\Pi _B\) results in conformal constraint equations that, for constant-mean curvature initial data, semi-decouple just as they do for the vacuum Einstein conformal constraint equations. We prove this result by establishing a structural property of the Einstein momentum constraint that is independent of the conformal method: For an Einstein-matter field theory which satisfies the conditions just stated, if B and \(\Pi _B\) satisfy the matter Euler–Lagrange equations, then (in suitable form) the right-hand side of the momentum constraint on each spatial slice depends only on B and \(\Pi _B\) and is independent of the spacetime metric. We discuss the details of our construction in the special cases of the following models: Einstein–Maxwell-charged scalar field, Einstein–Proca, Einstein-perfect fluid, and Einstein–Maxwell-charged dust. In these examples we find that our technique gives a theoretical basis for scaling rules, such as those for electromagnetism, that have worked pragmatically in the past, but also generates new equations with advantageous features for perfect fluids that allow direct specification of total rest mass and total charge in any spatial region.