2021/06/28 by James Isenberg, Isenberg, James, David Maxwell +1
Earth and Planetary Sciences · Physics and Astronomy · #Cosmology and Gravitation Theories #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Geophysics and Gravity Measurements #Relativity and Gravitational Theory
paper · pdf · doi:10.48550/arxiv.2106.15027
openalex publication_date 2021/06/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 space-time matter fields after a careful n+1 decomposition into spatial fields B and conjugate momenta ΠB, 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 space-time metric in the matter portion of the Lagrangian, then fixing B and ΠB 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 ΠB 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 ΠB and is independent of the space-time 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.