1999/07/28 by Arlen Anderson, Anderson, Arlen, Yvonne Choquet-Bruhat +5 · 1 citation
Physics and Astronomy · #Black Holes and Theoretical Physics #Cosmology and Gravitation Theories #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Noncommutative and Quantum Gravity Theories #gr-qc
paper · pdf · doi:10.48550/arxiv.gr-qc/9907099
30 pages, LaTeX2e, to be published in the Proceedings of the 2nd Samos Meeting. (Errors in previous version corrected)
openalex publication_date 1999/07/28 · arxiv created 1999/08/06 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We discuss several explicitly causal hyperbolic formulations of Einstein's dynamical 3+1 equations in a coherent way, emphasizing throughout the fundamental role of the ``slicing function,'' α---the quantity that relates the lapse N to the determinant of the spatial metric g through N = g1/2 α. The slicing function allows us to demonstrate explicitly that every foliation of spacetime by spatial time-slices can be used in conjunction with the causal hyperbolic forms of the dynamical Einstein equations. Specifically, the slicing function plays an essential role (1) in a clearer form of the canonical action principle and Hamiltonian dynamics for gravity and leads to a recasting (2) of the Bianchi identities ∇βGβ\mathstrutα≡ 0 as a well-posed system for the evolution of the gravitational constraints in vacuum, and also (3) of ∇βTβ\mathstrutα≡ 0 as a well-posed system for evolution of the energy and momentum components of the stress tensor in the presence of matter, (4) in an explicit rendering of four hyperbolic formulations of Einstein's equations with only physical characteristics, and (5) in providing guidance to a new ``conformal thin sandwich'' form of the initial value constraints.