4 Initial-Value Formulations for Einstein’s Equations

In this section, we apply the theory discussed in Section 3 to well-posed Cauchy formulations of Einstein’s vacuum equations. The first such formulation dates back to the 1950s [169Jump To The Next Citation Point] and will be discussed in Section 4.1. Since then, there has been a plethora of new formulations, which distinguish themselves by the choice of variables (metric vs. tetrad, Christoffel symbols vs. connection coefficients, inclusion or not of curvature components as independent variables, etc.), the choice of gauges and the use of the constraint equations in order to modify the evolution equations off the constraint surface. Many of these new formulations have been motivated by numerical calculations, which try to solve a given physical problem in a stable way.

By far the most successful formulations for numerically-evolving compact-object binaries have been the harmonic system, which is based on the original work of [169Jump To The Next Citation Point], and that of Baumgarte–Shapiro–Shibata–Nakamura (BSSN) [390Jump To The Next Citation Point, 44]. For this reason, we review these two formulations in detail in Sections 4.1 and 4.3, respectively. In Section 4.2 we also review the Arnowitt–Deser–Misner (ADM) formulation [30], which is based on a Hamiltonian approach to general relativity and serves as a starting point for many hyperbolic systems, including the BSSN one. A list of references for hyperbolic reductions of Einstein’s equations not discussed here is given in Section 4.4.

 4.1 The harmonic formulation
  4.1.1 Hyperbolicity
  4.1.2 Constraint propagation and damping
  4.1.3 Geometric issues
 4.2 The ADM formulation
  4.2.1 Algebraic gauge conditions
  4.2.2 Dynamical gauge conditions leading to a well-posed formulation
  4.2.3 Elliptic gauge conditions leading to a well-posed formulation
  4.2.4 Constraint propagation
 4.3 The BSSN formulation
  4.3.1 The hyperbolicity of the BSSN evolution equations
  4.3.2 Constraint propagation
 4.4 Other hyperbolic formulations

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