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 [169] 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 [169], and that of
Baumgarte–Shapiro–Shibata–Nakamura (BSSN) [390
, 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.
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