3+1 formalism and bases of numerical relativity - LUTh ...

Chapter 1
Introduction
The 3+1 formalism is an approach to general relativity and to Einstein equations that relies
on the slicing of the four-dimensional spacetime by three-dimensional surfaces (hypersurfaces).
These hypersurfaces have to be spacelike, so that the metric induced on them by the
Lorentzian spacetime metric [signature (−,+,+,+)] is Riemannian [signature (+,+,+)]. From
the mathematical point of view, this procedure allows to formulate the problem of resolution of
Einstein equations as a Cauchy problem with constraints. From the pedestrian point of view, it
amounts to a decomposition of spacetime into “space” + “time”, so that one manipulates only
time-varying tensor fields in the “ordinary” three-dimensional space, where the standard scalar
product is Riemannian. Notice that this space + time splitting is not an a priori structure of
general relativity but relies on the somewhat arbitrary choice of a time coordinate. The 3+1
formalism should not be confused with the 1+3 formalism, where the basic structure is a congruence
of one-dimensional curves (mostly timelike curves, i.e. worldlines), instead of a family
of three-dimensional surfaces.
The 3+1 formalism originates from works by Georges Darmois in the 1920’s [105], André
Lichnerowicz in the 1930-40’s [176, 177, 178] and Yvonne Choquet-Bruhat (at that time Yvonne
Fourès-Bruhat) in the 1950’s [127, 128] 1 . Notably, in 1952, Yvonne Choquet-Bruhat was able
to show that the Cauchy problem arising from the 3+1 decomposition has locally a unique
solution [127]. In the late 1950’s and early 1960’s, the 3+1 formalism received a considerable
impulse, serving as foundation of Hamiltonian formulations of general relativity by Paul A.M.
Dirac [115, 116], and Richard Arnowitt, Stanley Deser and Charles W. Misner (ADM) [23]. It
was also during this time that John A. Wheeler put forward the concept of geometrodynamics
and coined the names lapse and shift [267]. In the 1970’s, the 3+1 formalism became the basic
tool for the nascent numerical relativity. A primordial role has then been played by James W.
York, who developed a general method to solve the initial data problem [274] and who put the
3+1 equations in the shape used afterwards by the numerical community [276]. In the 1980’s
and 1990’s, numerical computations increased in complexity, from 1D (spherical symmetry) to
1 These three persons have some direct filiation: Georges Darmois was the thesis adviser of André Lichnerowicz,
who was himself the thesis adviser of Yvonne Choquet-Bruhat