# The 3+1 formalism One salient feature of General Relativity is the treatment of space and time in a unified manner. In practice, however, we still ask the basic and important question of how physical configurations at a given time evolve to the future (or the past); consequently, even though there are various analytical solutions in GR that can be studied as a whole, using GR to predict real world scenarios such as mergers of black holes still require us to break this unification of space and time. The 3+1 decomposition is what this is referred to as. It was first formulated in the period from 1920’s to 1950’s, by Darmois, Lichnerowicz, and Choquet-Bruhat. Then in the late 1950’s and early 1960’s, Dirac and Arnowitt-Deser-Misner (ADM) readdressed this problem with a Hamiltonian approach in an attempt to quantise gravity. ADM’s paper became the most frequently cited work in this aspect, and the canonical 3+1 equations are usually referred to as the **ADM equations**. The ADM equations were later reformulated by York Jr. (References are listed below.) However, for this formalism to have practical value, mere decomposition is not enough: the equations also need to be *well-posed*, which roughly speaking requires 1. existence of a solution given an initial data set; 2. uniqueness of the solution; 3. "stability": ability to define a norm $\|\cdot\|$ such that $\|u(t, x)\| \le ke^{\alpha t}\|u(0, x)\|$, with $k$ and $\alpha$ being constants independent of the initial data, and $u$ being some $n$-dimensional vector-valued function of time $t$ and space $x$. In other words, a small change in the initial data should not lead to a big change in the full solution. It was pointed out in [[NakamuraOoharaKojima1987]] that that ADM evolution equations are *not* stable. To save stability, notice that an arbitrary multiple of the constraint equations can always be added to the evolution equations without changing the physics, even though they could change the mathematical properties of the PDEs such as stability. Two particular approaches of achieving stability are known as KST and BSSN (see below for refs). ## Refs - reviews - [[2007#Gourgoulhon (Lectures)]] - [[BartnikIsenberg2004Review]][](https://arxiv.org/abs/gr-qc/0405092) - [[Carlotto2021]][](https://link.springer.com/article/10.1007/s41114-020-00030-z) (Living Reviews) - early developments of the 3+1 formalism - first formulations - [[Darmois1927]] - [[Lichnerowicz1939]] - [[Lichnerowicz1944]] - [[Lichnerowicz1952]] - [[Choquet-Bruhat1952]] - [[Choquet-Bruhat1956]] - readdressed using a Hamiltonian approach - [[Dirac1958]] - [[Dirac1959]] - [[ArnowittDeserMisner2004]] (republished version) - reformulation - [[YorkJr1979]] and [[YorkJrPiran1982]] - stable formulations - KST: [[KidderScheelTeukolsky2001]] - BSSN: - [[NakamuraOoharaKojima1987]] - [[ShibataNakamura1995]] - [[ShibataTaniguchiUryu2003]] - [[BaumgarteShapiro1998]] - [[YoBaumgarteShapiro2001]] - [[YoBaumgarteShapiro2002]] - [[AlcubierreBrugmannPollneySeidelTakahashi2001]]: gives some analytic understanding of the formulation - in AdS - [[AllenLeeMaxwell2022]][](https://arxiv.org/abs/2206.12854) and refs therein - initial data with [[0298 MOTS|MOTS]] - [[BrownYork1980]][](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.21.2047) - [[Thornburg1987]][](https://iopscience.iop.org/article/10.1088/0264-9381/4/5/013) - [[Dain2003]][](https://arxiv.org/abs/gr-qc/0308009) - [[Maxwell2003]][](https://arxiv.org/abs/gr-qc/0307117) - [[Maxwell2008]][](https://arxiv.org/abs/0804.0874) - [[HolstMeier2014]][](https://arxiv.org/abs/1403.4549) ## Solving initial data constraints: the conformal method The standard method of solving the initial data constraints is the conformal method by [[Lichnerowicz1944]] and [[Choquet-BruhatYork1980]]. Here, the free data are specified by the conformal metric, the divergence-free and traceless part of the conformal extrinsic curvature, and the mean curvature of the slice. This approach is good for initial data containing a black hole, where the [[0298 MOTS|MOTS]] is treated as an inner boundary and the conformal factor satisfies a Robin boundary condition at this boundary. ## Related - [[0277 Time symmetric initial data]] - [[0285 Brill-Lindquist initial data]] - [[0310 Initial data in AdS]]