The 3+1 Split and the Initial-Value Problem

The Einstein field equations and their action are manifestly four-dimensional and covariant — elegant, but silent about dynamics: given the state of the gravitational field "now", how does it evolve? Answering this requires breaking the covariance one has worked so hard to build, splitting spacetime into space evolving through time. The ADM formulation (Arnowitt–Deser–Misner, 1959) recasts general relativity as a Hamiltonian system with constraints, exposes which parts of the metric are dynamical and which are gauge, and defines the initial-value problem that makes numerical relativity — and the simulation of black-hole mergers — possible.

We use $c=1$, keep $G$ explicit, mostly-plus signature.

Foliating spacetime: lapse and shift

Assume spacetime is globally hyperbolic — that it can be sliced into a stack of spacelike hypersurfaces ${\Sigma }_t$ labelled by a time function $t$ (a foliation). On each slice lives an induced spatial metric ${\gamma }_{ij}$ (the geometry of space at that instant). Reconstructing the full spacetime metric from the stack requires two more pieces of data describing how neighboring slices are glued:

• the lapse function $N$ — the proper time elapsed per unit coordinate time for an observer moving normal to the slice: $d\tau =Ndt$;
• the shift vector $N^i$ — how spatial coordinates are dragged from one slice to the next.

The spacetime line element becomes the ADM form

$$\boxed{d{s}^2=-N^{2}d{t}^2+{\gamma }_{ij}(d{x}^i+N^{i}dt)(d{x}^j+N^{j}dt).}$$

Of the ten metric components, six are the spatial metric ${\gamma }_{ij}$, and four — the lapse $N$ and shift $N^i$ — are not dynamical: they encode the freedom to choose coordinates (how fast time advances, how space is labelled). This is the gauge freedom of general relativity, now made explicit as the four freely-specifiable functions $(N,N^i)$.

Extrinsic curvature: the "velocity" of space

The spatial metric ${\gamma }_{ij}$ tells how each slice is curved internally (its intrinsic geometry). How the slice is embedded — how it bends within the four-dimensional spacetime — is measured by the extrinsic curvature $K_{ij}$, essentially the time-derivative of ${\gamma }_{ij}$:

$$K_{ij}=-\frac{1}{2N}({\partial }_t{\gamma }_{ij}-D_i{N}_j-D_j{N}_i),$$

with $D_i$ the covariant derivative of the spatial metric. In the Hamiltonian picture ${\gamma }_{ij}$ is the configuration variable (position) and $K_{ij}$ (equivalently its conjugate momentum ${\pi }^{ij}$) is the velocity/momentum. The pair $({\gamma }_{ij},K_{ij})$ on a single slice is the gravitational initial data.

Constraints and evolution

Projecting the field equations onto and normal to the slices splits the ten equations into two very different kinds:

• Four constraint equations — containing no second time derivatives, they are conditions the initial data $({\gamma }_{ij},K_{ij})$ must satisfy on each slice:
  • the Hamiltonian constraint ${}^{(3)}R+K^2-K_{ij}{K}^{ij}=16\pi G\rho$ (the $\perp \perp$ projection);
  • the momentum constraint $D_j(K^{ij}-{\gamma }^{ij}K)=8\pi G{J}^i$ (the $\perp \parallel$ projection).
• Six evolution equations — giving ${\partial }_t{\gamma }_{ij}$ and ${\partial }_t{K}_{ij}$, propagating the data forward once the gauge $(N,N^i)$ is chosen.

This is the structure of a gauge theory with constraints, exactly parallel to electromagnetism, where Gauss's law $\nabla \cdot \mathbf{E}=\rho$ is a constraint on initial data and the remaining Maxwell equations evolve. A remarkable feature: the constraints are preserved by the evolution — if they hold on the initial slice, the evolution equations keep them satisfied — a consistency guaranteed by the contracted Bianchi identity. The four constraints reflect the four non-dynamical gauge functions, leaving the two physical degrees of freedom of the gravitational field (the two gravitational-wave polarizations).

The initial-value (Cauchy) problem

The 3+1 split turns general relativity into a Cauchy problem: specify constraint-satisfying data $({\gamma }_{ij},K_{ij})$ on an initial slice, choose a gauge $(N,N^i)$, and evolve. Two deep results make this rigorous:

• Well-posedness (Choquet-Bruhat, 1952). Given smooth initial data satisfying the constraints, there exists a solution of the Einstein equations, and it is unique up to diffeomorphism — the maximal Cauchy development. Gravity has a deterministic initial-value formulation despite its coordinate freedom.
• Constructing valid data. Because the constraints couple the data, one cannot freely specify ${\gamma }_{ij}$ and $K_{ij}$; the conformal method (Lichnerowicz–York) separates the freely-choosable parts from those fixed by solving the constraints, and is how initial data for binary black holes are built.

Determinism here is subtle: uniqueness holds only within the maximal development, and the strong cosmic censorship question (horizons) is precisely whether that development is inextendible — whether GR remains predictive, or whether Cauchy horizons let unpredictable data leak in.

Numerical relativity

The ADM system is the conceptual basis of numerical relativity — solving the field equations on a computer. The raw ADM equations turn out to be only weakly hyperbolic and numerically unstable; reformulations that add multiples of the constraints (the BSSN formulation, and the generalized-harmonic gauge) restore strong hyperbolicity. With these, plus careful gauge choices (moving punctures) and boundary treatment, the field simulations succeeded:

• 2005 breakthrough (Pretorius; Campanelli et al.; Baker et al.): the first stable, complete simulations of a binary black-hole merger — inspiral, merger, and ringdown.
• These waveforms are the templates that LIGO/Virgo match-filter against real data; the detection and interpretation of gravitational waves depends directly on numerical-relativity solutions of the ADM/BSSN system.

Toward canonical quantum gravity

Casting gravity in Hamiltonian form is also the gateway to canonical quantization. Promoting $({\gamma }_{ij},{\pi }^{ij})$ to operators and imposing the Hamiltonian constraint as an operator equation gives the Wheeler–DeWitt equation $\hat{\mathcal{H}}\Psi [\gamma ]=0$ — a "wavefunction of the universe" with no external time, since time evolution has become a constraint. This is the technical origin of the problem of time in quantum gravity. Recasting the same phase space in Ashtekar variables (a connection and its conjugate) turns gravity into a gauge theory resembling Yang–Mills and launches loop quantum gravity — see Frontiers.

Summary

• The ADM 3+1 split foliates spacetime into spatial slices ${\Sigma }_t$; the metric decomposes into the spatial metric ${\gamma }_{ij}$ plus the gauge functions lapse $N$ and shift $N^i$.
• Extrinsic curvature $K_{ij}$ is the "velocity" of ${\gamma }_{ij}$; the pair $({\gamma }_{ij},K_{ij})$ is the gravitational initial data.
• The field equations split into four constraints (Hamiltonian + momentum) on the data and six evolution equations; the constraints are preserved and reflect the four gauge functions, leaving two physical degrees of freedom.
• The initial-value problem is well-posed (Choquet-Bruhat): constraint-satisfying data determine a unique maximal Cauchy development — general relativity is deterministic.
• ADM (reformulated as BSSN / generalized-harmonic) is the basis of numerical relativity, which produced the binary-merger waveforms underpinning gravitational-wave detection, and the gateway to canonical quantum gravity (Wheeler–DeWitt, Ashtekar).

Next

The canonical formulation ends the classical development. Frontiers surveys the unsolved union of general relativity with quantum theory — why it is hard, the problem of time, and the main programs — and Experimental Status collects the theory's confirmations.