The ADM 3+1 formalism and the Hamiltonian constraint

Intuition Beginner

Picture spacetime as a loaf of bread. A numerical relativist cannot solve the whole loaf at once. Instead the loaf is cut into thin three-dimensional slices, one for each instant of time, and the geometry on each slice is tracked as it grows into the next. This is the $3+1$ split: three space dimensions plus one time direction. Every slice carries a three-dimensional geometry of its own, recorded by the induced metric. The way each slice bends inside the four-dimensional loaf is recorded by a second object, the extrinsic curvature. General relativity, restated slice by slice, becomes a rule for how the geometry on one slice evolves into the geometry on the next.

The slicing is not unique, and the formalism gives two control knobs. The lapse tells each slice how much proper time elapses before the next slice begins: a small lapse slows the simulation down in that region, a large lapse speeds it up. The shift lets the coordinate grid slide sideways from one slice to the next, so a point with a fixed label need not stand still in space. Choosing the lapse and the shift is the gauge freedom of the theory. Together with the induced metric and the extrinsic curvature, these four quantities rebuild the full four-dimensional spacetime without ambiguity.

Not every slice is allowed. The Einstein field equations 13.04.01 split into two kinds of statement. Evolution equations push the geometry forward in time. Constraint equations test whether a candidate slice could belong to any spacetime at all. The Hamiltonian constraint ties the spatial curvature of a slice to how fast volumes are changing on it. The momentum constraint ties the sideways shear of space to the flow of momentum through it. A simulation starts from one slice that satisfies both constraints, then evolves it forward; the constraints stay satisfied on every later slice.

Visual Beginner

The four-dimensional metric rewrites as a block matrix with three pieces. The lapse $N$ sits in the time-time slot. The shift vector $N^i$ couples time to space. The spatial block is the induced metric ${\gamma }_{ij}$. Two free functions (the lapse and the shift) encode the choice of slicing; the induced metric carries the physical geometry of each slice.

ADM variable	Meaning	Role
$N$ (lapse)	proper time per unit coordinate time	how fast the slice advances
$N^i$ (shift)	coordinate drift between slices	gauge slide of the grid
${\gamma }_{ij}$ (induced metric)	geometry within a slice	the physical 3-geometry
$K_{ij}$ (extrinsic curvature)	bending of a slice through spacetime	how the slice is embedded

Worked example Beginner

Take the expanding-universe line element $d{s}^2=-d{t}^2+a(t{)}^2(d{x}^2+d{y}^2+d{z}^2)$ with scale factor $a(t)=2t$. The lapse is $N=1$ and the shift is zero. The induced metric on each slice is $a(t{)}^2$ times the Euclidean metric, so at time $t=1$ the scale factor is $a=2$ and every spatial metric component equals $4$.

The extrinsic curvature measures how fast each metric component grows. Its value is $K_{ij}=-a$ times the growth rate of $a$, times the Euclidean metric. The growth rate of $a$ is $2$, so with $a=2$ every diagonal component is $K_{xx}=K_{yy}=K_{zz}=-4$. The trace is $K=-3$, which equals minus three times the Hubble rate $H=2/2=1$. A faster expansion makes $K$ more negative: each slice is bending outward through spacetime.

What this tells us: even a universe whose spatial slices are perfectly flat has nonzero extrinsic curvature whenever it expands. The bending lives in the time direction, not in the slices themselves.

Check your understanding Beginner

Formal definition Intermediate+

Let $(M,g_{\mu \nu })$ be a globally hyperbolic spacetime, so that $M$ admits a foliation by smooth spacelike Cauchy hypersurfaces ${\Sigma }_t$ 13.02.01. Let $t$ be a time function labelling the slices, with lapse $N>0$ and shift vector $N^i$, and let ${\gamma }_{ij}$ be the induced (positive-definite) metric on ${\Sigma }_t$. In adapted coordinates $(t,x^i)$ the four-metric decomposes as

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

Equivalently $g_{00}=-N^2+N_i{N}^i$, $g_{0i}=N_i={\gamma }_{ij}{N}^j$, $g_{ij}={\gamma }_{ij}$, and $g^{\mu \nu }$ has inverse $g^{00}=-1/N^2$, $g^{0i}=N^i/N^2$, $g^{ij}={\gamma }^{ij}-N^i{N}^j/N^2$. The future-directed unit normal to the slices is $n_{\mu }=(-N,\vec{0})$ and $n^{\mu }=N^{-1}(1,-N^i)$, with $n^{\mu }{n}_{\mu }=-1$. The projector onto the slices is ${\perp }_{\nu }^{\mu }={\delta }_{\nu }^{\mu }+n^{\mu }{n}_{\nu }$, which acting on spatial indices agrees with ${\gamma }_{\nu }^{\mu }$.

The extrinsic curvature of ${\Sigma }_t$ in $(M,g)$ is the symmetric tensor

$$\boxed{K_{ij}:=-\frac{1}{2N}\left({\partial }_t{\gamma }_{ij}-D_i{N}_j-D_j{N}_i\right)=-{\gamma }_i^{\alpha }{\gamma }_j^{\beta }{\nabla }_{\alpha }{n}_{\beta }=\frac{1}{2}{\mathcal{L}}_{\vec{n}}{\gamma }_{ij},}$$

where $D_i$ is the Levi-Civita connection of ${\gamma }_{ij}$, $\nabla$ the connection of $g$, and ${\mathcal{L}}_{\vec{n}}$ the Lie derivative along the unit normal. The first form is the evolution (or kinematical) expression; the second is the geometric (embedding) expression; the third identifies $K_{ij}$ as half the rate at which the spatial metric changes under transport along the normal. The trace $K={\gamma }^{ij}{K}_{ij}$ is the expansion of the normal congruence, with sign fixed so that $K>0$ corresponds to slices contracting in the future.

Projection of curvature: the Gauss--Codazzi relations

The Riemann tensor 13.03.01 and Ricci/Einstein tensor 13.03.02 of $(M,g)$ are not independent of the geometry of ${\Sigma }_t$. The orthogonal split of the 4D curvature into pieces tangential and normal to the slices is the content of the Gauss and Codazzi equations ^{[Gourgoulhon 2007]}:

$${}^{(4)}{R}_{\alpha \beta \gamma \delta }{\gamma }_i^{\alpha }{\gamma }_j^{\beta }{\gamma }_k^{\gamma }{\gamma }_l^{\delta }={}^{(3)}{R}_{ijkl}+K_{ik}{K}_{jl}-K_{il}{K}_{jk},\text{(Gauss)}$$ $${}^{(4)}{R}_{\alpha \beta \gamma \delta }{n}^{\beta }{\gamma }_i^{\alpha }{\gamma }_j^{\gamma }{\gamma }_k^{\delta }=D_j{K}_{ik}-D_i{K}_{jk}.\text{(Codazzi)}$$

Contracting the Gauss equation yields the scalar identity

$${}^{(4)}R+2{}^{(4)}{R}_{\mu \nu }{n}^{\mu }{n}^{\nu }={}^{(3)}R+K^2-K_{ij}{K}^{ij},$$

and contracting the Codazzi equation yields

$${}^{(4)}{R}_{\mu \nu }{n}^{\nu }{\gamma }_i^{\mu }=D^j{K}_{ij}-D_{i}K.$$

These two projected identities are the geometric raw material from which the constraint equations are read off by inserting the Einstein equations.

Counterexamples to common slips

• The lapse is not the time component of a four-vector. $N$ and $N^i$ are gauge variables encoding the coordinate map between slices; they carry no local stress-energy and vanish from all curvature invariants. Two slicings of the same spacetime differ in $(N,N^i)$ and agree in the induced geometry.
• Zero extrinsic curvature is not zero spatial curvature. A slice with $K_{ij}=0$ is momentarily static (a time-symmetric slice), not flat: the spatial metric can carry arbitrary ${}^{(3)}R$. Conversely a flat spatial slice (Minkowski at constant $t$) has $K_{ij}=0$ and ${}^{(3)}R=0$, but a flat slice inside a curved spacetime can have nonzero $K_{ij}$.
• The trace $K$ is a signed quantity. Flipping the orientation of the normal $n^{\mu }\to -n^{\mu }$ sends $K_{ij}\to -K_{ij}$ and $K\to -K$. The Hamiltonian constraint below depends on $K$ through $K^2$ (orientation-independent) and through $K$ itself in the momentum constraint, whose sign convention is fixed by the choice of future normal.

Key theorem with proof Intermediate+

Theorem (Hamiltonian and momentum constraints). Let $({\Sigma }_t,{\gamma }_{ij},K_{ij})$ be a $3+1$ slicing of a spacetime satisfying the Einstein equation ${}^{(4)}{G}_{\mu \nu }=8\pi T_{\mu \nu }$ 13.04.01. Define the energy and momentum densities measured by the normal observer,

$$\rho :=T_{\mu \nu }{n}^{\mu }{n}^{\nu },j_i:=-{\gamma }_i^{\mu }{T}_{\mu \nu }{n}^{\nu }.$$

Then the induced data on every slice obey

$$\boxed{{}^{(3)}R+K^2-K_{ij}{K}^{ij}=16\pi \rho }\text{(Hamiltonian constraint)}$$ $$\boxed{D_j{K}^{ij}-D^{i}K=8\pi j^i.}\text{(Momentum constraint)}$$

These are four elliptic equations that the data $({\gamma }_{ij},K_{ij})$ must satisfy on each slice; they contain no time derivatives and therefore act as integrability conditions, not as evolution equations ^{[Gourgoulhon 2007]}.

Proof of the Hamiltonian constraint. Start from the contracted Gauss identity

$${}^{(4)}R+2{}^{(4)}{R}_{\mu \nu }{n}^{\mu }{n}^{\nu }={}^{(3)}R+K^2-K_{ij}{K}^{ij}.$$

Insert the Einstein equation ${}^{(4)}{G}_{\mu \nu }={}^{(4)}{R}_{\mu \nu }-\frac{1}{2}{}^{(4)}R{g}_{\mu \nu }=8\pi T_{\mu \nu }$ 13.03.02. Contract once with $n^{\mu }{n}^{\nu }$. Using $n^{\mu }{n}_{\mu }=-1$,

$${}^{(4)}{G}_{\mu \nu }{n}^{\mu }{n}^{\nu }={}^{(4)}{R}_{\mu \nu }{n}^{\mu }{n}^{\nu }+\frac{1}{2}{}^{(4)}R=8\pi \rho .$$

Solving for ${}^{(4)}{R}_{\mu \nu }{n}^{\mu }{n}^{\nu }$ gives ${}^{(4)}{R}_{\mu \nu }{n}^{\mu }{n}^{\nu }=8\pi \rho -\frac{1}{2}{}^{(4)}R$. Substitute into the contracted Gauss identity:

$${}^{(4)}R+2\left(8\pi \rho -\frac{1}{2}{}^{(4)}R\right)={}^{(3)}R+K^2-K_{ij}{K}^{ij}.$$

The left-hand side collapses to $16\pi \rho$, giving ${}^{(3)}R+K^2-K_{ij}{K}^{ij}=16\pi \rho$. $\square$

Proof of the momentum constraint. Contract the Codazzi relation ${}^{(4)}{R}_{\mu \nu }{n}^{\nu }{\gamma }_i^{\mu }=D^j{K}_{ij}-D_{i}K$. The Ricci side is related to the Einstein tensor by ${}^{(4)}{R}_{\mu \nu }{n}^{\nu }{\gamma }_i^{\mu }={}^{(4)}{G}_{\mu \nu }{n}^{\nu }{\gamma }_i^{\mu }+\frac{1}{2}{}^{(4)}R{g}_{\mu \nu }{n}^{\nu }{\gamma }_i^{\mu }$. The cosmological term vanishes because $g_{\mu \nu }{n}^{\nu }=-n_{\mu }$ is purely normal, so ${\gamma }_i^{\mu }{g}_{\mu \nu }{n}^{\nu }=-{\gamma }_i^{\mu }{n}_{\mu }=0$. Thus ${}^{(4)}{R}_{\mu \nu }{n}^{\nu }{\gamma }_i^{\mu }={}^{(4)}{G}_{\mu \nu }{n}^{\nu }{\gamma }_i^{\mu }=8\pi T_{\mu \nu }{n}^{\nu }{\gamma }_i^{\mu }=-8\pi j_i$. Equating and raising the index with ${\gamma }^{ij}$,

$$D_j{K}^{ij}-D^{i}K=8\pi j^i.\square$$

Why the constraints are elliptic, not evolutionary

The Hamiltonian and momentum constraints contain no time derivatives of ${\gamma }_{ij}$ or $K_{ij}$: the Hamiltonian constraint is a nonlinear elliptic equation coupling the spatial scalar curvature to the squared extrinsic curvature, and the momentum constraint is a linear elliptic equation for the trace-free part of $K^{ij}$. A triple $({\gamma }_{ij},K_{ij},\text{matter})$ that satisfies both constraints is admissible initial data. The evolution equations (the remaining components of the Einstein equations) then propagate the data forward in time, and the Bianchi identity 13.03.02 guarantees that the constraints remain satisfied. This split --- constraints select the initial slice; evolution carries it forward --- is the structural backbone of numerical relativity ^{[Baumgarte & Shapiro 2010]}.

Bridge. The Gauss--Codazzi projection builds toward the Cauchy problem for the Einstein equations, in which the Hamiltonian and momentum constraints fix admissible initial data and the evolution system carries it forward; it appears again in 13.04.02 as the canonical (Hamiltonian) form of the Einstein--Hilbert action, where the constraints emerge as the vanishing bulk pieces of the ADM Hamiltonian. The central insight is that four of the ten Einstein equations are not dynamical but integrability conditions on each slice, and putting these together identifies the freely specifiable data --- the spatial metric and the trace-free extrinsic curvature --- that the conformal method of the next section will parametrise. The bridge is the recognition that the Einstein equations split cleanly into constraints and evolution, with the Bianchi identity as the mechanism that keeps them consistent.

Exercises Intermediate+

Advanced results Master

Theorem 1 (York conformal decomposition and the Lichnerowicz equation ^{[York 1971]}). Write the spatial metric in conformal form ${\gamma }_{ij}={\psi }^4{\tilde{\gamma }}_{ij}$ with $\det \tilde{\gamma }=1$, and decompose the extrinsic curvature into its trace and trace-free part, $K_{ij}=A_{ij}+\frac{1}{3}{\gamma }_{ij}K$, ${\tilde{\gamma }}^{ij}{A}_{ij}=0$. Then the Hamiltonian constraint becomes the semilinear elliptic Lichnerowicz equation

$$\boxed{8{\tilde{D}}^k{\tilde{D}}_k\psi -{}^{(3)}\tilde{R}\psi +\frac{2}{3}{K}^2{\psi }^5-{\tilde{A}}_{ij}{\tilde{A}}^{ij}{\psi }^{-7}=-16\pi \rho {\psi }^{-3},}$$

where tildes denote quantities computed with the conformal metric ${\tilde{\gamma }}_{ij}$. Given a choice of ${\tilde{\gamma }}_{ij}$, $K$, and ${\tilde{A}}^{ij}$, this equation determines $\psi$ and hence the physical metric ${\gamma }_{ij}$.

The Lichnerowicz equation is the workhorse of initial-data construction. Its structure --- a Laplacian plus competing powers ${\psi }^5$ and ${\psi }^{-7}$ of the conformal factor --- produces a unique positive solution under broad conditions (Cantor--Brill 1971), and its nonlinearity encodes the gravitational self-interaction that pure linear elliptic theory misses. York's contribution ^{[York 1972]} was to show that the momentum constraint, together with the requirement that the trace-free part of $K_{ij}$ be decomposed conformally, linearises in a conformal Killing vector, making the momentum constraint solvable in closed form once a momentum-rescaled weight ${\tilde{A}}^{ij}$ is fixed. The pair (Lichnerowicz for $\psi$, vector elliptic equation for the momentum potential) constitutes the York conformal method, the standard recipe for initial data.

Theorem 2 (Bowen--York black-hole initial data ^{[Bowen & York 1980]}). On a conformally flat background ${\tilde{\gamma }}_{ij}=f_{ij}$ (flat), with maximal slicing $K=0$, the momentum constraint $D_j{K}^{ij}=0$ admits the analytic solution

$${\tilde{A}}_{(P,S)}^{ij}=\frac{3}{2r^2}[2P^{⟨i⟩}{n}^{j⟩}+(2P^{(i}{ϵ}^{j)kl}{S}_k{n}_l)/r]{\psi }_{\mathrm{BL}}^{-2},$$

where $P^i$ is the linear momentum, $S^i$ the spin (angular momentum), $n^i$ the radial unit vector, brackets denote the symmetric trace-free part, and ${\psi }_{\mathrm{BL}}$ is the Brill--Lindquist conformal factor. Inserting this into the Lichnerowicz equation yields the puncture initial data used in the majority of binary black-hole simulations.

The Bowen--York solution supplies, in closed form, the momentum constraint contribution of a spinning or boosted black hole. It is the seed for the binary black-hole initial data evolved in the LIGO/Virgo waveform catalogue 13.07.01: two punctures, each carrying its own $(P,S)$, are superposed on a conformally flat maximal slice, and the Lichnerowicz equation is solved once on the initial surface. The constraints are then satisfied to machine precision at $t=0$ and preserved under evolution by the Bianchi identity.

Theorem 3 (Conformal thin-sandwich formulation). Instead of specifying $K_{ij}$, prescribe the time derivative ${\bar{u}}_{ij}={\partial }_t{\tilde{\gamma }}_{ij}$ of the conformal metric and a trace $K$. The momentum constraint plus the condition that the trace-free part of ${\partial }_t{\gamma }_{ij}$ match the prescribed ${\bar{u}}_{ij}$ close into a coupled elliptic system for the shift $N^i$ and the conformal factor $\psi$, with the lapse determined by an auxiliary elliptic equation. This conformal thin-sandwich (CTS) system fixes quasi-equilibrium initial data for binaries.

The CTS formulation, developed by York and finalised in the 1990s ^{[Baumgarte & Shapiro 2010]}, is preferred when one wants initial data that approximately solve the equilibrium equations --- the natural requirement for a binary in a slow inspiral. By prescribing the shear of the slicing rather than the extrinsic curvature, CTS makes the residual time-derivative of the geometry an input, so that a quasi-stationary binary emerges when that input is set to the corotating-frame value.

Synthesis. The foundational reason the ADM Hamiltonian is a boundary term and the bulk is a constraint combination is the general covariance of the Einstein--Hilbert action 13.04.02: a diffeomorphism-invariant theory has no local gauge-invariant energy density, so the local dynamics collapses into constraints while the energy escapes to infinity as a surface integral. This is exactly the mechanism by which the ten Einstein equations split into four elliptic constraints plus six hyperbolic evolution equations, and the York conformal method generalises the Lichnerowicz equation into a complete, decoupled initial-data recipe that makes the constraints solvable on a computer. The picture appears again in 13.04.02, where the canonical Hamiltonian reduction of the action reproduces the same constraint functions as Noether identities of diffeomorphism invariance, and the bridge is the recognition that the Hamiltonian and momentum constraints are not auxiliary conditions but the very signature of general covariance made visible by the $3+1$ split. The whole machinery is what makes binary black-hole simulation, and therefore gravitational-wave astronomy, computable.

Full proof set Master

Proposition 1 (Contracted Gauss identity). The orthogonal projection of the four-dimensional scalar curvature onto a spacelike slice satisfies

$${}^{(4)}R+2{}^{(4)}{R}_{\mu \nu }{n}^{\mu }{n}^{\nu }={}^{(3)}R+K^2-K_{ij}{K}^{ij}.$$

Proof. Begin from the Gauss equation

$${}^{(4)}{R}_{\alpha \beta \gamma \delta }{\gamma }^{\alpha i}{\gamma }^{\beta j}{\gamma }^{\gamma k}{\gamma }^{\delta l}={}^{(3)}{R}_{ijkl}+K_{ik}{K}_{jl}-K_{il}{K}_{jk}.$$

Contract with ${\gamma }^{ik}{\gamma }^{jl}={\gamma }^{ij}$:

$${}^{(4)}{R}_{\alpha \beta \gamma \delta }{\gamma }^{\alpha i}{\gamma }^{\beta j}{\gamma }_i^{\gamma }{\gamma }_j^{\delta }={}^{(3)}R+K^2-K_{ij}{K}^{ij}.$$

The projection operator satisfies ${\gamma }^{\alpha i}{\gamma }^{\beta j}{\gamma }_i^{\gamma }{\gamma }_j^{\delta }=({\gamma }^{\alpha \gamma }-ϵn^{\alpha }{n}^{\gamma })({\gamma }^{\beta \delta }-ϵn^{\beta }{n}^{\delta })$ with the sign $ϵ$ fixed by the signature; here $ϵ=-1$ since $n^{\mu }{n}_{\mu }=-1$. Expanding and using ${\gamma }^{\alpha \gamma }{\gamma }_{\gamma }^{\delta }={\gamma }^{\alpha \delta }$ together with the antisymmetry properties of the Riemann tensor,

$${}^{(4)}{R}_{\alpha \beta \gamma \delta }{\gamma }^{\alpha \gamma }{\gamma }^{\beta \delta }={}^{(4)}R+2{}^{(4)}{R}_{\mu \nu }{n}^{\mu }{n}^{\nu },$$

where the two cross-terms collapse by the pair-symmetry of Riemann and the pure-spatial projection of Riemann yields ${}^{(3)}R$. Combining the two displayed identities gives the result. $\square$

Proposition 2 (Contracted Codazzi identity). The mixed normal--spatial projection of the four-dimensional Ricci tensor satisfies

$${}^{(4)}{R}_{\mu \nu }{n}^{\nu }{\gamma }_i^{\mu }=D^j{K}_{ij}-D_{i}K.$$

Proof. Start from the Codazzi equation

$${}^{(4)}{R}_{\alpha \beta \gamma \delta }{n}^{\beta }{\gamma }_i^{\alpha }{\gamma }_j^{\gamma }{\gamma }_k^{\delta }=D_j{K}_{ik}-D_i{K}_{jk}.$$

Contract with ${\gamma }^{jk}$:

$${}^{(4)}{R}_{\alpha \beta \gamma \delta }{n}^{\beta }{\gamma }_i^{\alpha }{\gamma }_j^{\gamma }{\gamma }^{\delta j}=D^k{K}_{ik}-D_{i}K.$$

The projector ${\gamma }_j^{\gamma }{\gamma }^{\delta j}={\gamma }^{\gamma \delta }$, and the contraction ${}^{(4)}{R}_{\alpha \beta \gamma \delta }{n}^{\beta }{\gamma }^{\gamma \delta }{\gamma }_i^{\alpha }={}^{(4)}{R}_{\alpha \beta }{n}^{\beta }{\gamma }_i^{\alpha }$, since the Riemann contraction over a tangential pair returns the Ricci tensor. The right-hand side is $D^j{K}_{ij}-D_{i}K$ after relabelling the dummy index. $\square$

Proposition 3 (On-shell vanishing of the bulk ADM Hamiltonian). Let $H_{\mathrm{ADM}}={\int }_{\Sigma }(N\mathcal{H}+N^i{\mathcal{H}}_i)d^{3}x$ with $\mathcal{H}={}^{(3)}R+K^2-K_{ij}{K}^{ij}-16\pi \rho$ and ${\mathcal{H}}_i=D_j{K}_i^j-D_{i}K-8\pi j_i$. Then on any slice satisfying the constraints, $H_{\mathrm{ADM}}=0$.

Proof. The constraints give $\mathcal{H}(x)=0$ and ${\mathcal{H}}_i(x)=0$ pointwise on $\Sigma$. The integrand $N(x)\mathcal{H}(x)+N^i(x){\mathcal{H}}_i(x)$ therefore vanishes at every $x\in \Sigma$, regardless of the (smooth, positive) values of the lapse $N$ and the shift $N^i$. Integrating a pointwise-zero function over the compact slice returns zero, so $H_{\mathrm{ADM}}=0$. The lapse and shift, being gauge variables, enter the Hamiltonian only as Lagrange multipliers enforcing the constraints; on the constraint surface they carry no independent weight. $\square$

Proposition 4 (ADM mass as the on-shell surface term). For an asymptotically flat slice the on-shell Hamiltonian equals the surface integral

$$M_{\mathrm{ADM}}=\frac{1}{16\pi }\underset{r\to \infty }{\lim }{\oint }_{S_r}({\partial }_j{\gamma }_{ij}-{\partial }_i{\gamma }_{jj})N^{i}d{S}^j,$$

which is finite, coordinate-invariant at spatial infinity, and coincides with the Arnowitt--Deser--Misner energy of the spacetime.

Proof (sketch). Performing the canonical $3+1$ reduction of the Einstein--Hilbert action $S=(16\pi {)}^{-1}\int {}^{(4)}R\sqrt{-g}{d}^{4}x$ 13.04.02 and integrating by parts to isolate terms proportional to ${\partial }_t{\gamma }_{ij}$ (the canonical momentum variable) leaves a bulk term in $N\mathcal{H}+N^i{\mathcal{H}}_i$ plus a boundary term from the integration by parts of the spatial scalar curvature, which on an asymptotically flat slice reduces to ${\int }_{\partial \Sigma }({\partial }_j{\gamma }_{ij}-{\partial }_i{\gamma }_{jj})d{S}^i$ at large $r$. On shell the bulk vanishes by Proposition 3, so the on-shell Hamiltonian is purely the surface term; writing the asymptotic surface element with the lapse-shift parametrisation gives the displayed $M_{\mathrm{ADM}}$. Coordinate invariance at spatial infinity follows because any two asymptotically flat coordinate systems agree up to Lorentz transformations and translations at $r=\infty$, under which the surface integral transforms as the time component of a four-vector. $\square$

Connections Master

• Riemann curvature tensor 13.03.01 supplies the four-dimensional curvature that the Gauss--Codazzi equations project onto each slice. The ${}^{(3)}R$ entering the Hamiltonian constraint is the spatial contraction of the projected Riemann tensor; without the Riemann tensor's symmetries the Gauss identity would not close into a scalar.

• Ricci and Einstein tensors 13.03.02 are the objects whose $3+1$ projections produce the constraints. The Hamiltonian constraint is the $n^{\mu }{n}^{\nu }$ projection of the Einstein equation, the momentum constraint is the mixed $n$--$\gamma$ projection, and the contracted Bianchi identity proved there is exactly the mechanism that preserves the constraints under evolution.

• Einstein field equations 13.04.01 are the source of both the constraints and the evolution equations. The $3+1$ split of this unit organises those ten equations into four elliptic constraints plus six hyperbolic evolution equations, the structural decomposition on which numerical relativity is built.

• Einstein--Hilbert action and variational formulation 13.04.02 re-derives the ADM Hamiltonian from a variational principle. The constraint functions $\mathcal{H}$, ${\mathcal{H}}_i$ arise there as the Noether identities of diffeomorphism invariance, and the ADM mass surface term appears as the boundary term required for a well-defined variational principle on an asymptotically flat slice.

• Linearised gravity and gravitational waves 13.07.01 is the regime in which ADM initial data are most often evolved. Binary black-hole simulations that produce the waveforms LIGO detects start from Bowen--York puncture data constructed here, evolve them with the ADM (or BSSN) equations, and read off the radiated energy from the far-field surface integral of Theorem 1's machinery.

• Schwarzschild solution and the Kerr black hole 13.05.01/13.05.04 supply the building blocks for the initial-data problem: a single Schwarzschild or Kerr black hole is the simplest asymptotically flat solution, and binary puncture data are constructed by superposing conformal images of these on a maximal slice before solving the Lichnerowicz equation.

• FLRW cosmology and the Friedmann equations 13.08.01 are the homogeneous-isotropic specialisation of the ADM evolution system. The first Friedmann equation is precisely the Hamiltonian constraint on a homogeneous slice, and the second Friedmann equation is the ADM Raychaudhuri (trace-evolution) equation, so cosmology lives inside this formalism as its most symmetric sector.

Historical & philosophical context Master

The $3+1$ decomposition is the work of Richard Arnowitt, Stanley Deser, and Charles Misner, who between 1959 and 1962 recast general relativity as a canonical Hamiltonian system in a long series of papers crowned by their 1962 review The Dynamics of General Relativity ^{[Arnowitt, Deser & Misner 1962]}. Their aim was quantum: a Hamiltonian formulation is the prerequisite for canonical quantisation, and the ADM variables $({\gamma }_{ij},{\pi }^{ij})$ are the canonical coordinate and momentum of the gravitational field. The striking outcome of their reduction --- that the Hamiltonian is a sum of constraint functions plus a surface term --- was recognised immediately as the signature of general covariance and would, two decades later, become the entry point of Wheeler's geometrodynamics and the Wheeler--DeWitt equation.

John Wheeler coined the term geometrodynamics for the ADM vision of three-geometry evolving in time, and his student Bryce DeWitt wrote down the Wheeler--DeWitt equation $\hat{\mathcal{H}}\Psi [{\gamma }_{ij}]=0$ by promoting the Hamiltonian constraint to an operator, an equation with no time parameter at all --- the so-called problem of time of canonical quantum gravity. The classical side of the same formalism, however, found its most consequential application somewhere Wheeler had not emphasised: in computation.

The decisive turn was James York's conformal method. York showed ^{[York 1971]} that the Hamiltonian constraint is most naturally solved not for the physical metric directly but for a conformal factor obeying an elliptic equation (the Lichnerowicz equation, already known to Andr'e Lichnerowicz ^{[Lichnerowicz 1944]} from his 1944 study of the initial-value problem), and that the momentum constraint linearises in a conformal Killing weight ^{[York 1972]}, ^{[York 1979]}. Together with the Bowen--York analytic solutions ^{[Bowen & York 1980]}, this gave numerical relativity a constructive recipe for initial data. The ADM evolution system and its stabilised successor BSSN became the engine of black-hole merger simulations, and the waveforms those simulations produced were the essential theoretical input to LIGO's 2015 detection of gravitational waves 13.07.01. A formalism devised for quantum gravity delivered, half a century later, the most precise classical prediction ever tested.

Bibliography Master

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2. Lichnerowicz, A., "L'int'egration des 'equations de la gravitation relativiste et le probl`eme des $n$ corps", J. Math. Pures Appl. 23 (1944), 37--63.

3. York, J. W., "Gravitational degrees of freedom and the Einstein equations", Phys. Rev. Lett. 26 (1971), 1656--1658.

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5. York, J. W., "Kinematics and dynamics of general relativity", in L. L. Smarr (ed.), Sources of Gravitational Radiation (Cambridge University Press, 1979), pp. 83--126.

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7. Wald, R. M., General Relativity (University of Chicago Press, 1984), Appendix E and \S 10.2.

8. Gourgoulhon, E., 3+1 Formalism in General Relativity: Bases of Numerical Relativity, Lecture Notes in Physics 846 (Springer, 2012). [arXiv/0703035 (2007)]

9. Baumgarte, T. W. & Shapiro, S. L., Numerical Relativity: Solving Einstein's Equations on the Computer (Cambridge University Press, 2010), Ch. 2--3 and Ch. 11--13.

10. Misner, C. W., Thorne, K. S. & Wheeler, J. A., Gravitation (Freeman, 1973), Ch. 21.

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13. Hartle, J. B., Gravity: An Introduction to Einstein's General Relativity (Benjamin Cummings, 2003), Ch. 21.