ADM 3+1 Decomposition of Spacetime

General relativity is a theory of four-dimensional spacetime geometry, yet almost every question we ask of it is a question about time: given the gravitational field now, what will it be later? The ADM decomposition makes that question legitimate. Because the field equations $G_{\mu\nu}=8\pi T_{\mu\nu}$ are generally covariant, spacetime hands us no preferred clock; the 3+1 program supplies one by hand, choosing a smooth time function $t$ whose level sets $\Sigma_t$ are spacelike slices that stack up to fill (a globally hyperbolic patch of) spacetime. On a single slice, geometry is the induced Riemannian metric $\gamma_{ij}$ ; the relation between neighbouring slices is carried by two more objects — the lapse $N$ , measuring proper time along the normal, and the shift $N^i$ , measuring how the spatial coordinate grid slides sideways from one slice to the next. The payoff, worked out by Arnowitt, Deser and Misner between 1959 and 1962, is that the ten components of $g_{\mu\nu}$ separate cleanly into six that evolve and four that are pure convention. Only the spatial metric $\gamma_{ij}$ and its conjugate momentum — built from the extrinsic curvature $K_{ij}$ — carry dynamics; the lapse and shift enter the action with no time derivatives, so they are Lagrange multipliers whose variation delivers the four constraints. This single move turns Einstein's equations into a well-posed initial-value problem, defines the conserved global charges of an isolated system, and provides the scaffolding on which numerical relativity simulates colliding black holes.

Key takeaways

The ADM (Arnowitt-Deser-Misner) 3+1 decomposition slices spacetime $(\mathcal{M},g_{\mu\nu})$ into a family of spacelike hypersurfaces $\Sigma_t$ labelled by a global time function $t$ , recasting Einstein's equations as evolution in $t$ .
The geometry needs three objects: the induced spatial metric $\gamma_{ij}$ , the lapse $N$ (proper time per unit $t$ along the normal), and the shift $N^i$ (sideways drift of the coordinate grid between slices).
The four-metric reassembles as $\mathrm{d}s^2=-N^2\mathrm{d}t^2+\gamma_{ij}(\mathrm{d}x^i+N^i\mathrm{d}t)(\mathrm{d}x^j+N^j\mathrm{d}t)$ , and the extrinsic curvature $K_{ij}=-\tfrac12\mathcal{L}_n\gamma_{ij}$ records how each slice bends inside spacetime, playing the role of the velocity $\dot\gamma_{ij}$ .
Of Einstein's ten equations, four carry no second time derivative — the Hamiltonian and momentum constraints restricting the data on any slice — and six are evolution equations for $\gamma_{ij}$ and $K_{ij}$ ; lapse and shift are freely chosen gauge.
This split underlies the ADM Hamiltonian, the Cauchy problem, the definition of ADM mass, and essentially all of numerical relativity (BSSN, generalized harmonic, moving punctures).
Key idea: Foliate spacetime by spacelike slices $\Sigma_t$ . Split the four-metric into the slice's intrinsic geometry $\gamma_{ij}$ (six dynamical components) plus the embedding data — lapse $N$ and shift $N^i$ (four pure-gauge components). The pair $(\gamma_{ij},K_{ij})$ is the canonical phase-space data; lapse and shift are Lagrange multipliers enforcing the Hamiltonian and momentum constraints.

Where this lives today

The 3+1 decomposition is the language every gravitational-wave observation is ultimately decoded in. When LIGO and Virgo recorded GW150914 in 2015, the black-hole masses and spins were read off by matching the data against template waveforms calibrated by numerical-relativity simulations that integrate the 3+1 evolution equations forward from constraint-satisfying initial data. Bare ADM evolution, equation (7) taken literally, is only weakly hyperbolic and blows up numerically; the field's decade-long struggle with this ended around 2005 when Pretorius (generalized harmonic) and, independently, the Brownsville and Goddard groups (BSSN with moving punctures) achieved the first stable binary-black-hole mergers. Both approaches keep the 3+1 core — slices, lapse, shift, extrinsic curvature — while repackaging the variables to cure the ill-posedness. Today open-source codes such as the Einstein Toolkit evolve these equations on adaptive meshes to model not only black-hole binaries but the neutron-star merger GW170817, whose electromagnetic counterpart confirmed that such collisions forge heavy elements — with the puncture initial data seeding every run generated by solving the same constraints on a single slice via York's conformal method.

The physical picture

Picture spacetime as a stack of thin spatial slices, like the pages of a book. Each page $\Sigma_t$ carries its own intrinsic geometry $\gamma_{ij}$ — the distances an observer confined to that page would measure. To turn one page you need two more things. The lapse $N$ says how much proper time a clock carried straight up, perpendicular to the pages, records: proper time is $N\,\mathrm{d}t$ . The shift $N^i$ says how far the coordinate grid on the next page has slid sideways relative to that straight-up direction. Together, lapse and shift are the freedom to relabel spacetime points — they are gauge, not physics. The one object that cannot be read off from inside any single slice is the extrinsic curvature $K_{ij}$ : how the slice is bent within the surrounding spacetime, information a flatlander on the page can never obtain. A sheet of paper rolled into a cylinder has zero intrinsic curvature — you can unroll it without stretching — yet obvious extrinsic curvature. In the 3+1 language $K_{ij}$ is the "velocity" of the spatial geometry: up to lapse and shift it is $\dot\gamma_{ij}$ , and it supplies half the canonical momentum conjugate to $\gamma_{ij}$ . The first figure fixes this geometry — the step between slices splits into a normal piece set by the lapse and a tangential piece set by the shift.

Figure

Foliation of spacetime into spacelike slices. The coordinate-time vector t splits into a normal part N n (the lapse, giving proper time N dt) and a tangential part with components N^i dt (the shift), so that t = N n + shift and ds² = −N² dt² + γ_ij(dx^i + N^i dt)(dx^j + N^j dt)

The deepest surprise is that four of Einstein's ten equations contain no second time derivative at all. These are constraints — conditions the data on a single slice must already satisfy, exactly like $\nabla\cdot\mathbf{E}=\rho$ in electromagnetism, which restricts fields on a slice rather than telling them how to march. The other six are true evolution equations for $\partial_t\gamma_{ij}$ and $\partial_t K_{ij}$ . Because the contracted Bianchi identity keeps the constraints satisfied under that evolution, one solves them once, on the initial slice, and integrates forward. The second figure makes this counting exact, and it ends where LIGO begins: two physical polarizations of the gravitational field.

Figure

The bookkeeping of the 3+1 split. The 10 components of the metric split as 6 + 1 + 3; Einstein's 10 equations split into 4 constraints and 6 evolution equations; the 12-dimensional phase space of γ_ij and π^ij, minus 4 first-class constraints and their 4 gauge partners, leaves 2 physical polarizations — the two graviton states LIGO records

The core equations

The whole apparatus flows from one identity: the four-metric written in foliation data. Everything below is a piece of $g_{\mu\nu}$ read off from this line element, or a projection of Einstein's equations through it. The lapse squares the timelike direction, the shift threads the spatial coordinates, and the spatial block is the induced metric.

\mathrm{d}s^2=-N^2\,\mathrm{d}t^2+\gamma_{ij}\,(\mathrm{d}x^i+N^i\,\mathrm{d}t)(\mathrm{d}x^j+N^j\,\mathrm{d}t) \qquad (1)

where: $N$ — lapse function, $N>0$ ; proper time along the normal per unit coordinate time; $N^i$ — shift vector, 3 components; sideways drift of the coordinates between slices; $\gamma_{ij}$ — induced spatial metric on $\Sigma_t$ ; positive-definite, 6 components; $x^i$ — spatial coordinates, $i=1,2,3$ ; $t$ — global time function labelling the foliation. Reading it: set $N=1,N^i=0$ and it collapses to the familiar $-\mathrm{d}t^2+\gamma_{ij}\,\mathrm{d}x^i\mathrm{d}x^j$ ; lapse and shift are the corrections that let the slices be spaced and threaded arbitrarily. Each slice is a level set of $t$ , so its normal is the normalized gradient of $t$ . The Eulerian (normal) observer is the one whose worldline is this normal — at rest with respect to the foliation, ticking proper time $N\,\mathrm{d}t$ .

n_\mu=-N\,\nabla_\mu t=(-N,0,0,0),\qquad n^\mu=\frac{1}{N}\,(1,-N^i) \qquad (2)

where: $n_\mu$ — future-directed unit normal covector, $n_\mu n^\mu=-1$ ; $n^\mu$ — unit normal vector, the four-velocity of the Eulerian observer; $\nabla_\mu t$ — gradient of the time function, everywhere orthogonal to $\Sigma_t$ . Reading it: the shift appears in $n^\mu$ but not in $n_\mu$ — the normal direction is fixed by the slicing alone. Any spacetime tensor now splits into a piece along $n^\mu$ and a piece tangent to the slice. The tool is the projector $\gamma^\mu{}_\nu$ , which annihilates the normal and is idempotent; its fully-lowered form is the spatial metric as a four-tensor.

\gamma_{\mu\nu}=g_{\mu\nu}+n_\mu n_\nu,\qquad {\gamma^\mu}_\nu=\delta^\mu_\nu+n^\mu n_\nu \qquad (3)

where: $\gamma_{\mu\nu}$ — orthogonal projector onto $\Sigma_t$ ; equals the induced metric on spatial indices; ${\gamma^\mu}_\nu$ — projection operator, satisfying ${\gamma^\mu}_\nu n^\nu=0$ and ${\gamma^\mu}_\alpha{\gamma^\alpha}_\nu={\gamma^\mu}_\nu$ . Reading it: adding $n_\mu n_\nu$ removes exactly the timelike direction, turning four-dimensional geometry into slice geometry. The new geometric datum, invisible from within a slice, is the extrinsic curvature: the rate at which the unit normal tilts as you slide along the surface, equal to minus half the Lie derivative of $\gamma_{ij}$ along the normal.

K_{ij}=-\tfrac{1}{2}\,\mathcal{L}_n\gamma_{ij}=\frac{1}{2N}\left(-\partial_t\gamma_{ij}+D_iN_j+D_jN_i\right) \qquad (4)

where: $K_{ij}$ — extrinsic curvature (second fundamental form), symmetric, measures how $\Sigma_t$ bends in $\mathcal{M}$ ; $\mathcal{L}_n$ — Lie derivative along $n^\mu$ ; $D_i$ — covariant derivative compatible with $\gamma_{ij}$ ; $K=\gamma^{ij}K_{ij}$ — mean curvature (trace), equal to $-\nabla_\mu n^\mu$ , the expansion of the normal congruence. Reading it: solving for $\partial_t\gamma_{ij}=-2NK_{ij}+D_iN_j+D_jN_i$ gives the first evolution equation — the metric changes at a rate set by $K_{ij}$ , corrected by the shift's dragging of coordinates. Now project $G_{\mu\nu}=8\pi T_{\mu\nu}$ . The double-normal projection, via the Gauss equation, has no second time derivative — it is the Hamiltonian constraint, binding intrinsic and extrinsic curvature to the local energy density.

{}^{(3)}R+K^2-K_{ij}K^{ij}=16\pi\rho \qquad (5)

where: ${}^{(3)}R$ — Ricci scalar of the spatial metric $\gamma_{ij}$ ; $\rho=n_\mu n_\nu T^{\mu\nu}$ — energy density measured by the Eulerian observer; $K^{ij}=\gamma^{ik}\gamma^{jl}K_{kl}$ . Reading it: in vacuum it becomes ${}^{(3)}R=K_{ij}K^{ij}-K^2$ , a purely geometric condition the initial slice must obey — you cannot paint any $(\gamma_{ij},K_{ij})$ you like. The mixed normal-tangential projection, via the Codazzi equation, gives the three momentum constraints, tying the divergence of the trace-reversed extrinsic curvature to the momentum density through the slice.

D_j\!\left(K^{ij}-\gamma^{ij}K\right)=8\pi S^i \qquad (6)

where: $D_j$ — spatial covariant derivative; $S^i=-{\gamma^i}_\mu n_\nu T^{\mu\nu}$ — momentum density measured by the Eulerian observer. Reading it: in vacuum the right-hand side vanishes, and the constraint restricts the transverse-traceless part of $K_{ij}$ — the raw material York's conformal method later solves for. The purely spatial projection gives the second evolution equation, for $K_{ij}$ itself — the dynamical heart of the system, and where the lapse gradient (the gravitational acceleration) enters.

\partial_t K_{ij}=-D_iD_jN+N\Big({}^{(3)}R_{ij}+KK_{ij}-2K_{ik}{K^k}_j\Big)-8\pi N\!\left(S_{ij}-\tfrac12\gamma_{ij}(S-\rho)\right)+\mathcal{L}_{\vec N}K_{ij} \qquad (7)

where: ${}^{(3)}R_{ij}$ — Ricci tensor of the spatial metric; $S_{ij}={\gamma_i}^\mu{\gamma_j}^\nu T_{\mu\nu}$ — spatial stress, $S=\gamma^{ij}S_{ij}$ ; $-D_iD_jN$ — second covariant derivative of the lapse, the gauge term. Reading it: the $-D_iD_jN$ term is why a clever lapse can freeze evolution near a singularity, bleeding coordinate time out of the strong-field region. Finally, the Hamiltonian picture: the Legendre transform identifies the momentum conjugate to $\gamma_{ij}$ and shows the bulk Hamiltonian is a sum of constraints weighted by lapse and shift, plus a boundary term equal to the total energy.

\pi^{ij}=\sqrt{\gamma}\,(K^{ij}-\gamma^{ij}K),\qquad H_{ADM}=\int_{\Sigma_t}\!\mathrm{d}^3x\,\big(N\,\mathcal{H}+N^i\,\mathcal{H}_i\big)+\oint_{\partial\Sigma}(\ldots) \qquad (8)

where: $\pi^{ij}$ — momentum conjugate to $\gamma_{ij}$ , a tensor density of weight $+1$ ; $\sqrt{\gamma}$ — spatial volume element; $\mathcal{H}\approx0,\ \mathcal{H}_i\approx0$ — Hamiltonian and momentum constraint densities, weakly zero on shell; the boundary term is the Gibbons-Hawking-York / ADM surface integral giving the ADM mass at infinity. Reading it: on the constraint surface the bulk integral vanishes, so all the energy of an isolated gravitating system lives in the boundary — a statement with no analogue in non-gravitational field theory.

Derivation AAssembling the line element from lapse, shift, and the slice metric

The goal is to show that the four-metric is completely rebuilt from the triple $(N,N^i,\gamma_{ij})$ , so that lapse and shift are the only non-spatial data in $g_{\mu\nu}$ . We take the coordinate-time vector $\partial_t$ , decompose it into a normal part and a tangential part, and evaluate the metric on the coordinate basis vectors block by block; completing the square reproduces the ADM form and the volume element $\sqrt{-g}=N\sqrt{\gamma}$ that the action needs.

Step 1. Write the coordinate-time vector as $t^\mu=(\partial_t)^\mu=N\,n^\mu+N^\mu$ , with $N^\mu=(0,N^i)$ tangent to the slice and orthogonal to the normal, $n_\mu N^\mu=0$ . Why: this is the definition of lapse and shift — the lapse is how far you step along the normal per unit $t$ , the shift is how far you slide tangentially, and every foliation is specified by exactly this split.

Step 2. Compute $g_{tt}=g_{\mu\nu}t^\mu t^\nu=N^2\,g_{\mu\nu}n^\mu n^\nu+g_{\mu\nu}N^\mu N^\nu=-N^2+\gamma_{ij}N^iN^j$ , the cross term dropping because $N^\mu\perp n^\mu$ and using $n_\mu n^\mu=-1$ . Why: this is the only metric component where the lapse appears; the timelike interval between slices is set by $N$ , corrected by the sideways shift energy $\gamma_{ij}N^iN^j$ .

Step 3. Compute the mixed and spatial blocks. Since the coordinate basis vectors $(\partial_i)^\mu$ are tangent to the slice, $g_{ti}=g_{\mu\nu}t^\mu(\partial_i)^\nu=\gamma_{ij}N^j=N_i$ and $g_{ij}=\gamma_{ij}$ . Why: tangential vectors never see the normal direction, so only the shift survives in $g_{ti}$ and the spatial block is simply the induced metric — proving $\gamma_{ij}$ really is "the metric you would measure on the slice."

Step 4. Substitute $g_{tt},g_{ti},g_{ij}$ into $\mathrm{d}s^2=g_{\mu\nu}\mathrm{d}x^\mu\mathrm{d}x^\nu$ and complete the square in $\mathrm{d}x^i+N^i\mathrm{d}t$ to recover equation (1). Inverting the block matrix gives $g^{00}=-1/N^2$ , $g^{0i}=N^i/N^2$ , $g^{ij}=\gamma^{ij}-N^iN^j/N^2$ , and the determinant yields $\sqrt{-g}=N\sqrt{\gamma}$ . Why: the factorized volume element is what makes the action split into a spatial integral with $N$ out front, the seed of the entire Hamiltonian analysis. *Verified: matches MTW eq. (21.40), the inverse-metric block and $\sqrt{-g}=N\sqrt{\gamma}$ of Baumgarte-Shapiro eq. (2.94) and Gourgoulhon (gr-qc/0703035) §4, and Wald Appendix E.2.*

Go deeper · Derivation B · 1 minFrom Einstein's equations to the Hamiltonian and momentum constraints

Derivation BFrom Einstein's equations to the Hamiltonian and momentum constraints

The aim is to show why four of Einstein's ten equations become instantaneous constraints rather than evolution equations, and to derive their explicit form. The Gauss and Codazzi relations of hypersurface theory express projections of the four-curvature through the slice's intrinsic Ricci curvature and its extrinsic curvature $K_{ij}$ ; feeding these into $G_{\mu\nu}=8\pi T_{\mu\nu}$ and taking the projections that involve the normal produces relations with no second time derivative — they constrain the data on a slice instead of evolving it.

Step 1. Contract the Gauss equation twice with the normal to obtain the scalar identity ${}^{(3)}R+K^2-K_{ij}K^{ij}=2\,G_{\mu\nu}n^\mu n^\nu$ . Why: the double-normal projection of the four-curvature is exactly the combination of intrinsic and extrinsic curvature on the left, so this equation carries only data living on the slice — no information about how the slice will move.

Step 2. Insert the field equations: $G_{\mu\nu}n^\mu n^\nu=8\pi T_{\mu\nu}n^\mu n^\nu=8\pi\rho$ , where $\rho$ is the Eulerian energy density, giving ${}^{(3)}R+K^2-K_{ij}K^{ij}=16\pi\rho$ . Why: this is the Hamiltonian constraint — a scalar elliptic relation on each slice that forbids arbitrary initial data and, in the Hamiltonian picture, is the statement that the Hamiltonian density vanishes.

Step 3. Take the mixed normal-tangential projection using the contracted Codazzi equation, ${\gamma^\mu}_i\,n^\nu G_{\mu\nu}=D_j(K^{ij}-\gamma^{ij}K)$ . Why: this projection also lacks second time derivatives, and it isolates the divergence of the trace-reversed extrinsic curvature — the geometric object that will play the role of a momentum.

Step 4. Insert the field equations, ${\gamma^\mu}_i n^\nu G_{\mu\nu}=8\pi S_i$ , with momentum density $S^i=-{\gamma^i}_\mu n_\nu T^{\mu\nu}$ , yielding the three momentum constraints $D_j(K^{ij}-\gamma^{ij}K)=8\pi S^i$ . Why: together with the Hamiltonian constraint these four equations are precisely the projections of Einstein's equations that reference the normal direction, so exactly four of the ten are constraints and the remaining six — the purely spatial projection — are the evolution equations for $K_{ij}$ . *Verified: reproduces Wald (1984) eqs. (E.2.30)-(E.2.34) and Gourgoulhon (gr-qc/0703035) eqs. (10.36)-(10.39); consistent with MTW Box 21.2 up to their sign convention for $K$ .*

Go deeper · Derivation C · 1 minWhy lapse and shift carry no dynamics

Derivation CWhy lapse and shift carry no dynamics

Here the goal is to prove the central structural claim of the ADM formulation: the lapse and shift are not solved for by the field equations but are freely specifiable gauge functions. The route is the Hamiltonian analysis — identify the momentum conjugate to $\gamma_{ij}$ , note that none is conjugate to $N$ or $N^i$ since their time derivatives never appear in the action, Legendre-transform, and read off that lapse and shift multiply the constraints. Varying with respect to them then returns the constraints as their "equations of motion," confirming they encode pure coordinate freedom.

Step 1. From the ADM Lagrangian density $\mathcal{L}=\sqrt{\gamma}\,N\big({}^{(3)}R+K_{ij}K^{ij}-K^2\big)$ , define $\pi^{ij}=\partial\mathcal{L}/\partial(\partial_t\gamma_{ij})=\sqrt{\gamma}\,(K^{ij}-\gamma^{ij}K)$ . Why: this is the canonical momentum of the true dynamical field $\gamma_{ij}$ ; crucially $\partial_t N$ and $\partial_t N^i$ are absent from $\mathcal{L}$ , so $\pi_N\approx0$ and $\pi_{N^i}\approx0$ are primary constraints — there is nothing for lapse and shift to be canonically paired with.

Step 2. Legendre transform, $\mathcal{H}_{\text{density}}=\pi^{ij}\partial_t\gamma_{ij}-\mathcal{L}$ , and substitute $\partial_t\gamma_{ij}=-2NK_{ij}+D_iN_j+D_jN_i$ . Why: the substitution makes $N$ and $N^i$ appear only as overall multiplicative coefficients, never differentiated in time, which is the algebraic signature of a Lagrange multiplier.

Step 3. Collect terms into $\mathcal{H}_{\text{density}}=N\,\mathcal{H}+N^i\,\mathcal{H}_i+\partial_i(\ldots)$ , with $\mathcal{H}=\sqrt{\gamma}\big(-{}^{(3)}R+\gamma^{-1}(\pi_{ij}\pi^{ij}-\tfrac12\pi^2)\big)$ in vacuum and $\mathcal{H}_i=-2D_j{\pi^j}_i$ . Why: the bulk Hamiltonian is thus a sum of constraints weighted by lapse and shift, so on the constraint surface $\mathcal{H}=\mathcal{H}_i=0$ it vanishes and only the boundary term survives — this is the origin of the ADM mass as a surface integral.

Step 4. Vary the Hamiltonian with respect to the multipliers: $\delta H/\delta N=\mathcal{H}=0$ and $\delta H/\delta N^i=\mathcal{H}_i=0$ . Why: the "equations of motion" for lapse and shift are just the constraints again, not evolution equations for $N$ or $N^i$ — so they are undetermined by the dynamics and must be prescribed by hand as gauge conditions (geodesic, maximal, $1+\log$ , and so on). *Verified: equivalent to ADM (1962, reprinted Gen. Rel. Grav. 40, 1997); Wald Appendix E.2; the vanishing on-shell bulk Hamiltonian is the standard "Hamiltonian constraint generates time reparametrization" result, Gourgoulhon §4.*

Limiting cases

Particular choices of lapse and shift — or particular spacetimes — collapse the general 3+1 machinery onto familiar or simplified structures. Each row is a gauge or symmetry specialization, not a new physical theory.

Case	Result	Meaning
Geodesic (synchronous) slicing	$N=1,\ N^i=0$	Normal observers free-fall and $t$ is their proper time. Simple but self-destructing: geodesics focus and the coordinates hit caustics in finite time, useless near strong fields.
Maximal slicing	$K=\gamma^{ij}K_{ij}=0$	Demanding $\partial_t K=0$ gives the elliptic lapse equation $D^iD_iN=N(K_{ij}K^{ij}+4\pi(\rho+S))$ ; the lapse collapses near a black hole, freezing time before the singularity — the classic singularity-avoiding gauge.
Static / stationary spacetime	$\partial_t\gamma_{ij}=0$	A static metric admits a slicing with $K_{ij}=0$ ; stationary Kerr keeps $\partial_t\gamma_{ij}=0$ with $K_{ij}\ne0$ absorbed into the shift, recovering the timelike Killing structure.
Asymptotically flat slice	$\gamma_{ij}\to\delta_{ij}+O(1/r),\ K_{ij}\to O(1/r^2)$	The fall-off at spatial infinity makes the ADM mass, momentum and angular momentum finite surface integrals; without it the charges diverge.
Weak field, slow motion	$N\approx1+\Phi,\ \gamma_{ij}\approx(1-2\Phi)\delta_{ij}$	The lapse is $1+\Phi$ with $\Phi$ the Newtonian potential, and equation (7) reproduces $\nabla^2\Phi=4\pi\rho$ — the lapse gradient $-D_iD_jN$ is Newtonian gravity.

Common trapThe lapse and shift are outputs of Einstein's equations

They are not. The lapse $N$ and shift $N^i$ are freely specifiable gauge functions that you must prescribe — geodesic slicing $N=1$ , maximal slicing, $1+\log$ slicing $\partial_t N=-2NK$ , a Gamma-driver shift, whatever suits the problem. Einstein's equations never determine them; they determine only $(\gamma_{ij},K_{ij})$ . A frequent beginner error is to treat $\partial_t N$ or $\partial_t N^i$ as dynamical and hunt for their "evolution equations" — there are none, because $N$ and $N^i$ enter the action without time derivatives. A second trap is the sign convention for $K_{ij}$ : this article uses $K_{ij}=-\tfrac12\mathcal{L}_n\gamma_{ij}$ (Wald / Gourgoulhon), whereas MTW take the opposite sign. Always check the convention before quoting the trace $K$ , the momentum constraint, or the $1+\log$ condition, or your singularity-avoiding gauge will drive the lapse the wrong way.

Worked examples

1 · The Friedmann equation from the Hamiltonian constraint (fully worked)

Take a spatially flat expanding universe in the slicing $\mathrm{d}s^2=-\mathrm{d}t^2+a(t)^2\delta_{ij}\,\mathrm{d}x^i\mathrm{d}x^j$ , so $N=1$ , $N^i=0$ , and $\gamma_{ij}=a(t)^2\delta_{ij}$ . Show that the Hamiltonian constraint (5) is exactly the Friedmann equation, and evaluate the present critical density for $H_0=67.4\ \mathrm{km\,s^{-1}Mpc^{-1}}$ .

Step 1. With $N=1$ and $N^i=0$ , equation (4) gives $K_{ij}=-\tfrac12\partial_t\gamma_{ij}=-a\dot a\,\delta_{ij}$ . Tracing with $\gamma^{ij}=a^{-2}\delta^{ij}$ : $K=-3\dot a/a=\mathbf{-3H}$ , with $H=\dot a/a$ . The mean curvature of a cosmological slice is minus three times the Hubble rate.

Step 2. Form the curvature scalars: $K^2=9H^2$ and $K_{ij}K^{ij}=\gamma^{ik}\gamma^{jl}K_{ij}K_{kl}=3\dot a^2/a^2=3H^2$ , so $K^2-K_{ij}K^{ij}=\mathbf{6H^2}$ . The flat slices have ${}^{(3)}R=0$ , so the whole left side of the constraint is extrinsic.

Step 3. Substitute into (5): ${}^{(3)}R+K^2-K_{ij}K^{ij}=16\pi\rho$ becomes $6H^2=16\pi\rho$ , i.e. $\mathbf{H^2=\tfrac{8\pi}{3}\rho}$ (restoring $G$ , $H^2=\tfrac{8\pi G}{3}\rho$ ). The first Friedmann equation is nothing but the Hamiltonian constraint of an isotropic 3+1 slicing.

Step 4. Set $H^2=\tfrac{8\pi G}{3}\rho_c$ . With $H_0=67.4\ \mathrm{km\,s^{-1}Mpc^{-1}}=2.18\times10^{-18}\ \mathrm{s^{-1}}$ and $G=6.674\times10^{-11}$ , $\rho_c=3H_0^2/(8\pi G)=\mathbf{8.5\times10^{-27}\ kg\,m^{-3}}$ — about 5 hydrogen atoms per cubic metre, the density that makes space flat.

2 · Reading the lapse and shift off a black hole (Painlevé-Gullstrand)

The Schwarzschild geometry in Painlevé-Gullstrand coordinates is $\mathrm{d}s^2=-\mathrm{d}t^2+\big(\mathrm{d}r+\sqrt{2M/r}\,\mathrm{d}t\big)^2+r^2\mathrm{d}\Omega^2$ . Match it to the ADM form (1) and interpret the pieces at the horizon $r=2M$ .

Step 1. Expand the square and compare with (1): grouping the cross term $2\sqrt{2M/r}\,\mathrm{d}r\,\mathrm{d}t$ into a perfect square gives $\mathbf{N=1}$ , $\mathbf{N^r=\sqrt{2M/r}}$ , and $\gamma_{rr}=1$ , $\gamma_{\theta\theta}=r^2$ , $\gamma_{\phi\phi}=r^2\sin^2\theta$ . The spatial slices are exactly flat Euclidean 3-space, ${}^{(3)}R=0$ — the famous "flat slicing" of Schwarzschild.

Step 2. With $N=1$ the Eulerian observer's proper time equals coordinate time, and the shift $N^r=\sqrt{2M/r}$ drags the spatial grid radially inward at coordinate speed $\sqrt{2M/r}$ — the Newtonian free-fall velocity from rest at infinity. These Eulerian observers are the "raindrop" observers who fall freely from infinity.

Step 3. At $r=2M$ the shift magnitude is $\sqrt{2M/2M}=\mathbf{1}$ : the free-fall speed reaches $c$ exactly at the horizon, while the slices stay perfectly regular flat space. This is why flat-slicing coordinates penetrate the horizon smoothly where Schwarzschild's static lapse ( $N^2=1-2M/r\to0$ ) collapses. The horizon is where the shift reaches the speed of light, not where the geometry fails.

Physics of the portrait

The 3+1 decomposition is inseparable from three names — Richard Arnowitt, Stanley Deser and Charles Misner — who between 1959 and 1962 turned Dirac's abstract constraint analysis of gravity into the concrete canonical formulation the whole field now uses.

Origins · 1 minArnowitt, Deser and Misner and how this was discovered

Arnowitt, Deser and Misner

1959–1962, "The Dynamics of General Relativity" · canonical gravity and the birth of the 3+1 formalism

When Richard Arnowitt, Stanley Deser and Charles Misner began their collaboration at the end of the 1950s, general relativity was widely regarded as geometry too beautiful to be practical: elegant, generally covariant, and almost impossible to solve. Their papers, culminating in the 1962 review "The Dynamics of General Relativity," asked a physicist's question rather than a geometer's — what are the true degrees of freedom of the gravitational field, and what is its energy? Building on Dirac's 1958 Hamiltonian treatment of constrained systems and the century-old embedding theory of Gauss and Codazzi, they foliated spacetime into spacelike slices, isolated the spatial metric $\gamma_{ij}$ and its conjugate momentum $\pi^{ij}$ as the genuine dynamical variables, and exposed the lapse and shift as Lagrange multipliers enforcing four constraints. Two results have outlived everything else in the work. First, the counting: the field carries exactly two propagating degrees of freedom per point — the two polarizations LIGO later measured. Second, the ADM energy — a surface integral at spatial infinity giving an unambiguous, positive total mass to an isolated system, resolving decades of confusion over whether gravitational energy could even be defined. Their formalism, once mathematical housekeeping, is now the operating system of numerical relativity.

History

1915

— Einstein's field equations fix spacetime geometry without singling out any time coordinate

1958

— Dirac recasts general relativity as a constrained Hamiltonian system

1962

— Arnowitt, Deser and Misner publish the canonical 3+1 formulation and the ADM energy

1971

— York develops the conformal method for solving the initial-data constraints

2005

— Pretorius achieves the first stable binary-black-hole merger simulation

Going further

Texts. Misner, Thorne & Wheeler, Gravitation (1973), Ch. 21 · Wald, General Relativity (1984), Ch. 10 and Appendix E.2 · Carroll, Spacetime and Geometry (2004), Ch. 4 and the initial-value appendix · Baumgarte & Shapiro, Numerical Relativity (2010), Ch. 2-3 · Alcubierre, Introduction to 3+1 Numerical Relativity (2008), Ch. 2-4 · Gourgoulhon, 3+1 Formalism in General Relativity, arXiv:gr-qc/0703035.

Provenance. Signature $(-,+,+,+)$ throughout, with $K_{ij}=-\tfrac12\mathcal{L}_n\gamma_{ij}$ (Wald / Gourgoulhon convention; MTW use the opposite sign for $K$ ) and geometrized units $G=c=1$ except where SI values are quoted. A verifier should check: (1) the inverse-metric blocks and $\sqrt{-g}=N\sqrt{\gamma}$ against MTW Box 21.2; (2) the Hamiltonian and momentum constraints against Wald eqs. (E.2.30)-(E.2.34); (3) the $K_{ij}$ evolution equation against Gourgoulhon eqs. (10.43)-(10.44); (4) the FRW result $6H^2=16\pi\rho$ and $\rho_c=8.5\times10^{-27}\ \mathrm{kg\,m^{-3}}$ at $H_0=67.4\ \mathrm{km\,s^{-1}Mpc^{-1}}$ ; (5) the Painlevé-Gullstrand identification $N=1$ , $N^r=\sqrt{2M/r}$ , shift $=1$ at $r=2M$ . The count $12-4-4=4=2\times2$ is standard.

Prerequisites. Lorentzian manifolds and the metric tensor · Covariant derivative and curvature tensors · Lie derivatives · Einstein field equations · Hypersurfaces and embeddings. Leads to. Gauss-Codazzi-Mainardi equations · Hamiltonian and momentum constraints · ADM Hamiltonian formulation · Cauchy problem in general relativity · ADM mass, momentum and angular momentum · BSSN formulation · Numerical relativity and binary black hole mergers · Gauge conditions in numerical relativity.

Ask the tutor: Ask me to slice a specific spacetime — Schwarzschild, Kerr, or a Friedmann universe — and read off its lapse, shift, and extrinsic curvature step by step.

What is the difference between intrinsic and extrinsic curvature on a slice?

Intrinsic curvature is everything an observer trapped on the slice can measure: the Riemann curvature of $\gamma_{ij}$ , revealed by parallel transport around loops confined to $\Sigma_t$ . Extrinsic curvature $K_{ij}=-\tfrac12\mathcal{L}_n\gamma_{ij}$ is invisible from inside — it records how the slice is bent within the four-dimensional spacetime, the rate at which the normal tilts as you move along the surface. The two are independent: a slice can be perfectly flat intrinsically ( ${}^{(3)}R_{ij}=0$ ) yet have $K_{ij}\ne0$ , as for the flat expanding slices of a Friedmann universe.

Why are the lapse and shift free rather than determined by physics?

Because they are labels, not fields. The lapse sets how much proper time separates neighbouring slices; the shift sets how the spatial coordinates are carried forward. Both are aspects of choosing coordinates, which general covariance makes arbitrary. In the action this is concrete: $N$ and $N^i$ appear with no time derivatives, so they cannot be dynamical — varying them yields the four constraints, not evolution equations for themselves. The prescription still matters enormously in practice: singularity-avoiding gauges like maximal or $1+\log$ slicing are what let a simulation march past black-hole formation without crashing.

Is the ADM system what codes actually evolve for black-hole mergers?

Not in its raw form. The bare ADM equations (5)-(7) are only weakly hyperbolic, so tiny errors grow without bound and the run blows up within a few light-crossing times. Every modern code evolves a strongly hyperbolic reformulation sharing the same 3+1 skeleton: BSSN conformally rescales $\gamma_{ij}$ and promotes the contracted connection to an independent variable, while the generalized-harmonic system imposes coordinate conditions that make Einstein's equations manifestly wave-like. Both keep the slices, lapse, shift and extrinsic curvature intact and add constraint damping — the 3+1 split is the foundation, BSSN or generalized harmonic the working engine.

What is the ADM mass and why is it a surface integral?

The ADM mass is the total energy of an isolated system, defined on an asymptotically flat slice as $M_{ADM}=\tfrac{1}{16\pi}\oint_\infty(\partial_j\gamma_{ij}-\partial_i\gamma_{jj})\,\mathrm{d}S^i$ . It is a surface integral for a deep reason: on the constraint surface the bulk Hamiltonian vanishes, so all the energy resides in the boundary term at spatial infinity. Unlike a non-gravitational field, whose energy is a volume integral of a local density, gravitational energy cannot be localized — the ADM mass measures only the total, read off from how the metric approaches flatness, and it satisfies the positive-energy theorem.

Does slicing spacetime destroy the covariance of general relativity?

No — it hides the covariance without breaking it. Choosing a foliation is choosing coordinates, and the physics is independent of that choice; different slicings of the same spacetime give identical invariants (curvature scalars, horizon areas, radiated energy). What the split trades away is manifest four-dimensional covariance in the equations: singling out a time function makes them look non-covariant in exchange for a well-posed evolution problem. The lapse and shift are exactly the bookkeeping of the discarded covariance — they parametrize every way of laying coordinates over the spacetime, so no physics is lost.

Why does one only solve the constraints once, on the initial slice?

Because the evolution preserves them automatically. The contracted Bianchi identity $\nabla_\mu G^{\mu\nu}=0$ , with $\nabla_\mu T^{\mu\nu}=0$ , guarantees that if the Hamiltonian and momentum constraints hold on the initial slice and the six evolution equations are satisfied, the constraints hold on every later slice — propagated for free. That consistency is what makes the initial-value formulation work: you need not re-impose four conditions at each step. Numerically it is subtler, since discretization error excites small constraint violations that can grow, which is why practical codes add constraint-damping terms to drive them back toward zero.