ADM Hamiltonian Formulation of General Relativity

The most vivid modern home of ADM is numerical relativity. The raw ADM evolution equations are only weakly hyperbolic and go numerically unstable within a few light-crossing times, which is why decades of effort produced no stable binary merger. The breakthroughs of 2005-2006 — Frans Pretorius with generalized harmonic gauge, and the Brownsville and Goddard groups with the "moving puncture" version of the BSSN reformulation (aq-gr-bssn-formulation) — all begin from ADM and repackage its variables to tame the instability. Those codes, now bundled in community platforms like the Einstein Toolkit and the SXS collaboration's simulations, generate the waveform template banks that LIGO and Virgo match-filter against; the 2015 detection GW150914 was confirmed as a black-hole merger by comparison with ADM-descended simulations. Upstream of evolution, the initial data on the first slice must satisfy the Hamiltonian and momentum constraints, and the industry-standard way to solve them — York and Lichnerowicz's conformal method (aq-gr-conformal-method-constraints) — is a direct descendant of the ADM constraint split. The formalism also frames the mathematical Cauchy problem (aq-gr-cauchy-problem-general-relativity) and gives the rigorous definitions of total mass, momentum, and angular momentum (aq-gr-adm-mass-momentum-angular-momentum) that enter the positive-energy theorem. And on the quantum side, the Hamiltonian constraint promoted to an operator, $\hat{\mathcal{H}}\Psi=0$, is the Wheeler-DeWitt equation, still the starting point for canonical quantum gravity and quantum cosmology.

Picture spacetime as a deck of cards, each card a spacelike slice $\Sigma_t$. The intrinsic metric $h_{ij}$ fixes the geometry within a card — distances and angles a surveyor confined to the slice would measure. The extrinsic curvature $K_{ij}$ records how the card is bent inside the four-dimensional spacetime, that is, how the geometry differs from one card to the next; it is the "velocity" of the 3-geometry. Two more fields glue the deck together. The lapse $N$ says how much proper time elapses as you step perpendicular from one card to the one above, $\mathrm{d}\tau=N\,\mathrm{d}t$; the shift $N^i$ says how far the coordinate grid slides sideways in the process. The first figure is exactly this decomposition of the flow of coordinate time.

Because you may shuffle, tilt, or re-grid the cards however you like without touching the physical spacetime they build, "pushing forward one card" is a choice, not a physical process. This is why the generator of that push — the Hamiltonian — is forced to vanish in the bulk: moving through the deck moves you nowhere gauge-invariant. The genuine, frame-independent quantities (total energy, momentum, angular momentum) can only be read off at the edge of the deck, infinitely far away, where an asymptotic observer has fixed a reference frame once and for all. The second figure makes the accounting quantitative: of the twelve numbers per point that specify a card and its velocity, only four survive as physics.

The constraints are not restrictions imposed by hand. They fall out automatically as the Euler-Lagrange equations of the lapse and shift, which appear in the action with no velocities to their name. In Dirac's language they are first-class constraints: their Poisson brackets close on the constraint surface, and they generate the spacetime diffeomorphisms — normal deformations of a slice for $\mathcal{H}$, tangential ones for $\mathcal{H}_i$. That is the canonical face of general covariance. And the momentum $\pi^{ij}\propto(K^{ij}-h^{ij}K)$ plays exactly the role $p=m\dot q$ plays in mechanics: it encodes the velocity of the 3-geometry, so the constraint $\mathcal{H}=0$ is a quadratic relation between position-like $h_{ij}$ and momentum-like $\pi^{ij}$, the geometric analogue of an energy equation fixing which slices are allowed.

The starting point is the ADM parametrization of the metric. Rather than treat the ten components $g_{\mu\nu}$ democratically, one splits them into the lapse, the shift, and the spatial metric, so that the geometry of, and the stacking of, the slices are made explicit. This single change of variables is what exposes the constrained structure.

where: $N$ — lapse function, proper time per unit coordinate time normal to the slice; $N^i$ — shift vector, sideways drag of spatial coordinates between slices; $h_{ij}$ — induced (intrinsic) Riemannian 3-metric on the slice $\Sigma_t$. Reading it: set $N=1,\ N^i=0$ and you recover $\mathrm{d}s^2=-\mathrm{d}t^2+h_{ij}\,\mathrm{d}x^i\mathrm{d}x^j$, a synchronous stack of slices; the general form just lets the ticking rate and the grid slide from point to point. The velocity of the geometry is the extrinsic curvature. It measures how the unit normal $n^\mu$ to a slice tips as you move across the slice — equivalently, the rate at which the 3-metric is dragged along that normal.

where: $K_{ij}$ — extrinsic curvature, the embedding of $\Sigma_t$ in spacetime; $n^\mu$ — future-pointing unit normal, $n_\mu=-N\,\partial_\mu t$; $D_i$ — Levi-Civita covariant derivative of $h_{ij}$; $\mathcal{L}_n$ — Lie derivative along $n^\mu$. We use the convention $K_{ij}=+\tfrac12\mathcal{L}_n h_{ij}$, the one that goes with equation (3); much of the numerical-relativity literature takes the opposite sign, which flips $K$ and $\pi^{ij}$ but leaves $K^2-K_{ij}K^{ij}$ unchanged. Reading it: the term $\partial_t h_{ij}$ is the raw change of the metric, while the shift pieces $D_iN_j+D_jN_i$ subtract off the part of that change that is mere coordinate sliding, isolating the honest bending. Passing to the Hamiltonian picture requires the momentum conjugate to $h_{ij}$. Because the Lagrangian depends on $\dot h_{ij}$ only through $K_{ij}$, the momentum is a specific densitized, trace-reversed combination of the extrinsic curvature.

where: $\pi^{ij}$ — momentum conjugate to $h_{ij}$, a tensor density of weight $1$; $K$ — mean curvature, the trace $K=h^{ij}K_{ij}$; $\sqrt{h}$ — square root of $\det h_{ij}$; $G$ — Newton's constant. Reading it: the $\sqrt h$ makes $\pi^{ij}$ a density (so $\int\pi^{ij}\dot h_{ij}\,\mathrm{d}^3x$ is coordinate-invariant), and the $-Kh^{ij}$ trace reversal is dictated by the $K^2$ term in the action. Varying the lapse gives the Hamiltonian, or scalar, constraint — the projection of the Einstein equation along the normal $n^\mu$. It is the dynamical heart of the theory: a nonlinear relation the initial data must satisfy on every slice.

where: $\mathcal{H}$ — Hamiltonian constraint density (the super-Hamiltonian); $\pi$ — trace $\pi=h_{ij}\pi^{ij}$; ${}^{(3)}R$ — Ricci scalar of the intrinsic metric $h_{ij}$. Reading it: the momentum term is the gravitational "kinetic energy" and the $-{}^{(3)}R$ term the "potential"; their sum vanishing is the geometric statement $G_{\mu\nu}n^\mu n^\nu=0$ in vacuum, or $=8\pi G\,\rho$ with matter. Varying the shift gives the momentum, or vector, constraint — the projection of the Einstein equation along the slice. It is linear in $\pi^{ij}$ and says the momentum is transverse.

where: $\mathcal{H}_i$ — momentum constraint density (the super-momentum); $D_j$ — 3-covariant derivative compatible with $h_{ij}$. Reading it: $\mathcal{H}_i$ generates spatial diffeomorphisms of the data on the slice, so it is the gravitational analogue of Gauss's law demanding transverse momentum, and it must hold before any evolution can begin. Assembling the Legendre transform, the full Hamiltonian is a linear combination of these two constraints, smeared by the lapse and shift, plus a boundary term. This is the central structural result of canonical gravity.

where: $H_{\partial}$ — Regge-Teitelboim surface term making the variation well-defined; $N,N^i$ — lapse and shift acting as Lagrange multipliers. Reading it: on any solution the integrand vanishes pointwise, so the bulk Hamiltonian is zero and the entire numerical value of $H$ comes from $H_\partial$ at the boundary. The equations of motion are Hamilton's equations for the canonical pair, generated by this same $H$. They are equivalent to the six spatial Einstein equations once the constraints hold.

where: $\dot{(\,)}$ — partial derivative with respect to coordinate time $t$; $\delta/\delta$ — functional (variational) derivative. Reading it: the first equation just re-derives $K_{ij}$ from $\pi^{ij}$; the second is the genuine evolution, feeding the intrinsic curvature and momentum back into the change of momentum. The constraints obey a closed Poisson-bracket algebra — but not an ordinary Lie algebra, because one of the "structure constants" is actually a function of the metric. This is the hypersurface-deformation (Dirac) algebra.

where: $C_\mu$ — the smeared constraints, $C_0=\int N\mathcal{H}$ and $C_i=\int N^i\mathcal{H}_i$; $\{\,,\}$ — Poisson bracket on phase space; $f^\rho$ — structure functions depending on $h_{ij}$, not constants. Reading it: the bracket of two Hamiltonian constraints returns a momentum constraint weighted by $h^{ij}$, so the algebra "knows" the geometry it acts on — the hallmark that these are the deformations of a hypersurface, not symmetries of a fixed background. Finally, the boundary term evaluated at spatial infinity with unit lapse is the ADM energy, the rigorous total energy of an isolated system. It is built from the falloff of the spatial metric alone.

where: $S_r$ — sphere of coordinate radius $r$ in the asymptotic region; $n^i$ — outward unit normal of $S_r$ within the slice; $\mathrm{d}A$ — area element on $S_r$. Reading it: only the $1/r$ Coulomb-like tail of $h_{ij}$ survives the limit, and its coefficient is the mass — the same $M$ that a distant satellite would infer from Kepler's third law.

The ADM Hamiltonian specializes and simplifies in several physically important regimes, each of which isolates one facet of the general structure.

1 · The ADM mass of a Schwarzschild black hole (fully worked)

Take the Schwarzschild geometry in isotropic coordinates, whose spatial metric is $h_{ij}=\psi^4\delta_{ij}$ with conformal factor $\psi=1+GM/2r$ (units $c=1$ with $G$ displayed, so $GM$ is the geometric mass length and $M$ the physical mass). Compute the ADM energy from the boundary integral (9), and express it in physical units for a one-solar-mass black hole.

Step 1. Expand the conformal factor at large radius: $\psi^4=(1+GM/2r)^4=1+2GM/r+O(1/r^2)$, so the spatial metric has the Coulomb-like tail $h_{ij}\approx(1+2GM/r)\delta_{ij}$. Only this $1/r$ piece survives the $r\to\infty$ limit in (9); the higher terms fall off too fast to contribute.

Step 2. Form the integrand. With $h_{ij}=(1+2GM/r)\delta_{ij}$ the trace is $h_{jj}=3+6GM/r$, so $\partial_j h_{ij}=\partial_i(2GM/r)=-2GM x_i/r^3$ and $\partial_i h_{jj}=-6GM x_i/r^3$. The combination in the integral is $\partial_j h_{ij}-\partial_i h_{jj}=\mathbf{4GM\,x_i/r^3}$.

Step 3. Contract with the outward normal $n^i=x_i/r$ and integrate over the sphere: $(\partial_j h_{ij}-\partial_i h_{jj})n^i=4GM\,x_ix_i/r^4=4GM/r^2$, and $\oint_{S_r}\mathrm{d}A=4\pi r^2$, so the surface integral equals $4GM/r^2\cdot4\pi r^2=16\pi GM$, independent of $r$.

Step 4. Divide by the prefactor: $E_{\mathrm{ADM}}=\frac{1}{16\pi G}\cdot16\pi GM=\mathbf{M}$ — the mass a distant orbit would measure via Kepler's law. Restoring units for one solar mass, the geometric length is $GM_\odot/c^2=\mathbf{1.477\ km}$ (the Schwarzschild radius is twice this, $2.95\ \mathrm{km}$), and the energy is $M_\odot c^2\approx\mathbf{1.79\times10^{47}\ J}$. The bulk Hamiltonian is zero everywhere, yet the boundary integral delivers exactly the black hole's mass.

2 · The Hamiltonian constraint is the Friedmann equation

Apply the Hamiltonian constraint (4), in its geometric form $\,{}^{(3)}R+K^2-K_{ij}K^{ij}=16\pi G\,\rho$, to a spatially flat FLRW slice $h_{ij}=a(t)^2\delta_{ij}$ with $N=1$, $N^i=0$. Show it reproduces the Friedmann equation and evaluate today's critical density.

Step 1. Compute the extrinsic curvature of the slice. With $N=1$ and no shift, $K_{ij}=\tfrac12\partial_t h_{ij}=a\dot a\,\delta_{ij}$, so the trace is $K=h^{ij}K_{ij}=a^{-2}\delta^{ij}(a\dot a\,\delta_{ij})=3\dot a/a=3H$, where $H\equiv\dot a/a$ is the Hubble rate.

Step 2. Build the two quadratic invariants. Then $K^2=9H^2$, and $K_{ij}K^{ij}=(a\dot a)(\dot a/a^3)\,\delta_{ij}\delta^{ij}=3\dot a^2/a^2=3H^2$, so $K^2-K_{ij}K^{ij}=9H^2-3H^2=\mathbf{6H^2}$. The intrinsic curvature of a flat slice is ${}^{(3)}R=0$.

Step 3. Insert into the constraint: $0+6H^2=16\pi G\,\rho$, i.e. $H^2=\tfrac{8\pi G}{3}\rho$. Restoring spatial curvature ${}^{(3)}R=6k/a^2$ gives the full Friedmann equation $H^2=\tfrac{8\pi G}{3}\rho-\tfrac{k}{a^2}$. The Hamiltonian constraint, evaluated on a homogeneous slice, is exactly the Friedmann equation — cosmology is ADM in one line.

Step 4. Put numbers in. Setting $k=0$ and $\rho=\rho_c$ defines the critical density $\rho_c=3H_0^2/8\pi G$. With $H_0=67.4\ \mathrm{km\,s^{-1}\,Mpc^{-1}}=2.18\times10^{-18}\ \mathrm{s^{-1}}$ and $G=6.674\times10^{-11}\ \mathrm{m^3\,kg^{-1}\,s^{-2}}$, one finds $\rho_c\approx\mathbf{8.5\times10^{-27}\ kg\,m^{-3}}$ — about five hydrogen atoms per cubic metre. A cosmological constraint equation and a single small number both drop straight out of $\mathcal{H}=0$.

The ADM formulation is a rare case where a whole formalism is named for the three people who built it in a two-year burst, standing on ground Dirac had just cleared. The portrait, though, is best hung on Charles Misner, the youngest of the trio and the one who carried its ideas furthest — into geometrodynamics, into the textbook that taught a generation, and into the numerical relativity that finally made the equations compute.

Texts. Misner, Thorne & Wheeler, Gravitation, Ch. 21 (the original textbook account) · Wald, General Relativity, Ch. 10 and Appendix E · Poisson, A Relativist's Toolkit, Ch. 4 · Gourgoulhon, 3+1 Formalism in General Relativity (Lecture Notes in Physics 846) · Baumgarte & Shapiro, Numerical Relativity, Ch. 2 · Arnowitt, Deser & Misner, "The Dynamics of General Relativity" (1962; reprinted as arXiv:gr-qc/0405109). The key checkable results in this article are: (1) the determinant relation $\sqrt{-g}=N\sqrt h$ from the ADM line element; (2) the Gauss-Codazzi decomposition of $R$; (3) the Legendre transform giving $H=\int(N\mathcal{H}+N^i\mathcal{H}_i)$; (4) the Schwarzschild isotropic evaluation $E_{\mathrm{ADM}}=M$, obtained as $16\pi GM/16\pi G$; (5) the FLRW reduction $6H^2=16\pi G\rho\Rightarrow H^2=\tfrac{8\pi G}{3}\rho$ and $\rho_c=3H_0^2/8\pi G\approx8.5\times10^{-27}\ \mathrm{kg\,m^{-3}}$ at $H_0=67.4\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$. Geometrized factors of $16\pi G$ follow the $G=c=1$-restorable conventions of Wald and Poisson; no nonstandard numbers are introduced.

Prerequisites. ADM 3+1 Decomposition of Spacetime · Gauss-Codazzi-Mainardi Equations · Hamiltonian and Momentum Constraints. Leads to. The Cauchy Problem in General Relativity · ADM Mass, Momentum and Angular Momentum · The BSSN Formulation · The Conformal Method for Constraint Solving.

Ask the tutor: Want me to walk through how the ADM constraints become the initial data a black-hole merger simulation actually starts from, or how the same $\mathcal{H}=0$ becomes the Wheeler-DeWitt equation?