Conformal Method for Constraint Solving

Cast general relativity as an initial-value problem and the Einstein equations split into two kinds. Six are evolution equations that march the spatial metric $\gamma_{ij}$ and the extrinsic curvature $K_{ij}$ forward in time; the other four are constraints — one scalar Hamiltonian and a three-vector momentum equation — that the data must already satisfy on the very first slice. The Bianchi identities guarantee that data satisfying the constraints once satisfy them forever, so the whole problem of what counts as a physically admissible snapshot of the gravitational field reduces to those four equations. The trouble: four equations for the twelve components of $(\gamma_{ij},K_{ij})$ neither determine the data nor isolate the two true wave degrees of freedom, and as a coupled system they are of mixed, ill-posed character. André Lichnerowicz in 1944 and James York across 1971–73 turned this awkward system into a well-posed elliptic boundary-value problem by exploiting the one piece of gravitational freedom that carries no dynamics: the local scale of space. Rescale the metric by a positive conformal factor, $\gamma_{ij}=\psi^{4}\tilde\gamma_{ij}$ , split the traceless extrinsic curvature into a transverse-traceless part and a longitudinal part built from a vector potential, and the constraints reorganize themselves. The momentum constraint becomes a linear elliptic equation for the vector; the Hamiltonian constraint becomes a single quasilinear elliptic equation for $\psi$ — the Lichnerowicz equation. The conformal geometry, the wave tensor, the mean curvature and the matter are freely chosen; the scale and the longitudinal momentum are solved for. This is the engine underneath every binary-black-hole and binary-neutron-star simulation run today.

Key takeaways

The four constraints of general relativity — one Hamiltonian, three momentum — must hold on every initial slice; they are underdetermined, four equations for the twelve fields $(\gamma_{ij},K_{ij})$ .
The Lichnerowicz–York method rescales $\gamma_{ij}=\psi^{4}\tilde\gamma_{ij}$ and splits the data into freely specifiable conformal pieces and constrained pieces fixed by elliptic PDEs.
The momentum constraint becomes a linear vector-Laplacian equation for a potential $W^{i}$ ; the Hamiltonian constraint becomes the scalar quasilinear Lichnerowicz equation for $\psi$ .
On a constant-mean-curvature (CMC) slice the two constraints decouple, and the Yamabe classification settles existence and uniqueness for closed manifolds.
Explicit conformally-flat solutions — Brill–Lindquist and Bowen–York — give black-hole data with prescribed mass, momentum and spin, and seed the puncture and conformal-thin-sandwich methods.
Key idea: the conformal factor and a longitudinal vector are the 'constrained' degrees of freedom — solving the momentum constraint for the vector and the Hamiltonian constraint for $\psi$ projects arbitrary free conformal data onto the physical constraint surface, turning four mixed PDEs into a coupled elliptic system with clean existence theory.

The core insight

The conformal factor and a longitudinal vector field are the constrained degrees of freedom: solving the momentum constraint for the vector and the Hamiltonian constraint for the conformal factor (the Lichnerowicz equation) projects arbitrary free conformal data onto the physical constraint surface, converting four mixed PDEs into a coupled elliptic system with good existence and uniqueness theory.

Where this lives today

Every gravitational-wave detection begins, computationally, with a solution of the conformal constraints. The moving-puncture revolution of 2005 — the breakthrough that first evolved a binary-black-hole merger stably — starts from Brandt–Brügmann puncture data, where $\psi=1+\sum_a m_a/(2\lvert x-x_a\rvert)+u$ absorbs the singular pieces analytically and the Lichnerowicz equation is solved only for the bounded remainder $u$ ; superposed Bowen–York terms give each hole its momentum and spin. For the high-accuracy templates in the LIGO–Virgo–KAGRA catalogues, the SpEC and SpECTRE codes solve the conformal-thin-sandwich system on a superposed-Kerr–Schild background instead, giving low-eccentricity, low-junk data for inspirals of tens of orbits. Binary-neutron-star data — seeds for GW170817-type modelling — couple the constraints to a relativistic Euler/Bernoulli integral for fluid equilibria. Underpinning all of it: the Yamabe classification and weighted-Sobolev existence theory guarantee the data actually exist.

The physical picture

A spatial slice carries two kinds of information at once: its shape — the conformal class of the metric — and its size, the local scale. York's insight is that the physical wave content of gravity lives entirely in the shape, together with a transverse-traceless piece of the extrinsic curvature that plays the role of the field's momentum. The size is not yours to choose: the Hamiltonian constraint pins it down by demanding that the energy in curvature and extrinsic curvature balance the matter energy density, which is why that constraint collapses to a single equation for the scalar $\psi$ . The momentum constraint is the gravitational cousin of $\nabla\!\cdot\!\mathbf{E}=0$ — it singles out a longitudinal-versus-transverse split of the traceless extrinsic curvature just as the Helmholtz theorem splits a vector field, the transverse-traceless part free radiation data and the longitudinal part forced by the momentum sources.

Figure

Counting the degrees of freedom of an initial slice: the twelve components of the pair spatial-metric and extrinsic-curvature split into eight freely specified conformal fields plus four constrained fields fixed by the elliptic constraints, so 12 = 8 free + 4 constrained

The conformal weights that make this work look arbitrary but are forced. The fourth power $\gamma_{ij}=\psi^{4}\tilde\gamma_{ij}$ turns the scalar-curvature transformation into a Laplacian on $\psi$ ; the weight $A^{ij}=\psi^{-10}\tilde A^{ij}$ makes the momentum-constraint divergence carry no derivatives of $\psi$ , so it stops feeding the Hamiltonian one. On a CMC slice ( $\partial_i K=0$ ; maximal means $K=0$ ) the last coupling term dies and the elliptic system triangulates: solve the linear vector equation for $W^{i}$ , build the traceless extrinsic curvature, then solve the Lichnerowicz equation for $\psi$ . The simplest nontrivial solution is a single black hole in isotropic coordinates, where $\psi$ is a bare harmonic function whose far-field $1/r$ coefficient reads off the total mass.

Figure

Conformal factor of a single Schwarzschild black hole in isotropic coordinates, ψ = 1 + M/2r with M = 1: it is harmonic away from the puncture, diverges at r → 0, and its 1/r tail fixes the ADM mass at exactly M = 1, with an Einstein–Rosen throat at r = M/2 where ψ = 2

The core equations

The starting point is the pair of constraints in raw geometric form — the ADM restriction of the Einstein equations to a single slice, mixing intrinsic and extrinsic curvature with the matter seen by the normal observer.

\mathcal{H}\equiv {}^{(3)}\!R + K^2 - K_{ij}K^{ij} = 16\pi\rho,\qquad \mathcal{M}^i\equiv D_j\!\left(K^{ij}-\gamma^{ij}K\right)=8\pi j^i \qquad (1)

where: ${}^{(3)}\!R$ — Ricci scalar of $\gamma_{ij}$ ; $K_{ij}$ — extrinsic curvature, $K=\gamma^{ij}K_{ij}$ its trace; $D_j$ — covariant derivative of $\gamma_{ij}$ ; $\rho,\,j^i$ — energy and momentum density of the normal observer, $\rho=n_\mu n_\nu T^{\mu\nu}$ . Reading it: the Hamiltonian constraint balances curvature energy against matter energy; the momentum constraint ties the divergence of $K_{ij}$ to matter momentum flow. The first move is the conformal rescaling. In three dimensions the exponent four is not a choice — it is the unique weight rendering the curvature transformation an elliptic operator on $\psi$ .

\gamma_{ij}=\psi^4\,\tilde\gamma_{ij},\qquad \psi>0 \qquad (2)

where: $\psi$ — conformal factor, the constrained scalar degree of freedom; $\tilde\gamma_{ij}$ — freely specifiable conformal background metric. Positivity of $\psi$ keeps the rescaled metric a genuine Riemannian metric everywhere. Next, split the extrinsic curvature into its trace $K$ and traceless part $A_{ij}$ , and give the traceless part the conformal weight that decouples the momentum constraint from $\psi$ .

K_{ij}=A_{ij}+\tfrac13\gamma_{ij}K,\qquad A^{ij}=\psi^{-10}\tilde A^{ij} \qquad (3)

where: $A_{ij}$ — tracefree extrinsic curvature, $\gamma^{ij}A_{ij}=0$ ; $K$ — mean curvature, freely specifiable; $\tilde A^{ij}$ — conformal tracefree tensor. The weight $-10$ is the unique one making $D_jA^{ij}$ a conformal divergence with no stray factors of $\psi$ . York's orthogonal decomposition splits the conformal tracefree tensor into a divergence-free transverse-traceless part and a longitudinal part from a vector potential — the gravitational Helmholtz theorem.

\tilde A^{ij}=\tilde A^{ij}_{\rm TT}+(\tilde L W)^{ij},\qquad (\tilde L W)^{ij}=\tilde D^i W^j+\tilde D^j W^i-\tfrac23\tilde\gamma^{ij}\tilde D_k W^k \qquad (4)

where: $\tilde A^{ij}_{\rm TT}$ — transverse-traceless part, $\tilde D_j\tilde A^{ij}_{\rm TT}=0$ , the free gravitational-wave data; $(\tilde L W)^{ij}$ — conformal Killing (longitudinal) operator on $W^i$ ; $W^i$ — vector potential fixed by the momentum constraint. If $\tilde L W=0$ then $W$ is a conformal Killing vector, the kernel of the operator. Feeding this into the momentum constraint yields a linear elliptic equation — the conformal vector Laplacian — sourced by the mean-curvature gradient and the matter momentum.

\tilde\Delta_L W^i\equiv \tilde D_j(\tilde L W)^{ij}=\tfrac23\psi^{6}\,\tilde D^i K+8\pi\,\psi^{10}\,\tilde j^{\,i} \qquad (5)

where: $\tilde\Delta_L$ — conformal vector Laplacian, an elliptic operator; $\tilde D^i K$ — gradient of the mean curvature, vanishing on a CMC slice and decoupling the system; $\tilde j^{\,i}$ — conformally rescaled momentum density. On a CMC vacuum slice the source is zero, so $\tilde A^{ij}=\tilde A^{ij}_{\rm TT}$ . The Hamiltonian constraint becomes the Lichnerowicz–York equation — one quasilinear scalar elliptic equation for $\psi$ , the equation the whole method exists to produce, with signs subtle enough to derive in full below.

\tilde\Delta\psi-\tfrac18\tilde R\,\psi+\tfrac18 \tilde A_{ij}\tilde A^{ij}\,\psi^{-7}-\tfrac{1}{12}K^2\,\psi^{5}=-2\pi\tilde\rho\,\psi^{-3} \qquad (6)

where: $\tilde\Delta=\tilde\gamma^{ij}\tilde D_i\tilde D_j$ — conformal Laplacian; $\tilde R$ — scalar curvature of $\tilde\gamma_{ij}$ ; $\tilde A_{ij}\tilde A^{ij}$ — squared tracefree tensor, the shear/momentum source; $\tilde\rho=\psi^{8}\rho$ — rescaled energy density. Reading it: the positive $\psi^{-7}$ term and the $K^2\psi^5$ term push $\psi$ in opposite directions, and their balance fixes the local scale. For quasi-equilibrium binaries one prefers the conformal-thin-sandwich reformulation, which trades the free tracefree tensor for a free time derivative of the conformal metric.

\tilde A^{ij}=\frac{1}{2\tilde N}\Big[(\tilde L W)^{ij}-\partial_t\tilde\gamma^{ij}\Big] \qquad (7)

where: $\tilde N$ — conformal lapse, $N=\psi^{6}\tilde N$ ; $\partial_t\tilde\gamma^{ij}$ — freely specified "velocity" data. Setting $\partial_t\tilde\gamma^{ij}=0$ selects an instantaneously stationary geometry, which is why CTS is the tool of choice for low-eccentricity inspiral data. Finally, once $\psi$ is known, the total energy is read straight off its asymptotic falloff.

M_{\rm ADM}=-\frac{1}{2\pi}\oint_{\infty}\tilde D^i\psi\,\mathrm{d}\tilde S_i \quad\big(\text{conformally flat, }\psi\to 1+\tfrac{M}{2r}\big) \qquad (8)

where: $M_{\rm ADM}$ — total energy measured at spatial infinity; $\mathrm{d}\tilde S_i$ — surface element of a large coordinate sphere in the conformal metric. The conformal factor's monopole moment is the mass.

Derivation AReducing the Hamiltonian constraint to the Lichnerowicz equation

The goal is to take the raw Hamiltonian constraint ${}^{(3)}R+K^2-K_{ij}K^{ij}=16\pi\rho$ and, by inserting the conformal rescaling and the trace split, collapse it into the single quasilinear scalar equation (6) for the conformal factor. Every sign and power matters here — two published forms of this equation differ and only one solves correctly — so we track each factor of $\psi$ explicitly and cross-check against the Bowen–York limit and the Baumgarte–Shapiro reference form. Throughout we work in three dimensions with $\gamma_{ij}=\psi^{4}\tilde\gamma_{ij}$ .

Step 1. Transform the scalar curvature. Under $\gamma_{ij}=\psi^{4}\tilde\gamma_{ij}$ the three-dimensional Ricci scalar obeys ${}^{(3)}R=\psi^{-4}\tilde R-8\psi^{-5}\tilde\Delta\psi$ . Why: the fourth power makes the conformal cross-term appear linearly as $\tilde\Delta\psi$ ; any other exponent leaves a nonlinear $(\tilde D\psi)^2$ term and spoils ellipticity.

Step 2. Split the extrinsic-curvature scalars. With $K_{ij}=A_{ij}+\tfrac13\gamma_{ij}K$ and $\gamma^{ij}A_{ij}=0$ , expand $K_{ij}K^{ij}=A_{ij}A^{ij}+\tfrac13K^2$ , so $K^2-K_{ij}K^{ij}=\tfrac23K^2-A_{ij}A^{ij}$ . Why: isolating the trace $K$ from the tracefree $A$ lets $K$ become free slicing data while $A$ carries the momentum.

Step 3. Insert the conformal weight of $A$ . Lowering indices with $\gamma=\psi^{4}\tilde\gamma$ gives $A_{ij}=\psi^{-2}\tilde A_{ij}$ , so with $A^{ij}=\psi^{-10}\tilde A^{ij}$ the scalar is $A_{ij}A^{ij}=\psi^{-12}\tilde A_{ij}\tilde A^{ij}$ . Why: the net weight $-12$ produces the $\psi^{-7}$ nonlinearity, the hallmark of the Lichnerowicz equation.

Step 4. Assemble the constraint. Substituting Steps 1–3 into ${}^{(3)}R+\tfrac23K^2-A_{ij}A^{ij}=16\pi\rho$ gives $\psi^{-4}\tilde R-8\psi^{-5}\tilde\Delta\psi+\tfrac23K^2-\psi^{-12}\tilde A_{ij}\tilde A^{ij}=16\pi\rho$ . Why: this is the full constraint in mixed variables, before clearing the negative powers of $\psi$ .

Step 5. Multiply through by $-\tfrac18\psi^{5}$ and set $\tilde\rho=\psi^{8}\rho$ . The factors combine to $\tilde\Delta\psi-\tfrac18\tilde R\,\psi+\tfrac18\tilde A_{ij}\tilde A^{ij}\,\psi^{-7}-\tfrac{1}{12}K^2\,\psi^{5}=-2\pi\tilde\rho\,\psi^{-3}$ — equation (6), the exponents $-7,+5,-3$ coming from $-12+5$ , $0+5$ , $-8+5$ . Why: these signs — positive $\psi^{-7}$ , negative $K^2\psi^5$ , negative matter — are the physically correct ones, reducing in the flat maximal case to $\nabla^2\psi=-\tfrac18\sigma^2\psi^{-7}$ , matching Bowen–York exactly. *Verified: reproduces Baumgarte and Shapiro eq. (3.110) and Wald ch. 10; the flat-maximal reduction $\nabla^2\psi=-\tfrac18\sigma^2\psi^{-7}$ checked against Bowen–York.*

Go deeper · Derivation B · 1 minThe York decomposition and the linear momentum constraint

Derivation BThe York decomposition and the linear momentum constraint

The aim is to show why the momentum constraint, nonlinear in raw form, collapses to a single linear elliptic equation (5) for a vector potential once the conformal weight $A^{ij}=\psi^{-10}\tilde A^{ij}$ and York's transverse–longitudinal split are imposed. The payoff is large: a linear equation with a self-adjoint elliptic operator has textbook existence theory, and on a CMC slice it decouples from the scalar equation, so the two constraints solve one after the other rather than as a coupled tangle.

Step 1. Rescale the momentum constraint. Insert $K^{ij}=\psi^{-10}\tilde A^{ij}+\tfrac13\psi^{-4}\tilde\gamma^{ij}K$ into $D_j(K^{ij}-\gamma^{ij}K)=8\pi j^{i}$ ; the weight- $(-10)$ tensor satisfies $D_j(\psi^{-10}\tilde A^{ij})=\psi^{-10}\tilde D_j\tilde A^{ij}$ , with no $\psi$ -derivatives surviving. Why: this is the entire reason the weight is $-10$ — any other weight leaves $\partial_j\psi$ terms that recouple the constraint to $\psi$ .

Step 2. Handle the trace term. The remaining piece gives $\tfrac23\psi^{6}\tilde D^i K$ plus the rescaled matter momentum $8\pi\psi^{10}\tilde j^{i}$ , so $\tilde D_j\tilde A^{ij}=\tfrac23\psi^{6}\tilde D^i K+8\pi\psi^{10}\tilde j^{i}$ . Why: the mean-curvature gradient is the sole term still carrying $\psi$ , so $\partial_iK=0$ (CMC) makes the equation independent of the scalar.

Step 3. Apply the transverse–longitudinal split. With $\tilde A^{ij}=\tilde A^{ij}_{\rm TT}+(\tilde L W)^{ij}$ and $\tilde D_j\tilde A^{ij}_{\rm TT}=0$ , only the longitudinal piece survives the divergence: $\tilde D_j\tilde A^{ij}=\tilde\Delta_L W^i$ . Why: the transverse-traceless part is divergence-free by construction — free radiation data, unconstrained by the momentum equation.

Step 4. Read off the operator. The result $\tilde\Delta_L W^i=\tfrac23\psi^{6}\tilde D^i K+8\pi\psi^{10}\tilde j^{i}$ is linear elliptic, its principal part $\tilde\Delta_L$ self-adjoint and non-negative with kernel the conformal Killing vectors. Why: self-adjoint ellipticity gives Fredholm existence — a solution exists whenever the source is orthogonal to that kernel, automatic for physical momentum data. *Verified: matches York (1973) and the CTT decomposition in Cook's Living Review; the operator's conformal-Killing kernel is the standard integrability condition.*

Go deeper · Derivation C · 1 minBowen–York data and its ADM linear momentum

Derivation CBowen–York data and its ADM linear momentum

The goal is to exhibit the most-used explicit solution of the conformal method — the Bowen–York extrinsic curvature — and verify by direct computation that it solves the flat momentum constraint, is trace-free, and carries exactly the momentum $P^i$ appearing as its parameter. This is the building block of puncture data: superposing several Bowen–York pieces gives each hole a prescribed momentum and spin, which the Lichnerowicz equation then dresses into constraint-satisfying data. Take $\tilde\gamma_{ij}=\delta_{ij}$ , $K=0$ , vacuum, and $n^i=x^i/r$ the outward unit radial vector.

Step 1. Test the constraint. With $\tilde A^{ij}=\tfrac{3}{2r^2}\big(P^i n^j+P^j n^i-(\delta^{ij}-n^i n^j)P_k n^k\big)$ , the flat maximal momentum constraint is $\partial_j\tilde A^{ij}=0$ ; direct differentiation with $\partial_j n^i=(\delta^{ij}-n^i n^j)/r$ shows every term cancels for $r>0$ . Why: a closed-form solution means no vector equation need be solved numerically for the momentum sector.

Step 2. Confirm trace-freeness. Contracting with $\delta_{ij}$ , the first two terms give $2P_k n^k$ and the third $-2P_kn^k$ , so $\delta_{ij}\tilde A^{ij}=0$ . Why: trace-freeness is required for $\tilde A^{ij}$ to be a legitimate tensor feeding the Lichnerowicz equation.

Step 3. Compute the source. The squared tensor is $\tilde A_{ij}\tilde A^{ij}=\tfrac{9}{2r^4}\big(P^2+2(P_k n^k)^2\big)=\tfrac{9P^2}{2r^4}\big(1+2\cos^2\theta\big)$ , falling as $r^{-4}$ . Why: this is the $\sigma^2$ sourcing $\nabla^2\psi=-\tfrac18\sigma^2\psi^{-7}$ , whose bounded solution near the puncture regularizes the data.

Step 4. Extract the ADM momentum. Evaluating $P^i_{\rm ADM}=\tfrac{1}{8\pi}\oint_\infty(K^{ij}-\gamma^{ij}K)\,\mathrm{d}S_j$ over a large sphere returns exactly the parameter $P^i$ . Why: the abstract parameter is a measurable, gauge-invariant momentum, which is why numerical relativists dial it directly. *Verified: Bowen and York, Phys. Rev. D 21, 2047 (1980); divergence-free, trace-free, $\sigma^2=\tfrac{9P^2}{2r^4}(1+2\cos^2\theta)$ , and the ADM integral reproduced by direct symbolic and numerical computation.*

Limiting cases

The conformal system collapses to a handful of classic special cases, each a workhorse of numerical relativity.

Case	Result	Meaning
Maximal / CMC slice	$K=0\Rightarrow\tilde D_j\tilde A^{ij}=0,\ \ \tilde\Delta\psi-\tfrac18\tilde R\,\psi+\tfrac18\tilde A_{ij}\tilde A^{ij}\psi^{-7}=0$	Constraints decouple in vacuum: solve momentum first, then Lichnerowicz for $\psi$ .
Conformally flat, time-symmetric	$\tilde\gamma_{ij}=\delta_{ij},\ K_{ij}=0\Rightarrow\nabla^2\psi=0$	Hamiltonian constraint is the flat Laplace equation; Brill–Lindquist data $\psi=1+\sum_a m_a/(2\lvert x-x_a\rvert)$ .
Bowen–York momentum data	$\tilde A^{ij}=\tfrac{3}{2r^2}\big(P^i n^j+P^j n^i-(\delta^{ij}-n^i n^j)P_k n^k\big)$	Analytic conformally-flat solution of the flat momentum constraint carrying $P^i$ ; a spin analogue uses $S^i$ .
Weak field / linearized	$\psi=1+\tfrac{\phi}{2},\ \ \nabla^2\phi\simeq-\tfrac14\big(\delta\tilde R+\tilde A_{ij}\tilde A^{ij}\big)$	A Poisson equation sourced by matter-plus-wave energy; ties $\psi$ to the Newtonian potential and ADM mass.

Common trapThe conformal weights are just a convenient rescaling you can pick freely

They are not free — each weight is forced, and a naive one silently recouples the constraints. The metric weight $4/(n-2)$ is $4$ in three dimensions precisely so the scalar curvature transforms into a Laplacian on $\psi$ ; in another dimension the exponent changes. The tracefree weight must be exactly $A^{ij}=\psi^{-10}\tilde A^{ij}$ so the physical divergence $D_jA^{ij}=\psi^{-10}\tilde D_j\tilde A^{ij}$ carries no derivatives of $\psi$ — any other choice leaves $\partial_j\psi$ terms that weld the momentum constraint back to the Hamiltonian one. And clean decoupling demands constant mean curvature, $\partial_iK=0$ ; for non-CMC data the system stays coupled and existence and uniqueness become delicate, with known counterexamples. Treating the method as "just multiply by some power of $\psi$ " is the fastest way to write a system that looks elliptic but does not decouple.

Worked examples

1 · Boosted single black hole — Bowen–York momentum data (fully worked)

Give a single black hole linear momentum $P^i=(P,0,0)$ on a flat, maximal, vacuum slice. Confirm the Bowen–York extrinsic curvature is admissible, and find the momentum source $\sigma^2$ driving the Lichnerowicz equation, on the axis and in the equatorial plane at radius $r$ .

Step 1. Admissibility. On $\tilde\gamma_{ij}=\delta_{ij}$ with $K=0$ the momentum constraint is $\partial_j\tilde A^{ij}=0$ ; differentiating the ansatz with $\partial_j n^i=(\delta^{ij}-n^in^j)/r$ confirms it vanishes for $r>0$ , and $\delta_{ij}\tilde A^{ij}=0$ . The data is divergence-free and trace-free.

Step 2. Momentum source. From Derivation C, $\sigma^2=\tfrac{9P^2}{2r^4}(1+2\cos^2\theta)$ with $\theta$ from the momentum axis. On the axis ( $\cos^2\theta=1$ ): $\sigma^2=\tfrac{27P^2}{2r^4}$ ; in the equatorial plane ( $\cos^2\theta=0$ ): $\sigma^2=\tfrac{9P^2}{2r^4}$ . The source is three times larger along the motion than across it.

Step 3. The scalar solve. With $\tilde R=0$ the maximal Lichnerowicz equation is $\nabla^2\psi=-\tfrac18\sigma^2\psi^{-7}$ ; near the puncture the singular piece $\psi\sim1+m/2r$ dominates and one solves for a bounded correction $u$ . The boost raises the ADM energy above the bare mass, $E_{\rm ADM}\approx\sqrt{M_{\rm irr}^2+P^2}$ . The boost lives entirely in $\tilde A^{ij}$ while $\psi$ adjusts to satisfy the constraint.

2 · A single Schwarzschild hole — mass from the conformal factor

Take conformally-flat, time-symmetric vacuum data ( $\tilde\gamma_{ij}=\delta_{ij}$ , $K_{ij}=0$ ). Find $\psi$ for one black hole and read off its ADM mass; locate the throat.

Step 1. The equation. With $\tilde R=0$ , $\tilde A_{ij}=0$ , $K=0$ the Lichnerowicz equation is the flat Laplace equation $\nabla^2\psi=0$ ; the spherically symmetric single-pole solution regular at infinity is $\psi=1+M/2r$ . This is isotropic-coordinate Schwarzschild.

Step 2. The mass. With $\tilde D^i\psi\to-\tfrac{M}{2r^2}n^i$ and $\mathrm{d}\tilde S_i=n_i r^2\,\mathrm{d}\Omega$ , equation (8) gives $M_{\rm ADM}=-\tfrac{1}{2\pi}(-\tfrac{M}{2})(4\pi)=M$ . The $1/r$ coefficient of $\psi$ is exactly the mass.

Step 3. The throat. The areal radius $R=\psi^2 r=r(1+M/2r)^2$ has $\mathrm{d}R/\mathrm{d}r=0$ at $r=M/2$ , where $\psi=2$ and $R=2M$ — the minimal surface (Einstein–Rosen throat) joining two asymptotically flat sheets ( $r\to\infty$ and $r\to0$ ). The horizon sits at $r=M/2$ , $\psi=2$ .

Physics of the portrait

The conformal method is the achievement of one mathematically-minded relativist who spent a career turning the initial-value problem of general relativity from a philosophical worry into a computable elliptic system.

Origins · 1 minJames York and how this was discovered

James York

1939–2009 · the initial-value problem and numerical relativity

James Wesley York Jr. reshaped how general relativity begins. Building on André Lichnerowicz's wartime insight that the Hamiltonian constraint becomes elliptic under a conformal rescaling, York in a sequence of papers from 1971 to 1973 supplied the missing structural idea: the radiative degrees of freedom of the field are carried by the conformal class of the three-metric together with a transverse-traceless tensor, while the conformal factor and a longitudinal vector are precisely what the constraints determine. His orthogonal decomposition of symmetric tracefree tensors — the gravitational analogue of the Helmholtz theorem — cleanly separated free data from constrained data and turned the four coupled constraints into a well-posed elliptic problem. He went on to formulate the conformal-thin-sandwich system, which underlies today's quasi-equilibrium binary initial data. The through-line of his work is a conviction that the right variables make hard problems easy: choose the conformal factor and the longitudinal momentum as unknowns, and the initial-value problem practically solves itself.

History

1944

— Lichnerowicz recasts the Hamiltonian constraint as an elliptic equation for a conformal factor

1971

— York identifies the conformal class plus a transverse-traceless tensor as the true gravitational data

1972

— York publishes the orthogonal (longitudinal–transverse) decomposition of symmetric tracefree tensors

1973

— York gives the full conformal transformation of the constraints and the Lichnerowicz–York equation

1980

— Bowen and York construct explicit conformally-flat data carrying linear and angular momentum

1995

— Isenberg completes the Yamabe classification of CMC solutions on closed manifolds

Going further

Texts. Misner, Thorne and Wheeler, Gravitation (1973), ch. 21 · Wald, General Relativity (1984), ch. 10 · York, "Conformally invariant orthogonal decomposition of symmetric tensors...", J. Math. Phys. 14, 456 (1973) · Lichnerowicz, J. Math. Pures Appl. 23, 37 (1944) · Cook, "Initial Data for Numerical Relativity", Living Rev. Relativity 3, 5 (2000) · Baumgarte and Shapiro, Numerical Relativity (2010), ch. 3 · Choquet-Bruhat, General Relativity and the Einstein Equations (2009), ch. VII · Bowen and York, Phys. Rev. D 21, 2047 (1980).

Provenance. Constraints and conformal transformations checked against Wald ch. 10, MTW ch. 21, and Baumgarte–Shapiro ch. 3. The Lichnerowicz–York equation (6) was re-derived from scratch and its signs corrected to $+\tfrac18\tilde A^2\psi^{-7}$ , $-\tfrac1{12}K^2\psi^5$ , $-2\pi\tilde\rho\psi^{-3}$ , matching Baumgarte–Shapiro eq. (3.110) and the flat-maximal reduction $\nabla^2\psi=-\tfrac18\sigma^2\psi^{-7}$ . The weights ( $\psi^4$ metric, $\psi^{-10}$ tracefree $A$ ) and the longitudinal–TT split follow York (1973). Bowen–York's divergence-free and trace-free properties, its source $\sigma^2=\tfrac{9P^2}{2r^4}(1+2\cos^2\theta)$ , and the ADM integrals were verified by direct symbolic and numerical computation; CMC existence uses Isenberg (1995) and Choquet-Bruhat (2009).

Builds on: ADM 3+1 Decomposition of Spacetime · Hamiltonian and Momentum Constraints · Gauss–Codazzi–Mainardi Equations · Initial-Value (Cauchy) Problem in General Relativity. Leads to: BSSN Formulation · Black-Hole Excision and Puncture Methods · Numerical Relativity and Binary-Black-Hole Mergers · Apparent Horizons and Horizon Finders · ADM Mass, Momentum and Angular Momentum.

Ask the tutor: Want to walk through solving the Lichnerowicz equation for a specific configuration — a boosted puncture, a Brill–Lindquist pair, or a CMC slice in a nonzero Yamabe class — and see exactly where the free data ends and the elliptic solve begins?

Why must the initial data satisfy constraints at all — is that not extra work?

Because the Einstein equations are not all evolution equations. Projecting them normal to a slice yields four combinations with no second time derivatives — the Hamiltonian and momentum constraints — conditions on the data at a single instant, not laws of motion. They are the analogue of $\nabla\!\cdot\!\mathbf{E}=\rho/\varepsilon_0$ , which must also hold on the initial slice. The Bianchi identities guarantee that if the constraints hold initially the evolution preserves them exactly, so the work is done once, up front. Skipping it means evolving data that is not a valid solution of general relativity, and the violation typically grows and wrecks the simulation.

What is actually free to choose, and what gets solved for?

Free: the conformal three-metric $\tilde\gamma_{ij}$ (shape, five functions before spatial gauge), a transverse-traceless tensor $\tilde A^{ij}_{\rm TT}$ (two, the wave degrees of freedom), the mean curvature $K$ (one, a slicing choice), and any matter. Solved for: the conformal factor $\psi$ (from the Hamiltonian constraint) and the longitudinal vector $W^i$ (three, from the momentum constraint). The bookkeeping closes as $12=8+4$ : twelve components of $(\gamma_{ij},K_{ij})$ split into eight free and four constrained, the two physical polarizations of gravity hidden inside the conformal class and the transverse-traceless tensor.

Why does constant mean curvature make everything easier?

Because it kills the one term that couples the two constraints. In the conformal momentum constraint the source $\tfrac23\psi^{6}\tilde D^i K$ vanishes when $K$ is constant, so in vacuum that constraint becomes a $\psi$ -independent linear equation for $W^i$ . You solve it first, build the tracefree extrinsic curvature, and only then face the scalar Lichnerowicz equation — a triangular system instead of a coupled nonlinear one. CMC slices also have clean existence theory via the Yamabe classification. For non-CMC data the coupling stays and both existence and uniqueness can fail, which is why CMC is the gauge of convenience.

Is conformal flatness a physical assumption or just convenient?

Convenient, and sometimes too convenient. Setting $\tilde\gamma_{ij}=\delta_{ij}$ makes the Hamiltonian constraint tractable and yields the Brill–Lindquist and Bowen–York solutions. But a conformally-flat slice cannot exactly represent a spinning Kerr black hole — Kerr has no conformally-flat maximal slice — so conformally-flat binary data carries spurious "junk" radiation that must burn off before a clean waveform emerges. High-accuracy codes therefore build $\tilde\gamma_{ij}$ from superposed Kerr–Schild metrics instead. It is a fine starting approximation and a superb teaching case, but a modelling choice with a measurable cost, not a feature of the physics.

How does the method handle black holes without hitting the singularity?

Two strategies, both descendants of the conformal method. Excision cuts a hole out of the domain inside the horizon, where no information can escape to the exterior, and imposes boundary conditions on the cut surface. The puncture method instead keeps the singular $1/r$ behaviour of the conformal factor analytically — $\psi=1+\sum_a m_a/(2\lvert x-x_a\rvert)+u$ — and solves the Lichnerowicz equation only for the smooth remainder $u$ , so the code never differentiates the singular part. Isotropic coordinates hide the singularity behind an Einstein–Rosen throat. Moving-puncture gauges let those punctures drift across the grid, which is how the 2005 breakthrough worked.

What guarantees a solution of the Lichnerowicz equation even exists?

For closed slices with constant mean curvature, the Yamabe classification settles it completely: the conformal metric belongs to a positive, zero, or negative Yamabe class, and existence and uniqueness of a positive $\psi$ follow from that class and whether the tracefree tensor $\sigma$ vanishes. In the negative class a unique solution always exists; in the positive class one needs $\sigma\not\equiv0$ . For asymptotically Euclidean data — isolated binaries — existence and uniqueness are proved on weighted Sobolev spaces encoding the $\psi\to1$ falloff. These theorems (Choquet-Bruhat, Isenberg) let a numerical relativist trust the solver is converging to something real.