Gauss-Codazzi-Mainardi Equations

The Gauss-Codazzi-Mainardi equations are the daily tools of numerical relativity, and every gravitational-wave detection rests on them. Before LIGO-Virgo-KAGRA can predict a binary black-hole waveform, someone must build initial data — a spatial metric $\gamma_{ij}$ and extrinsic curvature $K_{ij}$ satisfying the Hamiltonian and momentum constraints to machine precision. This is the conformal method: solving the elliptic constraint equations (contracted Gauss and Codazzi) for a conformal factor and a longitudinal vector, using Bowen-York or conformal-thin-sandwich data. The result is handed to an evolution code (BSSN or generalized-harmonic), whose equations come from the Ricci projection. In the continuum the constraints are preserved by the Bianchi identity $\nabla_\mu G^{\mu\nu}=0$; in practice discretization seeds violations that BSSN and constraint-damping suppress, so constraint monitors on equations (6) and (7) are a running simulation's standard health check. The same machinery drives the Israel junction conditions for thin shells, apparent-horizon finders solving $\Theta_{(\ell)}=0$, and — in mathematical relativity — the positive-mass theorem and the Penrose inequality.

Imagine an ant on an eggshell. It can never step off, yet it can still discover its world is curved: it walks a triangle and finds the angles sum to more than 180 degrees, or carries an arrow around a loop and finds it rotated on return. That is intrinsic curvature. A geometer floating above sees what the ant cannot: the shell bends away into the third dimension, and the direction "straight out" — the unit normal — swings around as you move from the fat end to the point. The rate of that swing is the extrinsic curvature $K_{ij}$. The first figure sets the two side by side and pins the sphere to a number.

The two curvatures are not independent — that is the whole subject. For a surface in flat space the ambient curvature is zero, so the intrinsic curvature is forced to equal a fixed combination of the extrinsic curvature: bend a flat sheet without stretching and its intrinsic geometry never changes, because rolling alters $K_{ij}$ but leaves $K^2 - K_{ij}K^{ij}$ alone. In general relativity the ambient space is not flat, and the same bookkeeping becomes physics: spacetime curvature, sourced by matter, ties the geometry of a spatial slice to how that slice sits in time. The second figure turns the three projections into the machinery of the initial-value problem.

Why should only some projections be constraints? The answer is causal. A constraint is imposed at a single instant, before evolution begins, so it must contain no second time derivatives. The Gauss and Codazzi projections involve $\gamma_{ij}$, $K_{ij}$, and their spatial derivatives only; since $K_{ij}$ is essentially the metric's first time derivative, these are at most first-order in time and checkable on the slice itself. The doubly-normal Ricci projection contains $\mathcal{L}_n K_{ij}$, a second time derivative of $\gamma_{ij}$ — it does not constrain the data, it evolves it. That distinction is why relativity has a well-posed Cauchy problem: four of the ten Einstein equations are constraints from Gauss and Codazzi, the remaining six push the data forward.

Everything begins with splitting the ambient metric into a tangent piece and a normal piece. Let $\Sigma$ be a spacelike hypersurface in $(\mathcal{M},g)$ of signature $(-,+,+,+)$, with future-directed unit normal $n^\mu$, $n_\mu n^\mu = -1$. The induced metric doubles as the orthogonal projector onto $\Sigma$.

where: $g_{\mu\nu}$ — ambient spacetime metric; $n^\mu$ — unit normal, $n_\mu n^\mu=\epsilon=-1$ for a spacelike slice ($+1$ for a timelike wall); $\gamma_{\mu\nu}$ — induced (spatial) metric on $\Sigma$; $\gamma^\mu{}_\nu$ — projector, $\gamma^\mu{}_\nu n^\nu=0$, $\gamma^\mu{}_\alpha\gamma^\alpha{}_\nu=\gamma^\mu{}_\nu$. Reading: any tensor splits into parts along $n$ and parts in $\Sigma$ by hitting each index with $\gamma$ or $n$. The extrinsic curvature measures how the normal fails to be parallel-transported within $\Sigma$ — equivalently minus half the Lie derivative of the spatial metric along the normal, which is why it acts as the metric's "velocity."

where: $K_{\mu\nu}$ — extrinsic curvature, symmetric and purely tangential ($K_{\mu\nu}n^\nu=0$); $\nabla_\alpha$ — ambient Levi-Civita connection; $\mathcal{L}_n$ — Lie derivative along $n$; $K\equiv\gamma^{\mu\nu}K_{\mu\nu}$ — mean curvature. Reading: $K_{\mu\nu}$ lives on $\Sigma$ yet remembers how $\Sigma$ sits in $\mathcal{M}$; it is symmetric because the normal is hypersurface-orthogonal. The Gauss equation is the fully tangential projection: the intrinsic Riemann tensor of $\Sigma$ differs from the projected ambient one by a quadratic in $K$.

where: ${}^{(4)}R$ — ambient Riemann tensor; ${}^{(3)}R$ — intrinsic Riemann tensor of $\Sigma$ (from the connection $D$ compatible with $\gamma$); the $KK$ terms are the "bending" correction. Reading: in general the quadratic term carries $-\epsilon$, so a spacelike slice ($\epsilon=-1$) shows the $+$ sign here while a Euclidean surface ($\epsilon=+1$) flips it — the single sign separating the two settings. The Codazzi-Mainardi equation is the one-index-normal projection, equating a mixed component of the ambient Riemann tensor to the antisymmetrized covariant derivative of $K$.

where: $n^\sigma$ — the single normal-projected index; $D_\alpha$ — the intrinsic (Levi-Civita) covariant derivative of $\gamma$; $D_\beta K_{\alpha\gamma}$ — spatial gradient of $K$. Reading: first order in $K$ and free of time derivatives of $K$, so it is a genuine constraint whose contraction is a momentum-balance law. The Ricci equation is the doubly-normal projection; it contains a time derivative of $K$ and so is dynamical.

where: $N$ — lapse function (proper time per unit coordinate time along $n$); $\mathcal{L}_n K_{\alpha\gamma}$ — the normal Lie derivative carrying $\partial_t K_{ij}$; $\tfrac{1}{N}D_\alpha D_\gamma N$ — acceleration term, $a_\mu=D_\mu\ln N$. Reading: rearranged, this is the ADM evolution equation for $K_{ij}$ — the six equations pushing data forward once the constraints hold. Contracting Gauss twice and inserting Einstein's equations yields the Hamiltonian constraint.

where: ${}^{(3)}R$ — intrinsic Ricci scalar of the slice; $K$ — mean curvature; $K_{ij}K^{ij}$ — squared extrinsic curvature; $\rho\equiv T_{\mu\nu}n^\mu n^\nu$ — Eulerian energy density. Reading: this ties the intrinsic curvature of "space now" to the matter energy density and how space embeds in time; in vacuum the right side vanishes. Contracting Codazzi once and inserting Einstein's equations yields the momentum constraint.

where: $j^i\equiv -\gamma^{i}{}_\mu n_\nu T^{\mu\nu}$ — Eulerian momentum density; $D_j$ — covariant derivative of $\gamma_{ij}$; $K^{ij}-\gamma^{ij}K$ — trace-reversed extrinsic curvature. Reading: the divergence of trace-reversed $K$ balances the momentum flux — the structure York's conformal method exploits.

Each special case recovers a landmark result.

1 · The Friedmann equation as a Hamiltonian constraint (fully worked)

Slice a flat Friedmann-Lemaître-Robertson-Walker universe along constant cosmic time and show the Hamiltonian constraint (6) is exactly the Friedmann equation.

Step 1. The flat FLRW slice has $\gamma_{ij}=a(t)^2\delta_{ij}$, lapse $N=1$, zero shift. It is flat space rescaled, so ${}^{(3)}R=0$ — the slices are intrinsically flat even though spacetime is curved; all the curvature is in how they stack in time.

Step 2. Compute the extrinsic curvature from (2): with $N=1$ and $\beta^i=0$, $K_{ij}=-\tfrac12\partial_t\gamma_{ij}=-\tfrac12\partial_t(a^2)\delta_{ij}=-a\dot a\,\delta_{ij}$. Its trace is $K=\gamma^{ij}K_{ij}=a^{-2}\delta^{ij}(-a\dot a\,\delta_{ij})=-3\dfrac{\dot a}{a}=\mathbf{-3H}$, where $H=\dot a/a$ is the Hubble rate.

Step 3. Form the two invariants. $K^2=9H^2$. For the norm, $K_{ij}K^{ij}=\gamma^{ik}\gamma^{jl}K_{ij}K_{kl}=a^{-4}(a\dot a)^2\,\delta^{ik}\delta^{jl}\delta_{ij}\delta_{kl}=3\,\dfrac{\dot a^2}{a^2}=3H^2$. Hence $K^2-K_{ij}K^{ij}=9H^2-3H^2=\mathbf{6H^2}$.

Step 4. Insert into the Hamiltonian constraint (6): $0+6H^2=16\pi\rho$, i.e. $H^2=\dfrac{8\pi}{3}\rho$ — the Friedmann equation ($G=c=1$). The central equation of cosmology is nothing but the twice-contracted Gauss equation of a comoving slice; the curvature term $k/a^2$ reappears once the slices carry ${}^{(3)}R=6k/a^2$.

2 · The Schwarzschild time-symmetric slice (fully worked)

Show that the $t=0$ slice of Schwarzschild in isotropic coordinates satisfies both constraints, so it is valid vacuum initial data.

Step 1. In isotropic coordinates the slice is conformally flat, $\gamma_{ij}=\psi^4\delta_{ij}$ with $\psi=1+\dfrac{M}{2r}$. It is a moment of time symmetry, so $K_{ij}=0$ — the geometry looks the same forward or backward in time, so its "velocity" $K_{ij}\propto\partial_t\gamma_{ij}$ vanishes.

Step 2. With $K_{ij}=0$ the momentum constraint (7) reads $D_j(0)=8\pi j^i$, and in vacuum $j^i=0$, so it is satisfied identically — every time-symmetric slice solves the momentum constraint.

Step 3. The Hamiltonian constraint (6) with $K_{ij}=0$ and $\rho=0$ collapses to ${}^{(3)}R=0$. For $\gamma_{ij}=\psi^4\delta_{ij}$ the intrinsic Ricci scalar is ${}^{(3)}R=-8\psi^{-5}\hat\nabla^2\psi$ with $\hat\nabla^2$ the flat Laplacian, so ${}^{(3)}R=0$ iff $\psi$ is flat-harmonic.

Step 4. Check that $\psi=1+\dfrac{M}{2r}$ is flat-harmonic for $r>0$: $\hat\nabla^2\left(1+\dfrac{M}{2r}\right)=\dfrac{M}{2}\hat\nabla^2\!\left(\dfrac1r\right)=0$, since $1/r$ is the Green's function of the 3D Laplacian. Both constraints hold, so the slice is legitimate vacuum initial data. This is the seed of the puncture method: a black hole is a harmonic conformal factor with a $1/r$ pole, and superposing poles gives constraint-satisfying multi-black-hole data.

3 · Gaussian curvature of a sphere (Theorema Egregium)

Recover the intrinsic curvature of a round 2-sphere of radius $r$ in flat space and check it against Gauss-Bonnet.

Step 1. For a sphere in Euclidean $\mathbb{R}^3$ the ambient Riemann tensor vanishes and $\epsilon=+1$, so the Gauss equation reads ${}^{(2)}R_{abcd}=K_{ac}K_{bd}-K_{ad}K_{bc}$ with equal principal curvatures $\kappa_1=\kappa_2=1/r$. Tracing gives ${}^{(2)}R=K^2-K_{ij}K^{ij}=2\kappa_1\kappa_2=\mathbf{2/r^2}$ and Gaussian curvature $\mathcal{K}=1/r^2$ — Gauss's remarkable result that the product of principal curvatures is intrinsic.

Step 2. Cross-check with Gauss-Bonnet: $\int_\Sigma \mathcal{K}\,dA=\dfrac{1}{r^2}\cdot 4\pi r^2=\mathbf{4\pi}=2\pi\chi$, giving Euler characteristic $\chi=2$ — correct for a sphere. The local Gauss equation and the global topological invariant agree.

The Gauss-Codazzi-Mainardi equations were essentially complete in classical surface theory a century before physics needed them — then turned out to be exactly the language relativity required to make sense of time.

Texts. MTW, Gravitation (1973), §21.5, Box 21.2 · Wald, General Relativity (1984), Ch. 10 and App. E · Carroll, Spacetime and Geometry (2004), App. D · Gourgoulhon, 3+1 Formalism in General Relativity (2012), Ch. 2-4 · Baumgarte & Shapiro, Numerical Relativity (2010), Ch. 2-3 · do Carmo, Differential Geometry of Curves and Surfaces (1976), Ch. 4. Key checks: (1) the Gauss equation follows from the antisymmetrized projected double derivative, with a $+$ sign for a spacelike slice; (2) the contracted Gauss relation gives ${}^{(3)}R + K^2 - K_{ij}K^{ij}=2G_{\mu\nu}n^\mu n^\nu=16\pi\rho$; (3) the FLRW reduction yields $K=-3H$, $K^2-K_{ij}K^{ij}=6H^2\Rightarrow H^2=\tfrac{8\pi}{3}\rho$; (4) $\psi=1+M/2r$ is flat-harmonic; (5) the sphere gives ${}^{(2)}R=2/r^2$ with Gauss-Bonnet $\int\mathcal{K}\,dA=4\pi$. No constants beyond $8\pi$/$16\pi$ appear.

Prerequisites. ADM 3+1 Decomposition · Riemann curvature and covariant derivatives · Hypersurfaces and the induced metric. Leads to. Hamiltonian and Momentum Constraints · Initial-Value (Cauchy) Problem · Conformal Method for Constraint Solving · BSSN Formulation · Numerical Relativity and Binary Black-Hole Mergers.

Ask the tutor: ask me to walk through any one projection index-by-index, or to build fresh constraint-satisfying initial data and watch the Hamiltonian constraint fall out.