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For the positive-expansion convention for extrinsic curvature, write ${\nabla }_{\mu }{n}_{\nu }=K_{\mu \nu }-n_{\mu }{a}_{\nu }$ with normal acceleration $a_{\nu }=n^{\rho }{\nabla }_{\rho }{n}_{\nu }$. The Ricci commutator applied to $n$ gives the difference of its two second-derivative contractions. Projecting tangentially, the acceleration term in that commutator cancels exactly the acceleration term converting a spacetime divergence of $K$ into its fully projected spatial covariant derivative. Einstein's equations then identify the displayed equation as the momentum constraint.

Using the lapse function $N$, shift vector $N^i$ and spatial metric $h_{ij}$, the Legendre transform in mechanics of the Einstein-Hilbert action gives, in units $16\pi G=1$,Here $\pi =h_{ij}{\pi }^{ij}$ and ${\pi }^{ij}$ is the canonical momentum of the spatial metric. Varying lapse and shift imposes the Hamiltonian constraint and momentum constraint. An asymptotic time translation requires an Arnowitt-Deser-Misner energy boundary term. The bulk constraints generate gauge evolution; the nonzero asymptotic energy comes from the boundary.

Gravitational initial data on a spatial manifold consist of a Riemannian metric $h$ and an extrinsic curvature $K$ satisfying the Hamiltonian constraint and momentum constraint. Matter theories require their own data and constraints as well. The data determine a maximal Cauchy development.

With negligible shear, a minimally coupled scalar field serving as a clock, and $\Pi \ne 0$, the Hamiltonian constraint and momentum constraint imply $\dot{H}=-4\pi GN{\Pi }^2$ and ${\partial }_{i}H=-4\pi G\Pi {\partial }_i\varphi$. Differentiating the local Friedmann relation yields $\dot{\Pi }=-N(3H\Pi +V_{,\varphi })$ and ${\partial }_i\Pi =-(3H+V_{,\varphi }/\Pi ){\partial }_i\varphi$. Inserting these identities into ${\dot{\zeta }}_i$ cancels all terms, including gradients of the lapse function. The conserved covector becomes the gradient of a uniform-density curvature perturbation at linear adiabatic order. Dropping the shear constraint is essential; this is not a proof of conservation in an arbitrary non-attractor phase.

The positive energy theorem concerns the total gravitational energy of isolated systems, rather than a pointwise gravitational energy density. Its standard asymptotically flat form isHere are the hypotheses used in a spinorial proof. Take smooth, complete, three-dimensional initial data in general relativity $(\Sigma ,h,k)$ with a spin structure, no inner boundary, and finitely many asymptotically Euclidean ends. Assume the usual differentiable falloff $h_{ij}-{\delta }_{ij}=O(r^{-1})$, $\partial h=O(r^{-2})$, $k_{ij}=O(r^{-2})$, with sufficiently decaying further derivatives and integrable matter densities. The Hamiltonian constraint and momentum constraint must hold, and the matter must satisfy the dominant energy condition. These hypotheses exclude a naked singularity or an unaccounted boundary flux from the argument. An oriented three-manifold admits a spin structure; in higher dimensions the spinorial proof needs this as an additional assumption. Work in signature $(-+++)$ and units $G=c=1$.

Let $n$ be the future unit normal, choose $k_{ij}=-g({\nabla }_{i}n,e_j)$, and define $\mu =T_{ab}{n}^a{n}^b$ and $J_i=-T_{ab}{n}^a{e}_i^b$. With this convention the Hamiltonian constraint and momentum constraint readThe dominant energy condition implies $\mu \ge |J{|}_h$. The asymptotic charges are the ADM energy and ADM momentum,These are boundary charges; a sign condition on $\mu$ alone does not visibly determine their sign.

Choose gamma matrices obeying $\{{\gamma }^a,{\gamma }^b\}=2g^{ab}$, with spatial matrices Hermitian and ${\gamma }^{\hat{0}}$ anti-Hermitian. The relevant derivative is the Sen spinor connection, not just the intrinsic spin connection:The symmetric extrinsic curvature makes $\mathcal{D}={\gamma }^i{D}_i^{(3)}-\frac{1}{2}(\mathrm{tr}k){\gamma }^{\hat{0}}$. This is an elliptic differential operator with the ordinary spatial Dirac operator as principal part. Solve the Witten spinor equation $\mathcal{D}ϵ=0$, with $ϵ\to {ϵ}_{\infty }$ constant at the end under consideration and zero asymptotic constants at any other ends.

The central algebraic ingredient is the Witten-Nester identity. Define $\overline{ϵ}=-{ϵ}^{†}{\gamma }^{\hat{0}}$, the future-directed Dirac current $K^a=\overline{ϵ}{\gamma }^aϵ$, and the Nester two-formIn an orthonormal frame $K^0=|ϵ{|}^2$. Since $-{\gamma }^{\hat{0}}{\gamma }^i{q}_i$ is Hermitian with eigenvalues $\pm 1$ for any unit spatial vector $q$, $|\mathbf{K}|\le K^0$. Thus the Dirac current is future causal. Differentiate the Nester two-form by the product rule. The antisymmetric second-derivative terms reduce using the spinor curvature identityThe Clifford algebra and the first Bianchi identity turn the curvature contraction into the Einstein tensor. The spatial derivative terms give the difference of two squares. Projecting onto $\Sigma$ therefore givesThe constraints are exactly the normal components of the Einstein field equations needed here. Moreover,For a solution of the Witten spinor equation, both terms left in this divergence are nonnegative.

There is an essential analytic step: an asymptotically constant spinor field solving the Witten spinor equation must actually exist. Extend the chosen constant smoothly from infinity to a field ${ϵ}_0$. Then $\mathcal{D}{ϵ}_0=O(r^{-2})$ is square-integrable. For compactly supported fields $\eta$, the same Witten-Nester identity has zero boundary flux and givesThe asymptotically Euclidean Hardy inequality in Euclidean space, local elliptic estimates and this identity supply the coercive estimate in the weighted first-order Sobolev space of decaying corrections. One way to see why a compact-region error cannot spoil coercivity is to assume the estimate fails and take a normalized sequence whose Dirac operator tends to zero. Local compactness and the estimate on the ends produce a decaying homogeneous solution. Integrating the identity with cutoffs shows that such a solution is parallel for the Sen spinor connection. A parallel field tending to zero at infinity is zero, contradicting its normalization. The end estimates exclude escape of the normalized sequence to infinity.

More constructively, on this completed correction space solvefor every compactly supported $\chi$, using the Lax-Milgram theorem. Set $ϵ={ϵ}_0+\eta$. Since $\mathcal{D}$ is formally skew-adjoint in the positive spatial spinor inner product, the residual $\xi =\mathcal{D}ϵ$ is a square-integrable weak solution of $\mathcal{D}\xi =0$. Elliptic regularity makes it smooth. Cutoff integration of the identity excludes a nonzero square-integrable homogeneous solution: it would be parallel, whereas a nonzero parallel field has nonzero limiting norm on an asymptotically Euclidean end and cannot be square-integrable. Hence $\xi =0$. Weighted elliptic estimates then give the required asymptotic decay of $\eta$. This supplies elliptic existence of a Witten spinor, rather than assuming it from the sign of a formal integral.

Finally integrate the Witten-Nester identity over $\Sigma$. The ADM boundary term of the Nester two-form isTo identify this boundary term, expand the asymptotic spin connection to first order in $h-\delta$. Its flux is half of $({\partial }_j{h}_{ij}-{\partial }_i{h}_{jj})s^i{K}_{\infty }^0$; the extrinsic curvature contribution is $-(k_{ij}-h_{ij}\mathrm{tr}k)s^j{K}_{\infty }^i$. Their coefficients give precisely the two charges displayed above. The derivative of the decaying spinor correction has zero leading integrated flux by antisymmetry, and the higher-order terms vanish with the falloff. ThusChoose a normalized asymptotic spinor field that is an eigenvector with eigenvalue $+1$ of $-{\gamma }^{\hat{0}}{\gamma }^i{q}_i$. Then $K_{\infty }^0=1$ and ${\mathbf{K}}_{\infty }=q$. Choosing $q=\mathbf{P}/|\mathbf{P}|$ proves $E\ge |\mathbf{P}|$; if $\mathbf{P}=0$, any such choice gives $E\ge 0$. This is positivity in every asymptotic frame, not only the zero-momentum case.

The zero-energy rigidity also follows from the vanishing integrals. If $E=0$, then $\mathbf{P}=0$ and the construction for a basis of asymptotic spinors produces a basis of parallel fields. Their spinor curvature identity forces the spatially pulled-back spacetime curvature to vanish; the nonnegative matter terms vanish too. The resulting Gauss–Codazzi equations are those of a hypersurface in Minkowski spacetime, and its vacuum development is flat. Completeness and the ordinary Euclidean asymptotic end rule out a nontrivial flat quotient. The initial hypersurface need not have $k=0$: a curved spacelike slice of Minkowski spacetime also has zero ADM energy. The proof works by converting an asymptotic energy charge into a nonnegative bulk integral, with the spinor equation and its existence supplying the crucial bridge.

The long-wavelength approximation in cosmology organizes equations in powers of $ϵ\sim k/(aH)\ll 1$, while allowing finite perturbation amplitudes. For fields and lapse varying on the same large spatial scale, a spatial derivative is of order $ϵ$ relative to a temporal expansion scale. Intrinsic spatial curvature, scalar gradient energy, and second spatial derivatives of the lapse are therefore second order and can be neglected at leading order. First-gradient constraints must still be retained: they relate neighboring locally homogeneous patches. During cosmic inflation, physical wavelengths stretch while the Hubble scale changes slowly, improving this expansion. Sharp spatial features or independently large anisotropic expansion require additional care.

Set the shift to zero, let $S^i{}_j={\tilde{K}}^i{}_j$, $S^2=S^i{}_j{S}^j{}_i$, and keep the lapse explicit. The normal scalar momentum is $\Pi =\dot{\varphi }/N$, and the shear-inclusive long-wavelength scalar equations obtained from the given Einstein and scalar equations areTerms with two spatial gradients have been omitted in the time equations; the last equation is the retained first-gradient momentum constraint. The Friedmann-type equation is the Hamiltonian constraint. To check the Hubble evolution, the trace equation gives $\dot{H}=-3N{H}^2+8\pi GNV$; inserting the Hamiltonian constraint gives the displayed kinetic and cosmological shear terms. Shear energy is not a spatial-gradient correction and cannot be discarded merely because the wavelength is large.

The trace-free evolution integrates immediately:The arbitrary spatial integration tensor is trace free and must obey the initial constraints. This inflationary shear damping suppresses anisotropic expansion exponentially during sustained inflation; its contribution $S^2$ falls as $a^{-6}$. It does not force the spatial conformal metric to be exactly flat: a frozen anisotropic shape or tensor perturbation can survive with negligible time-dependent cosmological shear.

After this decaying cosmological shear mode and its first-gradient contribution become negligible, the equations simplify to $3H^2=8\pi G({\Pi }^2/2+V)$ and $D_{i}H=-4\pi G\Pi D_i\varphi$. Spatially differentiating the first relation givesSubstitute the momentum constraint and divide by $\Pi \ne 0$ to obtain $D_i\Pi =-(3H+V_{,\varphi }/\Pi )D_i\varphi$. The time scalar equation, together with $\dot{\varphi }=N\Pi$, similarly becomes $\dot{\Pi }=-(3H+V_{,\varphi }/\Pi )\dot{\varphi }$. Finally $\dot{H}=-4\pi G\Pi \dot{\varphi }$. Thus the requested single-clock relations areThese retain arbitrary lapse and reduce to proper-time expressions when $N=1$. The division assumes the scalar is a valid local clock; turning points with $\Pi =0$ need another regular variable or a limiting treatment. The shear-neglect step, rather than the long-wavelength expansion alone, is what permits these single-clock spatial relations.

Let $a_{\mu }=n^{\nu }{\nabla }_{\nu }{n}_{\mu }$ be the normal acceleration. It is tangential by normalization. Resolving the derivative's first index into normal and tangential parts givesThe extrinsic curvature is symmetric because the normal is hypersurface orthogonal: locally $n_{\mu }=-N{\partial }_{\mu }t$, and the antisymmetric derivative of $n$ vanishes after projecting both indices tangentially.

Apply the supplied Ricci commutator to $n^{\nu }$ and project its free index. With the positive-expansion convention it givesTo identify the first two terms, expand the fully projected divergence:Tangency gives $n_{\lambda }{K}^{\lambda }{}_{\mu }=0$. Differentiating it shows that the last term is $-P^{\mu }{}_{\alpha }{K}^{\lambda }{}_{\mu }{n}^{\rho }{\nabla }_{\rho }{n}_{\lambda }=-K_{\alpha \lambda }{a}^{\lambda }$. Substituting into the commutator proves the contracted Codazzi equation with positive expansion:No assumption of geodesic normals was needed; the acceleration terms cancel rather than being discarded.

The metric-trace term in the Einstein field equations drops out of a tangential-normal projection, since $P^{\mu }{}_{\alpha }{g}_{\mu \nu }{n}^{\nu }=0$. Define the normal-frame momentum density $j_{\alpha }=-P^{\mu }{}_{\alpha }{T}_{\mu \nu }{n}^{\nu }$. The equation becomesThus Codazzi supplies the momentum constraint on each spatial slice. In flat homogeneous FRW, $K^i{}_j=H{\delta }^i{}_j$ and $H$ has no spatial gradient, so its left side vanishes. A comoving homogeneous perfect fluid likewise has $j_i=0$. The background satisfies the constraint automatically; the Friedmann expansion law is instead obtained from the normal-normal constraint.

Suitable scalar variables are lapse and shift potentials $A,B$, spatial curvature and scalar shear potentials $\psi ,E$, and a matter momentum potential $J$ with $j_i={\partial }_{i}J$. For example, chooseA density contrast and pressure perturbation can be added for the remaining scalar matter equations. With this positive-shift convention, the linear extrinsic-curvature perturbation isIts scalar shear cancels out of the momentum-constraint combination, leavingThese are the scalar momentum constraint with positive extrinsic curvature variables; a specified gauge can then remove two of the four scalar metric potentials.

Retain zeroth and first spatial-gradient orders in the long-wavelength approximation in cosmology. Spatial Ricci curvature, $D_i{D}_{j}N$, $q^2$ and scalar anisotropic stress are of second gradient order. Put $A^i{}_j={\tilde{K}}^i{}_j$ and $A^2=A^i{}_j{A}^j{}_i$. The supplied Einstein field equations then reduce toThe scalar-field stress projections show that the momentum constraint is a first-gradient equation and must be retained. The square $A^2$ is not a spatial derivative and cannot yet be dropped merely by invoking the long-wavelength approximation in cosmology. The time-derivative dot on the trace-free evolution equation is visible in the PDF and missing from the TeX aid.

Using $NH=\dot{a}/a$, the last equation becomes ${\partial }_t(a^3{A}^i{}_j)\simeq 0$. HenceThe time-independent spatial tensor must also satisfy the momentum constraint and the symmetry inherited from $K_{ij}$. This is inflationary shear damping: expanding regions lose anisotropic expansion, without requiring the frozen spatial metric shape to vanish. The matter evolution, useful below, is $\dot{\varphi }=N\Pi$ and $\dot{\Pi }\simeq -N(3H\Pi +V_{,\varphi })$, the leading Klein-Gordon equation in each local region.

Now impose the stipulated negligible shear and work where $\Pi \ne 0$. The Hamiltonian constraint, momentum constraint and local expansion evolution giveTime differentiation of the first equation, followed by $\dot{\varphi }=N\Pi$, gives $\dot{\Pi }=-N(3H\Pi +V_{,\varphi })$. Spatial differentiation of the same equation and use of the momentum constraint givesThese are the temporal and spatial identities needed for the nonlinear curvature covector. Differentiate its definition, keeping the spatially varying lapse function:The terms proportional to ${\partial }_{i}N$ cancel. The terms $-N{\partial }_{i}H$ and $-4\pi GN\Pi {\partial }_i\varphi$ cancel by the momentum constraint, and the remaining two terms cancel by the spatial momentum identity. Thereforeto the retained gradient order. No homogeneous-lapse assumption was needed.

For linear adiabatic cosmological perturbations, expansion of the local scale and scalar field gives ${\zeta }_i={\partial }_i[\Psi +\bar{H}\delta \varphi /\bar{\Pi }]$. For nonzero spatial modes, linearizing the Hamiltonian constraint and the momentum constraint givesThusThis uses the paper's overall sign for the uniform-density curvature perturbation; the opposite overall sign is also common. For an adiabatic single-field attractor, superhorizon conservation of single-field comoving curvature allows the primordial amplitude to be transported through later epochs without following every short-scale process. This proof specifically discarded the shear contribution to the momentum constraint and assumed a usable field clock. It is not a claim that every single-field background conserves every curvature mode: growing curvature perturbation in ultra-slow-roll inflation is a non-attractor counterexample, and $\Pi =0$ makes this particular covector undefined.

With vanishing extrinsic curvature, the momentum constraint is identically satisfied and the vacuum Hamiltonian constraint reduces to $\mathcal{R}=0$. To evaluate that Ricci scalar, write ${\psi }_i={\partial }_i\psi$, raise these intermediate indices with ${\delta }^{ij}$, and use ${\gamma }^{ij}={\psi }^{-4}{\delta }^{ij}$. The Levi-Civita connection isSubstitution into the stated curvature convention givesTracing with ${\gamma }^{ij}$ cancels the gradient-square terms:Here the Laplacian and norm on the right are those of the flat Euclidean metric. Thus the time-symmetric conformally flat vacuum initial data constraints becomeThe equivalence uses $\psi \ne 0$. This identity is also the three-dimensional specialization of scalar curvature under conformal rescaling; locally the conformal exponent is $2\log |\psi |$, so either fixed nonzero sign of $\psi$ gives the same metric.

Write $\mathcal{D}={\bar{N}}^{-1}{\partial }_t$, $\bar{\Pi }=\mathcal{D}\bar{\varphi }$ and $\Delta =a^{-2}{\nabla }^2$. Since the background scalar has no spatial gradient,Also $\delta P=\bar{\Pi }\mathcal{D}\delta \varphi -{\bar{\Pi }}^2\Psi -V_{,\varphi }\delta \varphi$. These are the linearized scalar-field matter projections; gradient energy first contributes quadratically.

There is a source convention defect in the shear formula. For the stated positive shift $N_i=a^2{B}_{,i}$ and spatial metric with $+2E_{,ij}$, direct substitution into the extrinsic curvature gives the positive-shift scalar shear conventionThere is no extra Laplacian in the definition of ${\chi }_{+}$. Equivalently use $\chi =-{\chi }_{+}$: then the trace-free term has the printed minus sign and $\kappa =3(\mathcal{D}\Phi +H\Psi )-\Delta \chi$. This latter convention retains the printed momentum-constraint form. The source mixes these two choices.

At first order $\delta (K^2)=-6H\kappa$ and $\delta (K^i{}_j{K}^j{}_i)=-2H\kappa$. Combining their difference with $\delta {}^{(3)}R=4\Delta \Phi$ givesFor the momentum constraint, differentiating the tensor above givesFor nonzero Fourier wavenumber, and absorbing the homogeneous integration mode into the background, the consistent momentum constraint is

Linearizing the scalar equation, the shift-advection term and lapse-gradient times scalar-gradient term vanish at this order because the background is spatially homogeneous. The remaining equation isInsert $\delta \Pi$, differentiate its lapse term, and use $\mathcal{D}\bar{\Pi }+3H\bar{\Pi }=-V_{,\varphi }$. The two background-acceleration terms combine into $2\Psi V_{,\varphi }+3H\bar{\Pi }\Psi$. Hence the perturbed Klein-Gordon equation with background lapse isThis scalar evolution equation and the Hamiltonian constraint agree with the displayed targets after the shear conventions are repaired.

Use signature $(+---)$, the positive spatial metric ${\gamma }_{ij}$, and a future-directed unit normal $n_{\mu }=(N,0)$, so $n^{\mu }=N^{-1}(1,-N^i)$. This fixes the sign of the normal to agree with the specified $\Pi$. The negative normal covector printed in the source would instead give $-\Pi$; its orientation and the displayed derivative cannot both be retained. Use the operational negative-expansion convention for the extrinsic curvature of a spatial hypersurface, $K_{ij}=-({\dot{\gamma }}_{ij}-D_i{N}_j-D_j{N}_i)/(2N)$, which is the convention giving the requested expanding-universe sign. With this future normal and signature it equals $n_{i;j}$; the source's $-n_{i;j}$ identity instead uses its past-pointing normal.

For the scalar-field matter projections, put $s_i=D_i\varphi$ and $s^2={\gamma }^{ij}{s}_i{s}_j$. Decomposing the scalar gradient into normal and tangent parts gives ${\partial }_{\mu }\varphi {\partial }^{\mu }\varphi ={\Pi }^2-s^2$. Contracting the Klein-Gordon scalar stress-energy tensor with the normal and the spatial projection tensor therefore yieldsIn particular, the positive spatial metric raises the $i,j$ indices in $s^2$; raising them with the four-metric would reverse that spatial sign. These quantities are the energy density, momentum density and spatial stress in the normal frame.

For the homogeneous Spatially flat FLRW metric, $N_i=0$, ${\gamma }_{ij}=a^2{\delta }_{ij}$ and $s_i=0$. Consequently $\Pi =\dot{\bar{\varphi }}/\bar{N}$, $J_i=0$, $\bar{\rho }={\Pi }^2/2+V$, $\bar{P}={\Pi }^2/2-V$, andThe intrinsic Ricci scalar is zero and $K^2-K_{ij}{K}^{ij}=9H^2-3H^2=6H^2$. The Hamiltonian constraint thus gives the scalar-field Friedmann equation:The momentum constraint is identically satisfied, since its homogeneous spatial derivatives and $J_i$ vanish. The scalar equation reduces to ${\bar{N}}^{-1}\dot{\Pi }+3H\Pi +V_{,\varphi }=0$. Substituting $\Pi$ and multiplying by ${\bar{N}}^2$ gives the background inflaton equation of motion:No derivation of the supplied gravitational constraints is needed.

For a closed universe, the spatial Cauchy hypersurface is compact without boundary. There is consequently no asymptotic surface term in the gravitational Hamiltonian. On solutions, the Hamiltonian constraint and momentum constraint giveThis is the vanishing canonical Hamiltonian of a closed universe: evolution generated by lapse and shift is a gauge transformation, and there is no asymptotic time-translation charge. It does not mean the local stress-energy tensor or Riemann curvature tensor is zero. In an asymptotically flat universe the same bulk constraint equations in field theory hold, but a boundary Arnowitt-Deser-Misner energy can be nonzero.

Perform the Legendre transform in mechanics, using ${\dot{h}}_{ij}=2N{K}_{ij}+D_i{N}_j+D_j{N}_i$. The trace contraction ${\pi }^{ij}{K}_{ij}=\sqrt{h}(K_{ij}{K}^{ij}-K^2)$ givesTo avoid differentiating a tensor density as though it were an ordinary tensor, set $p^{ij}={\pi }^{ij}/\sqrt{h}$. Integration by parts with the Levi-Civita connection of $h$ givesafter dropping the surface term. AlsoTherefore the requested Hamiltonian isVarying the lapse function and shift vector imposes the Hamiltonian constraint $\mathcal{H}=0$ and momentum constraint ${\mathcal{H}}_i=0$. The total canonical generator also contains multipliers for the vanishing lapse and shift canonical momenta. With an asymptotic boundary, surface terms must be restored to define the Arnowitt-Deser-Misner energy.

In the paper's convention, ${\dot{h}}_{ij}=2N{K}_{ij}+D_i{N}_j+D_j{N}_i$. The Legendre transform in mechanics therefore givesIndeed ${\pi }^{ij}{K}_{ij}=\sqrt{h}(K_{ij}{K}^{ij}-K^2)$, and inversion of the canonical momentum of the spatial metric givesTo integrate the shift term without ambiguity about tensor densities, put $p^{ij}={\pi }^{ij}/\sqrt{h}$. Metric compatibility of the spatial covariant derivative gives, up to a boundary term,Thus the Hamiltonian formulation of general relativity hasVariation of the lapse function imposes the Hamiltonian constraint $\mathcal{H}=0$, while variation of the shift vector imposes the momentum constraint ${\mathcal{H}}_i=0$.

A closed spatial Cauchy hypersurface is compact and has no boundary. Thus there is no asymptotic surface charge, and the complete gravitational Hamiltonian is the integral of the Hamiltonian constraint and momentum constraint derived above. Both constraints vanish on solutions, givingThis is the vanishing canonical Hamiltonian of a closed universe: canonical evolution is generated by constraints and there is no asymptotic time translation whose boundary charge could define an Arnowitt-Deser-Misner energy. It does not require local matter energy densities or local curvature to vanish. With matter included, the corresponding total constraints still vanish.

For scalar metric potentials $d{s}^2=a^2[-(1+2A)d{\tau }^2+2B_{,i}d\tau d{x}^i+((1-2\psi ){\delta }_{ij}+2E_{,ij})d{x}^{i}d{x}^j]$ and positive extrinsic curvature, $\delta K^i{}_j=a^{-1}[-({\psi }^{\prime }+\mathcal{H}A){\delta }^i{}_j+{\partial }^i{\partial }_j(E^{\prime }-B)]$. The scalar shear terms cancel between its divergence and trace gradient. With $j_i=-P^{\mu }{}_i{T}_{\mu \nu }{n}^{\nu }$, the momentum constraint becomes $2{\partial }_i({\psi }^{\prime }+\mathcal{H}A)/a=-8\pi G{j}_i$. Reversing the convention for $K$ reverses the geometric left side.

For zero shift vector, negative-expansion extrinsic curvature $K=-3H$ and $S^i{}_j={\tilde{K}}^i{}_j$, retain the leading spatial-gradient equations of a minimally coupled scalar. They give $\dot{\varphi }=N\Pi$, $\dot{\Pi }=-N(3H\Pi +V_{,\varphi })$, $\dot{H}=-4\pi GN{\Pi }^2-N{S}^i{}_j{S}^j{}_i/2$, and ${\dot{S}}^i{}_j=-3NH{S}^i{}_j$. The first-gradient momentum constraint is ${\partial }_{i}H=-4\pi G\Pi {\partial }_i\varphi -{\nabla }_j{S}^j{}_i/2$. Shear decays as $a^{-3}$ during expansion. Its neglect after inflationary damping yields the equations underlying conservation of the nonlinear curvature covector.

For a spatial initial slice with vanishing extrinsic curvature, the vacuum momentum constraint is automatic and the Hamiltonian constraint requires zero Ricci scalar. In three spatial dimensions, the indicated conformally flat metric has $\mathcal{R}=-8{\psi }^{-5}\Delta \psi$, with a nonzero conformal factor. Thus a harmonic function supplies the data on its domain. A positive example is $\psi =1+a/r$ on a punctured or exterior region. There is no nonconstant globally smooth example on all of Euclidean three-space tending to one at infinity, by the maximum principle for harmonic functions. Time symmetry of an initial slice does not imply that its entire evolved spacetime is static.