Descent of first-order relational scalars and the free-product obstruction Classical equation spaces, retarded sections, and compact Ward observables

We identify the classical equation spaces and gauge actions on which specified first-order relational scalar functionals descend. An exhibited Poisson-dressed symbol is not constant on the unchanged free product of linearized gravity with a massive scalar; the same two-representative gap survives spatial Fourier regularization. Moving the scalar does not repair that product: $X_{\xi }\varphi$ preserves every compact-data massive free solution only when $\xi$ is Killing. Retaining the forced correction $P{\varphi }_1=-L_h{\varphi }_0$ changes the equation space and supplies a genuine coefficient gauge action. A past-trivial retarded harmonic section then makes the complete first-order functional a quotient observable. For a triangular system with a fixed free scalar background, compact constrained de Donder Cauchy data modulo residual wave gauges form a Hausdorff, well-posed quotient. We construct a compact coupled Ward pair with strictly positive on-shell value, although the immediate adjoint-exact compact repair vanishes on shell. Poisson compensation, compensator choice, and canonical normalization are treated separately. Finally, the sourced successor is divided into invariant equations, a gauge-fixed forced response, and descent on a named quotient. None of those sourced constructions is completed here.

This is a classical quotient paper. It proves no physical Hilbert-space correspondence, measurement instrument, quantum record, Born theorem, contact-span separation, or interacting quantum coefficient. It does not solve the next matter-sourced gravitational reduction, construct a compact gravitational apparatus, or establish an asymptotic scattering theory. Companion manuscripts appear only as explicitly scoped comparisons.

A proposed relational expression must define a function on the equivalence classes actually declared. This question precedes any subsequent representation or measurement construction:

Only the first object and its classical descent are studied here.

Two failures must be distinguished. An expression can assign different values to equivalent representatives. Alternatively, its proposed transformation can fail to preserve the equation space, so that the intended quotient action is not defined. An off-shell cancellation excludes neither failure.

Relational and metric-dependent coordinate constructions have a substantial literature; see, for example, Brunetti, Fredenhagen, and Rejzner BFR. Explicit gravitational dressings and their nonlocal kernels are developed by Donnelly and Giddings DG. Our question is narrower: for the displayed first-order functionals, which equations, identifications, and support prescriptions make descent valid?

The first negative result fixes a free scalar and varies only its metric representative. A compact gauge change of a Poisson compensator produces

We choose $f,\varphi ,\xi$ once so that this integral is nonzero. No function of the unchanged label $([h],\varphi )$ can then agree with the symbol on both representatives. The same representatives yield a quantitative regulated obstruction.

The second negative result allows the scalar to move but retains its free equation. The required condition is

A solution-jet argument proves that this holds exactly for Minkowski Killing fields. In particular, no nonzero compactly supported gauge vector satisfies it.

The first positive result changes the equation space:

The coefficient transformation is an action on these equations. For a named past-trivial gauge group, a retarded harmonic compensator defines a unique orbit representative. The complete first-order scalar then descends.

The second positive result fixes a free background $\Phi$ and uses the linear triangular system

Compact constrained harmonic data modulo residual wave gauges form a Hausdorff Cauchy quotient. A compact coupled Ward pair has a strictly positive value on an explicit solution.

The retarded group and the C1 residual-wave group differ. Their intersection is zero when the retarded support requirement is imposed. The section proof for one group cannot be transplanted to the other.

Nor is C1 a linearization of the full Einstein–scalar system about a nonlinear solution $(\eta ,\Phi )$. Its metric equation is vacuum linearized gravity; the stress of $\Phi$ is not inserted into that equation. For the genuinely linearized Einstein–Klein–Gordon system on on-shell backgrounds, see Hack Hack. That system is not an additional premise here.

The differential identities, obstructions, orbit section, closed-range test, and compact Ward construction are proved below. The analytical background is smooth wave and Klein–Gordon Cauchy theory with finite propagation BGP; we include the estimates needed for its use.

The Coulomb comparison concerns precisely the kernels of the cited version of Donnelly and Giddings DG. Cheung's companion manuscripts are cited as versioned unpublished source records Registry. Their comparison role is collected in Section sec:nontransfer. They are not hypotheses of the five principal theorems: Theorems thm:product, thm:Killing, thm:jet-descent, thm:reduced-Cauchy, and thm:compact-pair.

On $M={\mathbb{R}}^{1+3}$, use

where $\Delta ={\partial }_i{\partial }_i$. All fields and gauge parameters are real unless Fourier transforms or complex-valued comparison functions are explicitly used. Indices are raised with $\eta$, and symmetrization includes the factor $1/2$.

For a covector ${\xi }_{\mu }$, set

The same letter denotes its raised vector when used in $X_{\xi }$.

For a symmetric covariant tensor $h$, define

and

In four dimensions, ${\mathsf{r}}^2=\mathrm{id}$. Our linearized Einstein normalization is

For smooth $\xi$ and $u$,

Moreover,

and therefore

Proof. In $\mathsf{D}K\xi$, the derivatives of ${\partial }_{\mu }{\xi }^{\mu }$ cancel, leaving ${\square }_{\eta }\xi$. Insert this and

into (eq:Glin) to obtain $G^{(1)}K\xi =0$.

Taking the divergence of (eq:Glin) gives

Apply the product rule twice to ${\square }_{\eta }({\xi }^{\nu }{\partial }_{\nu }u)$. The constant mass term commutes with $X_{\xi }$, proving (eq:commutator). Substituting $K\xi$ in (eq:Lh) gives its negative.

The consistent triangular signs are consequently

For first-order coefficients, use

Equality in $\mathbb{D}$ is coefficient equality, not a numerical remainder estimate.

In the two-representative estimates, $\kappa$ can instead be a real parameter in an exact degree-one polynomial. We indicate that use explicitly. A real representation or cutoff parameter in a different model is not automatically this nilpotent formal parameter.

Fix a reference time $t_0$. For this paper, a smooth field is called spacelike compact if, for some $R_0$,

This concrete cone condition is sufficient for every use below. It follows from compact Cauchy data for homogeneous wave and Klein–Gordon solutions.

Let ${\mathcal{E}}_{+}$ be the smooth fields supported in $J_{\eta }^{+}(K_0)$ for some compact spacetime set $K_0$. Use this notation componentwise for vectors and tensors. Such fields have zero data on a sufficiently early Cauchy slice and bounded spatial support on each finite time interval.

Compact spacetime support, compact Cauchy data, and future support from a compact set are different conditions.

For $\mu \ge 0$, the equation

has a unique global smooth solution for smooth compact Cauchy data and smooth forcing with bounded spatial support on every finite time interval. On such intervals, the solution depends continuously on data and forcing in their smooth topologies. Finite propagation gives the corresponding cone-support bounds when the data and source satisfy those bounds.

Proof. Let ${\Omega }_{\mu }=\sqrt{-\Delta +{\mu }^2}$. Spatial Fourier transformation gives

For $\mu =0$, the sine quotient has the continuous value $t-s$ at zero frequency. On a finite interval, the source has uniformly compact spatial support and bounded smooth seminorms, so the required Sobolev integrals are defined. Differentiation verifies the equation and data.

For an integer $s\ge 0$, use

Integration by parts and Cauchy–Schwarz give

The backwards estimate and Gronwall's inequality give uniqueness and continuous dependence. Increasing $s$, applying Sobolev embedding, and using the equation for time derivatives give smooth estimates.

For finite propagation, integrate the local energy identity outside an expanding ball. The outward characteristic flux has the required nonnegative sign. Where initial data and forcing vanish, the exterior energy remains zero. The same argument after a time translation proves the retarded support statement. These are the flat-space estimates underlying the standard general Cauchy theory BGP (Chapter 3).

Let

and

Choose any linear subspace ${\mathcal{G}}_0\subset {\mathcal{S}}_g$ containing

Define

Only the metric is quotiented.

A function $F$ on $X$ descends through $q:X\to Y$ if there is a function $\overline{F}$ on $Y$ with

Equivalently, $F$ is constant on every fiber of $q$. Failure of this set-theoretic condition excludes continuous or smooth descent as well.

For compact smooth or Schwartz spatial sources, define

The fundamental-solution identity gives

For the second equality, the difference is smooth harmonic and tends to zero at infinity, hence vanishes.

For smooth spacelike-compact $h$, apply $N$ at each time:

These potentials are smooth but need not be spatially compact. Their sources have uniformly bounded support on finite time intervals, so time differentiation is legitimate.

For compact smooth $\xi$,

Proof. Directly,

Therefore

Use linearity.

For real $f\in {C_c}^{\infty }(M)$, put

The displayed expression is an exact degree-one polynomial, not an all-orders invariant.

There exist fixed real

such that

and

For every real $\kappa \ne 0$, this fixed symbol does not descend through $q_0$. Its first-order coefficient does not descend either.

Proof. Choose compact scalar data at $t=0$:

where $\chi =1$ near the origin. The resulting free solution has ${\partial }_1\varphi (0,0)=1$, so this derivative is positive on some relatively compact spacetime neighborhood $U$.

Take nonnegative nonzero $f\in {C_c}^{\infty }(U)$. Choose a compact gauge vector equal to ${\partial }_1$ near $\mathrm{supp}f$. Then $J=\int f{\partial }_1\varphi >0$.

Both $0$ and $K\xi$ solve the vacuum metric equation, and they have the same quotient class. Their scalar labels agree. Since $w_{\mathrm{P}}(K\xi )=-\xi$, equation (eq:product-gap) follows. A descended function cannot distinguish these representatives.

The theorem exhibits one fixed symbol. It does not assert that every scalar functional fails to descend; metric-independent scalar functionals already descend through the metric quotient.

The argument uses only $w_{\mathrm{P}}(K\xi )=-\xi$. Adding a linear compensator term annihilating $K\xi$ does not remove this particular gap.

If $q(x_0)=q(x_1)$, then for $H:X\to \mathbb{C}$ and $G:Y\to \mathbb{C}$,

Proof. Insert the common value $G(q(x_0))=G(q(x_1))$ between the two values of $H$, and use the triangle inequality.

For the fixed data above and any $G:{\mathcal{P}}_0\to \mathbb{C}$,

Agreement to $o(|\kappa |)$ on both representatives is impossible.

Proof. Apply Lemma lem:two-point.

Use the unitary transform

Choose smooth $a,b:[0,\infty )\to [0,1]$ with

For $0<2\mu <\Lambda$, let

and

Replacing $N$ in (eq:Poisson) gives ${w_{\mathrm{P}}}_{\rho }$ and the symbol ${F^{\mathrm{P}}}_{\kappa ,\rho }$.

For compact smooth $\xi$,

For the fixed witness,

as $\mu \downarrow 0$, $\Lambda \uparrow \infty$.

Proof. The multiplier identity $N_{\rho }\Delta ={\chi }_{\rho }(D)$ proves the first claim.

On a compact time interval containing $\mathrm{supp}f$, the spatial Fourier transforms of $\xi$ have a uniform Schwartz bound. Thus

tends to zero uniformly on that interval. One can use a fixed integrable Schwartz envelope and dominated convergence. Integration against the fixed compact test factors gives the result.

For every family $G_{\rho }:{\mathcal{P}}_0\to \mathbb{C}$,

For fixed real $\kappa \ne 0$, the bound tends to $|\kappa |J/2>0$.

Proof. The symbol difference is $-\kappa J_{\rho }$, while the quotient labels remain identical. Apply Lemma lem:two-point and the convergence lemma.

This regularizes a classical symbol only. It does not concern an operator or measurement. It also does not obstruct a different quotient that changes the scalar identification along with the metric.

premises{ The product theorem fixes $\varphi$ and varies $h$. The present theorem moves $\varphi$ by $X_{\xi }\varphi$, while retaining the massive free equation. Neither statement is a reformulation of the other.}

At any $p\in M$, let $a\in \mathbb{R}$, $b_{\mu }\in {\mathbb{R}}^4$, and symmetric $H_{\mu \nu }$ satisfy

There is $\varphi \in {\mathcal{S}}_{\varphi }$ with

Proof. Translate $p$ to $(0,0)$. Choose compact smooth data $q,\pi$ with

Finite polynomials multiplied by a bump equal to one near the origin realize these data jets. The free equation supplies

which is the remaining required component.

For smooth $\xi$, the following are equivalent:

1. $P(X_{\xi }\varphi )=0$ for every $\varphi \in {\mathcal{S}}_{\varphi }$;

2. $K\xi =0$;

3. $${\xi }^{\mu }(x)=a^{\mu }+B^{\mu }{}_{\nu }{x}^{\nu }$$

   for constants with $B_{\mu \nu }=-B_{\nu \mu }$.

A field satisfying these conditions and vanishing on a nonempty open set is identically zero.

Proof. Assume (i). At a fixed point, (eq:commutator) and the jet lemma imply

for every allowed jet, where

Take $a=0,H=0$ and arbitrary $b$, obtaining ${\square }_{\eta }\xi =0$.

Take $a=0,b=0$ and arbitrary trace-free symmetric $H$. The annihilator of that codimension-one space is the span of the trace functional, so

at the point. Finally take $a=1,b=0$ and a symmetric $H$ of trace $m^2$. Then $c{m}^2=0$, hence $c=0$. The point was arbitrary, so $K\xi =0$.

Differentiating the Killing equation gives

Thus $\xi$ is affine with antisymmetric linear part.

Conversely, an affine Killing field has ${\square }_{\eta }\xi =0$ and vanishing symmetric first derivative. The commutator identity gives (i). An affine field vanishing on an open set is zero.

The prescription

does not define a tangent action on the unchanged massive free solution space for all compact smooth $\xi$. This conclusion does not depend on a proposed metric transformation.

Proof. The tangent space of the linear solution space is itself. Tangency at every solution is condition (i) above. A compactly supported Killing field vanishes on an open set and is therefore zero.

The assumption $m>0$ is used in eliminating $c$. No massless version of this classification is asserted.

There exist real compact-data $\varphi$, compact smooth $\xi$, and nonnegative $f\in {C_c}^{\infty }(M)$, $\int f=1$, such that

For ${\varphi }_{\kappa }=\varphi +\kappa X_{\xi }\varphi$,

Proof. Take $\beta \in {C_c}^{\infty }({\mathbb{R}}^3)$ equal to one near the origin and scalar data

For $a\in {C_c}^{\infty }(\mathbb{R})$ equal to one near zero, set

At the origin,

Thus $P(X_{\xi }\varphi )(0)=1$. It exceeds $1/2$ on a neighborhood. A normalized nonnegative test supported there proves the first claim; $P\varphi =0$ proves the second.

An additional regular formal coefficient of order ${\kappa }^2$ cannot remove this order-$\kappa$ equation defect. Numerical remainder claims would require additional estimates.

For $g=\eta +\kappa h$,

Substitution into

gives

Consequently $\varphi ={\varphi }_0+\kappa {\varphi }_1$ obeys the coefficient equations

The second scalar coefficient is forced, not independently free.

For field-independent $\xi$, define

This preserves (eq:jet-scalar) and $G^{(1)}(h)=0$, and

Proof. The transformed forced equation has left-hand side

The metric equation is preserved because $G^{(1)}K=0$. The group identities are immediate.

This is the additive action on retained coefficients. Commutators of diffeomorphisms generated by $\kappa \xi$ and $\kappa \zeta$ begin at order ${\kappa }^2$, which is not retained.

Let

For $F\in {\mathcal{E}}_{+}$, define $R_{\square }F$ to be the solution of

with zero data on a slice preceding $\mathrm{supp}F$. The forced Cauchy lemma constructs compatible solutions on all finite intervals, and finite propagation gives $R_{\square }F\in {\mathcal{E}}_{+}$. The prescription is independent of the earlier slice.

We have

The second equality follows because the difference is homogeneous with zero past data.

Define

and

In $K{w}_{\mathrm{R}}(h)$, lower the index first.

For $\xi \in {\mathcal{G}}_{\mathrm{R}}$,

Also,

and

Proof. Apply $\mathsf{D}K\xi ={\square }_{\eta }\xi$ and (eq:retarded-inverse). Then

The other projector identities follow, and $G^{(1)}K=0$ proves the final equality.

A homogeneous vector wave in ${\mathcal{G}}_{\mathrm{R}}$ is zero. Thus

Proof. A field in ${\mathcal{G}}_{\mathrm{R}}$ has zero data on a sufficiently early slice. Homogeneous wave uniqueness makes it zero.

Let $\mathcal{J}$ consist of triples satisfying

with ${\varphi }_1$ smooth and spacelike compact. Let $\mathcal{Y}\subset \mathcal{J}$ have $\mathsf{D}h=0$. The gauge group is ${\mathcal{G}}_{\mathrm{R}}$, acting by $T_{\xi }$.

There is no spacelike-compact requirement on $h$ in this definition. Its harmonic part may be noncompact. The retarded prescription needs the condition on $\mathsf{D}h$. The scalar forcing remains spacelike compact because its support is contained in that of ${\varphi }_0$.

Set

The map ${\mathcal{S}}_{\mathrm{R}}:\mathcal{J}\to \mathcal{Y}$ obeys:

1. ${\mathcal{S}}_{\mathrm{R}}x=T_{-w_{\mathrm{R}}(h)}x$;

2. ${\mathcal{S}}_{\mathrm{R}}{T}_{\xi }x={\mathcal{S}}_{\mathrm{R}}x$;

3. ${\mathcal{S}}_{\mathrm{R}}{|}_{\mathcal{Y}}=\mathrm{id}$;

4. every ${\mathcal{G}}_{\mathrm{R}}$-orbit meets $\mathcal{Y}$ exactly once.

It induces a bijection

This is an orbit-space assertion, not an infinite-dimensional quotient-manifold theorem.

Proof. Because $w_{\mathrm{R}}(h)\in {\mathcal{G}}_{\mathrm{R}}$, part (i) follows from the action. The equations are preserved by Proposition prop:jet-action. The scalar correction remains spacelike compact because its support is contained in that of ${\varphi }_0$. The retarded section identities supply $\mathsf{D}{\Pi }_{\mathrm{R}}h=0$.

Under $T_{\xi }$, the scalar correction is

The metric projection is unchanged as well, proving (ii). On the slice, $w_{\mathrm{R}}(h)=0$, proving (iii).

Part (i) gives existence. If $h$ and $h+K\xi$ are harmonic with $\xi \in {\mathcal{G}}_{\mathrm{R}}$, then ${\square }_{\eta }\xi =0$, hence $\xi =0$ by Lemma lem:gauge-intersection. The scalar correction is consequently unchanged.

For smooth $k$ satisfying $\mathsf{D}k=0$, $G^{(1)}(k)=0$, and ${\varphi }_0\in {\mathcal{S}}_{\varphi }$, every pair of compact smooth scalar data determines a unique smooth spacelike-compact solution

Proof. The source is smooth and supported in the cone of ${\varphi }_0$. The forced Cauchy lemma gives the solution. Propagation from both compact initial data and the source cone gives a cone of slope one with possibly enlarged initial radius.

For real compact $f$, define

The functional ${\mathcal{F}}_f$ descends to $\mathcal{J}/{\mathcal{G}}_{\mathrm{R}}$. If

then

Proof. Both coefficients are invariant by the section calculation, so the functional factors through the orbit map. On the slice, $w_{\mathrm{R}}(k)=0$.

Writing $w=w_{\mathrm{R}}(h)$,

The ${\varphi }_1$ term cannot be omitted from the declared coupled equation space. It provides both the forced equation and the transformation needed for cancellation.

There are slice data with $P{\psi }_1\ne 0$.

Proof. Let

and all other components vanish. This tensor is trace free, divergence free, and a componentwise wave, so it is harmonic vacuum data.

Use the jet lemma to choose ${\varphi }_0$ whose value and gradient vanish at the origin, with Hessian

and all other components zero there. Then $L_k{\varphi }_0(0)=-2$. Every corresponding correction satisfies $P{\psi }_1(0)=2$.

Fix the reduced metric $k$; do not vary it. Put