Relational Einstein–scalar observables at second order: classical descent and a compact-switch obstruction

We examine the compatibility of a finite-time harmonic relational prescription with compactly switched Einstein–scalar perturbation theory. For a specified second-order Einstein–scalar solution-jet space, we construct harmonic coordinate coefficients with fixed Cauchy data and prove that the resulting scalar functional descends under formal diffeomorphisms that are trivial near the initial surface. The metric is an unknown of the coefficient equations; an explicit sourced family establishes nonvacuity.

The same geometric dressing does not automatically descend after the scalar correction is replaced by the retarded response of a compactly switched vertex. We derive the exact first-order switch defect and give two representatives with a strictly nonzero smeared gap, even though the switch equals one near the observation smearing. No correction beginning at second order can remove this first-order defect while that representative identification and lower-order prescription are retained.

We distinguish the classical second-order coefficient from the four-term expansion of a renormalized interacting composite. We also state and prove the finite-dimensional certificate criterion, including its catalogue-completeness and finite-sampling requirements. No renormalized second-order relational product, complete finite-renormalization response space, or radiative response matrix is constructed here. The quantum certification problem is therefore -open. The closed result is the classical descent theorem together with the compact-switch obstruction, not a quantum renormalization separator.

{ {0.945} Scope. This paper proves a second-order classical descent statement and a first-order obstruction to a specified switched realization of that statement. It does not prove a quantum Ward identity or a scheme-independent radiative correction.

Excluded endpoints. No ultraviolet completion, adiabatic limit, interacting Hilbert-space construction, localized probe, dense transverse-traceless representability theorem, or global multi-graviton contact classification is asserted. }

A relational scalar functional can be defined geometrically before an interaction switch is introduced. A retarded interacting field can also be defined perturbatively after an interaction is multiplied by a compact test function. These operations do not commute merely because both use retarded inverses.

The distinction is between

and

The first map has a natural coefficient gauge action inherited from diffeomorphisms. The second requires an intertwining statement between that action and the switched dynamics. We prove the first statement through second order and exhibit failure of the second for a specified naive identification.

This obstruction must be resolved before a finite-renormalization separator is interpreted as a gauge-invariant response.

Figure fig:VI-two-maps separates the geometric map from the proposed switched realization and records the required logical order.

Relational constructions and second-order expansions are already part of perturbative gravitational field theory BFHPR,Frob. In particular, Fröb computed gauge-invariant one-loop scalar correlation functions and exhibited cancellation between gauge-dependent self-energy contributions and relational corrections Frob. Neither the existence of relational observables nor the existence of such a cancellation is claimed as a new result here.

The locally covariant, renormalized Batalin–Vilkovisky treatment provides a framework for perturbative effective gravity BFR. It does not, by citation alone, establish the Ward identity of a newly specified boundary prescription or its compatibility with a fixed compact switch.

Hack treats the linearized Einstein–Klein–Gordon system on on-shell backgrounds Hack. Our background specialization is $(\eta ,0)$. The nonlinear coefficient problem studied below is not a linearization about a fixed nonzero scalar on Minkowski spacetime.

Retarded products and the Action Ward Identity are developed in DF. The Action Ward Identity concerns the dependence on local functionals rather than their density representatives. It must not be confused with a proved anomaly-free diffeomorphism Ward identity for the composite considered here.

The following distinctions are needed before formulating the theorems.

1. The pair $(\eta ,0)$, with zero cosmological constant, is an exact Einstein–scalar background solution. A nonzero free scalar on a fixed Minkowski metric is generally not an exact background solution at nonzero gravitational coupling.

2. The bare dressing coefficient is not the first interacting coefficient. If both constructions exist, the first interacting coefficient is

   $${{\mathcal{O}}_f}^{(1)}+R_1(V_1;{{\mathcal{O}}_f}^{(0)}).$$

   An external-metric scalar response supplies only the retarded term.

3. Turning on a second-order source does not retrospectively alter a previously fixed first-order coefficient. It changes the equations and data needed at the next order.

4. A second-order interacting coefficient requires products involving two first-order vertices. Consequently, the required $T_2(V_1,V_1)$ and three-insertion constructions cannot all be postponed until after that coefficient has supposedly been proved.

5. A fit on finitely many smearings does not prove a global counterterm identity. Conversely, a separator is valid only against the complete allowed response space.

The unresolved entries are not filled by assigning them theorem names. The obstruction branch gives a definite result without claiming the unperformed radiative calculation.

Let

Fix a finite time $T>0$, an initial surface $t=0$, and the slab

All fields are smooth on a neighborhood of the slab. Compact test functions used for observations and interactions have support in its interior.

The metric is denoted by $\gamma$; the letter $g$ denotes only an interaction switch.

For a symmetric tensor $a$, write

Then

The scalar first-variation operator is

In particular,

These identities follow by expanding the derivatives.

The action is

The curvature convention is fixed by requiring

Thus

For the matter action in (eq:action),

Variation with respect to the inverse metric gives

The factor $1/4$ belongs to the normalization $2/{\kappa }^2$. It cannot be removed from a coefficient equation without redefining the coupling or the source.

Work coefficientwise modulo ${\kappa }^3$, and write

Define $G^{(2)}[h,h]$ by

Similarly, define ${\mathsf{M}}_h$ by

An explicit formula for ${\mathsf{M}}_h$ is given in Section sec:scalar-expansion.

The space ${\mathcal{J}}_T$ consists of real smooth tuples $(h,j,{\varphi }_0,{\varphi }_1,{\varphi }_2)$ on $M_T$ satisfying

No assertion that arbitrary unconstrained metric data extend to this space is included in the definition.

The coefficient ${\varphi }_0$ is a free field configuration in a coupling expansion. It is not being declared the nonzero scalar of an exact Minkowski Einstein–scalar background.

Figure fig:VI-coefficient-domain organizes the named coefficient equations and their action-dependent source normalization.

Let

Gauge transformations are formal pullbacks

The corresponding transformations are the identity near the initial surface. They form a group of formal second-order transformations under composition. Closure follows from the Baker–Campbell–Hausdorff formula and closure of ${\mathfrak{g}}_T$ under the Lie bracket.

The use of compact support is a convenient sufficient condition. No residual-wave gauge group is substituted for this group.

The action (eq:formal-pullback) induces

It preserves ${\mathcal{J}}_T$ and all coefficient Cauchy data at $t=0$.

Proof. Expand the exponential in (eq:formal-pullback) and collect powers of $\kappa$ after substituting (eq:jets). This gives the displayed formulas.

The tensor identity

and the scalar identity

preserve the coefficient equations. These identities follow from the tensorial definitions of the Einstein tensor and the scalar wave operator. Equivalently, they may be verified by substituting the pullback formulas into their coordinate definitions and applying the chain rule.

Since $\xi ,\zeta$ vanish on a neighborhood of $t=0$, all displayed changes and their time derivatives vanish there.

Figure fig:VI-gauge-action displays the full coefficient action and the initial-surface support restriction.

At $(\eta ,0)$, the ordinary linearized scalar gauge variation vanishes, because the background scalar has zero derivative. With canonical metric normalization, the free Hessian and gauge map are

Metric indices in the first entry are raised when it is paired with a covariant variation.

For compact tests $(J,c)$, use

The formal adjoint gauge map is

The quotient

is well defined. Its elements induce gauge-invariant linear functionals on free solutions.

Proof. The linearized Bianchi identity implies ${\mathcal{K}}_0^{\ast }{\mathcal{P}}_0=0$, so the denominator is contained in the numerator. The free Hessian is formally self-adjoint: it is the second variation of the quadratic action, and its explicit expression also verifies this by integration by parts. Consequently, a test in the denominator pairs to zero with every solution of ${\mathcal{P}}_0(a,u)=0$.

A numerator test annihilates every smooth pure gauge:

All integrations by parts are justified by compactness of the tests.

We do not claim that (eq:linear-quotient) is a complete classification of all continuous observables, that its induced pairing has a specified radical, or that it already carries the nonlinear relational observable.

Figure fig:VI-linear-quotient summarizes the algebraic linear-observable quotient and its limited claim.

The decoupling in (eq:free-operators) is a consequence of the chosen zero-scalar background. It is not a prescription for assigning a direct-product symplectic form to a different triangular system with a fixed nonzero scalar. The general coupled-background construction is treated in Hack.

The metric in Definition def:jet-space is an unknown, not an external coefficient. We next exhibit solutions in which it is nontrivially sourced.

For a spatial Schwartz source define the Newton inverse

Only this inverse branch is used below.

Let ${\varphi }_0$ be any real Klein–Gordon solution with compact smooth Cauchy data. There is a tuple in ${\mathcal{J}}_T$ with

and with $j$ satisfying

For nonzero scalar data the source is nonzero.

Proof. Put

The identity

shows that $Q$ is conserved. Finite propagation makes $Q$ spatially compact on every bounded time interval.

At $t=0$, set

and

Prescribe trace-reversed metric data by

Solve

The Newton data are smooth, and the finite-time wave Cauchy problem therefore gives a smooth solution. Existence can also be obtained directly from the cosine and sine propagators. The multipliers $|p{|}^{-2}$ in the position data are locally integrable in three spatial dimensions.

Write $d_{\nu }=-{\partial }_t{\bar{j}}_{0\nu }+{\partial }_i{\bar{j}}_{i\nu }$. The data give $d_{\nu }(0)=0$. Since

equation (eq:sourced-reduced) gives ${\partial }_t{d}_{\nu }(0)=0$. Conservation of $Q$ implies ${\square }_{\eta }{d}_{\nu }=0$. Wave uniqueness gives $D(j)=0$, and (eq:G1) then gives $G^{(1)}(j)=Q$.

Finally, solve

with zero Cauchy data. The forcing is smooth and spatially compact on finite slabs because it contains derivatives of ${\varphi }_0$. The resulting tuple satisfies all equations of Definition def:jet-space.

For nonzero scalar data,

Thus $Q\ne 0$.

This proposition supplies a family, not a general constrained Einstein–scalar Cauchy theorem. It permits the Newton tails required by its chosen data.

Figure fig:VI-sourced-family shows the explicit sourced family and the propagation of its harmonic constraints.

For $h$, define

The operator in (eq:wave-expansion) is

Proof. The inverse metric and volume density are

Hence

Substitute this and

into the divergence formula for ${\square }_{\gamma }$. The first coefficient is

The part linear in $j$ is $L_j$. The remaining part is

which equals (eq:M).

For a smooth source $b$, let $G_{\square ,0}b$ denote the solution of

On $0\le t\le T$,

For sources without spatial decay, the formula is interpreted locally using finite propagation and compatible compact cutoffs. Values on a compact target set depend only on a compact truncated past region. Ordinary wave Cauchy theory justifies this construction BGP.

No inverse from an infinite past is needed.

Define four scalar coordinate functions, with an external coordinate label $\mu$, by

They are required to satisfy

For every smooth $h,j$, the conditions (eq:harmonic-coordinate-data) have the unique coefficient solution

Both coefficients and their first time derivatives vanish at $t=0$.

Proof. Since ${\square }_{\eta }{x}^{\mu }=0$ and $L_h{x}^{\mu }=-D(h{)}^{\mu }$, the first coefficient of ${\square }_{\gamma }{X}^{\mu }$ is

The second coefficient is

The zero-data wave problem gives the formulas and uniqueness.

The functions $X^{\mu }$ are scalar functions under spacetime pullback. Their index $\mu$ labels the four coordinate functions; it does not mean that $X^{\mu }$ transforms as the components of a vector field.

Figure fig:VI-harmonic-coordinates summarizes the finite-time coordinate equations and their fixed Cauchy data.

For the transformations in Proposition prop:gauge-action,

Consequently,

Proof. Scalar covariance gives

Because $F$ is the identity on a neighborhood of the initial surface, $F^{\ast }{X_{\gamma }}^{\mu }$ has exactly the data in (eq:harmonic-coordinate-data). Uniqueness in Proposition prop:coordinates proves (eq:coordinate-covariance). Expanding the pullback gives (eq:w-transform)–(eq:v-transform).

The formal inverse coordinate map is

The formula follows by substituting it into $X_{\gamma }(Y_{\gamma }(X))$.

For $f\in {C_c}^{\infty }((0,T)\times {\mathbb{R}}^3;\mathbb{R})$, define

This is a formal coordinate-space smearing. Its definition does not assume that a numerical nonlinear coordinate map exists globally.

On the jet space ${\mathcal{J}}_T$, the functional (eq:observable) has coefficients

Each coefficient is invariant under the formal gauge group (eq:formal-pullback). Hence ${\mathcal{O}}_f$ descends to the orbit space of ${\mathcal{J}}_T$.

Proof. Taylor expansion of $\varphi (Y_{\gamma }(X))$, using (eq:jets) and (eq:inverse-coordinate), gives (eq:O0-jet)–(eq:psi2).

For invariance, equation (eq:coordinate-covariance) says $X_{{\gamma }^{\prime }}=X_{\gamma }\circ F$. Therefore

Since ${\varphi }^{\prime }=F^{\ast }\varphi =\varphi \circ F$,

Smearing with the same fixed function of the relational labels proves coefficientwise invariance.

All coefficients are integrals of smooth functions against compact $f$. The coordinate coefficients at points of $\mathrm{supp}f$ depend on finite-time wave problems, so the definition is meaningful on the declared smooth domain.

The invariance proof uses coordinate covariance and fixed coordinate Cauchy data, not the Einstein or scalar equations. It therefore also gives an off-shell formal invariance statement. Restriction to ${\mathcal{J}}_T$ makes it an observable on the named solution-jet quotient. This does not establish quantum renormalizability of the functional.

Figure fig:VI-classical-descent illustrates the composition identity underlying coefficientwise classical descent.

Changing coordinate Cauchy data is not a gauge transformation in the group used above. For example, if $w$ is replaced by $w+a$, where ${\square }_{\eta }a=0$, then at first order

This need not vanish.

The following operations must therefore remain separate:

1. applying a diffeomorphism in the named gauge group;

2. changing a gauge-fixing term used for a perturbative construction;

3. changing coordinate initial data or the relational prescription;

4. changing solution Cauchy data.

Theorem thm:descent concerns only the first operation with the other choices fixed.

Figure fig:VI-prescription-choices separates gauge descent from changes of the relational or dynamical data.

For perturbative quantization one usually expands a functional of independent fields $(h,\phi )$, with $\gamma =\eta +\kappa h$, before substituting interacting solutions. Let

The bare geometric coefficients are

Round parentheses label bare functional coefficients. Square brackets in Theorem thm:descent label coefficients after substitution of a solution jet.

The relation between the two expansions is explicit:

Substituting

into the bare expansion gives exactly (eq:O0-jet)–(eq:psi2). In particular, one must not insert a sourced $j$ into a bare second-order term and then count the same dynamical correction again through an interacting-field map.

Fix

For a smooth external tensor $h$, the matter vertex is

For $H=gh$, define

It is the scalar Hessian contribution of (eq:matter-vertex), and it is formally self-adjoint. Direct expansion gives

Hence, on a free scalar,

Let $G_{\mathrm{R}}$ be the retarded inverse of $P$, with time kernel

Since $g$ is compactly supported after $t=0$, the first scalar response with zero past data is

For $P{\varphi }_0=0$,

where