Observable Quotients and Exact Projected Dynamics: Closure, Memory, Minimal Dynamical Completion, and Effective Field Operators

Effective metric response

Jeremy RodgersSection 11 of 14Original PDF ↗Publication DOI ↗

Effective metric response

We finally allow the metric to participate, and ask what elimination of $\chi$ does to the gravitational field equations. The answer is a stationary-elimination (envelope) identity, which converts the eliminated sector into a covariant variational response rather than an unexplained additive term.

Assumption 10.1.

Let $M$ be a smooth spacetime manifold. Let the metrics, visible fields, and completion fields range over open subsets $\mathcal{G}$ , $\mathcal{F}_{\varphi}$ , $\mathcal{F}_{\chi}$ of Banach spaces (equivalently, over Banach manifolds modelled on them, for definiteness one may take suitable Sobolev or weighted Hölder sections of the relevant bundles over $M$ , with the boundary or fall-off conditions specified below built into the space). Let $S:\mathcal{G}\times\mathcal{F}_{\varphi}\times\mathcal{F}_{\chi}\longrightarrow\mathbb{R}$ be a real, diffeomorphism-invariant action functional which is continuously Fréchet differentiable ( $C^{1}$ ) on its domain, and suppose the boundary/fall-off conditions defining the spaces are such that all integrations by parts performed below produce no boundary contributions, and such that the Fréchet derivatives are represented by the densities $\delta S/\delta g^{\mu\nu}$ , $\delta S/\delta\varphi$ , $\delta S/\delta\chi$ in the usual way, $DS[g,\varphi,\chi]\,(h,\psi,\theta) =\int_{M}\Bigl(\frac{\delta S}{\delta g^{\mu\nu}}h^{\mu\nu} +\frac{\delta S}{\delta\varphi}\psi +\frac{\delta S}{\delta\chi^{I}}\theta^{I}\Bigr)d^{4}x .$ Suppose a branch $(g,\varphi)\ \longmapsto\ \chi_{*}[g,\varphi]\tag{10.1}$ of solutions of the eliminated field equation $\left.\frac{\delta S}{\delta\chi(x)}\right|_{\chi=\chi_{*}[g,\varphi]}=0 \qquad\text{for all }x\in M\tag{10.2}$ has been selected, and that this branch is a continuously Fréchet differentiable map $\chi_{*}:\mathcal{G}\times\mathcal{F}_{\varphi}\longrightarrow\mathcal{F}_{\chi} .$ (In practice such a branch is produced, where it exists, by the implicit function theorem in Banach spaces [18, Ch. 4], applied to the Euler map $(g,\varphi,\chi)\mapsto\delta S/\delta\chi$ at a solution where the linearization $\delta^{2}S/\delta\chi^{2}$ is a Banach-space isomorphism. That theorem requires the Euler map itself to be continuously Fréchet differentiable, which does not follow from $S\in C^{1}$ ; so for this remark, and for this remark only, assume additionally that $S\in C^{2}$ , or equivalently for present purposes that $(g,\varphi,\chi)\mapsto\delta S/\delta\chi$ is itself continuously Fréchet differentiable. Under that extra hypothesis the implicit function theorem also delivers the $C^{1}$ dependence of $\chi_{*}$ assumed above. We emphasize that this strengthening is not imposed on Assumption 10.1 or used in Proposition 10.2: the envelope identity needs only $S\in C^{1}$ together with the separately assumed $C^{1}$ branch $\chi_{*}$ , however that branch is obtained. We do not verify any of these hypotheses for a particular action; we assume them.) Define the reduced action $\Gamma_{\mathrm{eff}}[g,\varphi]:=S\bigl[g,\varphi,\chi_{*}[g,\varphi]\bigr].\tag{10.3}$

We stress the three hypotheses that are doing real work and that are frequently left tacit: (i) differentiability of the branch $\chi_{*}$ , a solution of (10.2) need not depend differentiably on $(g,\varphi)$ , and where several branches exist a choice has been made; (ii) boundary conditions, the variation $\delta\chi_{*}$ induced by a variation of $g$ must be an admissible variation, i.e. satisfy the same boundary conditions, or the envelope argument fails; (iii) equivariance of the branch, the assignment (10.1) must intertwine the diffeomorphism action, $\chi_{*}[\phi^{*}g,\phi^{*}\varphi] =\phi^{*}\chi_{*}[g,\varphi]$ for diffeomorphisms $\phi$ , if $\Gamma_{\mathrm{eff}}$ is to inherit the diffeomorphism invariance of $S$ . Without (iii), $S$ can be invariant while $\Gamma_{\mathrm{eff}}$ is not, because the selection of the branch may break the symmetry.

Proposition 10.2 (Stationary-elimination / envelope identity).

Under Assumption 10.1, $\boxed{\ \ \frac{\delta\Gamma_{\mathrm{eff}}[g,\varphi]}{\delta g^{\mu\nu}(x)} =\left.\frac{\delta S[g,\varphi,\chi]}{\delta g^{\mu\nu}(x)}\right|_{\chi=\chi_{*}[g,\varphi]} \ \ }\tag{10.4}$ and likewise $\dfrac{\delta\Gamma_{\mathrm{eff}}}{\delta\varphi(x)} =\left.\dfrac{\delta S}{\delta\varphi(x)}\right|_{\chi_{*}}$ . That is, the metric (and visible-field) derivative of the reduced action is computed by differentiating $S$ at fixed $\chi$ and then evaluating on the branch: the implicit dependence of $\chi_{*}$ on $g$ contributes nothing.

Proof. Since $S$ and $\chi_{*}$ are continuously Fréchet differentiable, so is the composite $\Gamma_{\mathrm{eff}}=S\circ(\mathrm{id},\mathrm{id},\chi_{*})$ , and the standard Fréchet chain rule for a composition of continuously Fréchet differentiable maps between Banach spaces [18, Ch. 4] applies without further hypotheses: $D\Gamma_{\mathrm{eff}}[g,\varphi]=D_{g,\varphi}S\big|_{\chi_{*}} +D_{\chi}S\big|_{\chi_{*}}\circ D\chi_{*}[g,\varphi] .$ Writing this in terms of the representing densities, with the spacetime integration and field indices explicit, and with $\chi$ carrying whatever index structure $I$ it possesses, it reads $\frac{\delta\Gamma_{\mathrm{eff}}[g,\varphi]}{\delta g^{\mu\nu}(x)} =\left.\frac{\delta S}{\delta g^{\mu\nu}(x)}\right|_{\chi_{*}} +\int_{M}\!d^{4}y\; \left.\frac{\delta S}{\delta\chi^{I}(y)}\right|_{\chi_{*}} \frac{\delta\chi_{*}^{I}(y)}{\delta g^{\mu\nu}(x)} ,\tag{10.5}$ with summation over $I$ . The induced variation $\delta\chi_{*}$ is an admissible variation, it lands in the tangent space to $\mathcal{F}_{\chi}$ , hence satisfies the boundary conditions built into that space, which is what licenses writing the second term as an integral against $\delta S/\delta\chi^{I}$ with no boundary contribution. By the eliminated field equation (10.2), the factor $\delta S/\delta\chi^{I}(y)\big|_{\chi_{*}}$ vanishes identically in $y$ , so the entire integral in (10.5) vanishes, leaving (10.4). The argument for $\delta/\delta\varphi$ is identical, replacing $g^{\mu\nu}(x)$ by $\varphi(x)$ . ◻

Proposition 10.2 is what makes the reduction usable: to compute the gravitational response of the reduced theory one does not need $\delta\chi_{*}/\delta g$ at all, an object which is typically the solution of a linearized equation and thoroughly inaccessible.

Definition 10.3 (Effective stress-energy).

Split the reduced action into a visible part and a correction, $\Gamma_{\mathrm{eff}}[g,\varphi]=S_{\mathrm{visible}}[g,\varphi]+\Gamma_{\mathrm{corr}}[g,\varphi],\tag{10.6}$ where, by declaration, $S_{\mathrm{visible}}$ contains the gravitational sector and the visible matter sector in the conventions $S_{\mathrm{visible}}[g,\varphi] =\frac{1}{16\pi G_{N}}\int_{M}\!\bigl(R-2\Lambda\bigr)\sqrt{-g}\,d^{4}x \;+\;S_{\mathrm{mat}}[g,\varphi], \qquad T^{\mathrm{visible}}_{\mu\nu} :=-\frac{2}{\sqrt{-g}}\frac{\delta S_{\mathrm{mat}}}{\delta g^{\mu\nu}},\tag{10.7}$ and $\Gamma_{\mathrm{corr}}:=\Gamma_{\mathrm{eff}}-S_{\mathrm{visible}}$ is whatever the elimination of $\chi$ has added. We assume, as part of the declaration (10.7), that the visible matter action $S_{\mathrm{mat}}$ is itself diffeomorphism invariant; the Einstein–Hilbert term is invariant automatically, so $S_{\mathrm{visible}}$ is an invariant functional. Consequently, whenever $\Gamma_{\mathrm{eff}}$ is invariant, which, by Assumption 10.1 and the equivariance of the branch, is the case under the hypotheses of Theorem 10.5, the correction $\Gamma_{\mathrm{corr}}=\Gamma_{\mathrm{eff}}-S_{\mathrm{visible}}$ is a difference of invariant functionals and is therefore separately invariant as well. This is what entitles us to call both $T^{\mathrm{visible}}_{\mu\nu}$ and $T^{\mathrm{eff}}_{\mu\nu}$ covariant tensorial responses in Remark 10.6: each is the metric variation of an invariant functional, and each therefore satisfies the equivariance law (10.17). Separate invariance of the two functionals must not, however, be confused with separate conservation of the two tensors, which Theorem 10.5 does not assert; the two sectors do each carry an off-shell Noether identity, but they are coupled on shell by an exchange current, as the proof of that theorem makes precise. Define the effective stress-energy tensor of the eliminated sector by the same variational rule: $\boxed{\ \ T^{\mathrm{eff}}_{\mu\nu} :=-\frac{2}{\sqrt{-g}}\,\frac{\delta\Gamma_{\mathrm{corr}}}{\delta g^{\mu\nu}} . \ \ }\tag{10.8}$

The point of (10.8) is that it is a definition by variation of an explicitly constructed functional $\Gamma_{\mathrm{corr}}$ , and (10.6) tells us exactly what that functional is. Contrast the common alternative: take a field equation one does not like, move the offending term to the right-hand side of Einstein’s equation, and call the result a stress-energy tensor. That rearrangement is not empty, it does define a symbol, and algebraically the resulting equation is true by construction, but it establishes, by itself, none of the three properties one wants of a stress-energy tensor: it exhibits no variational origin (no functional whose metric derivative the term is), it does not by itself certify that the object transforms as a $(0,2)$ -tensor, and it yields no conservation law. Definition 10.3 supplies all three: the functional is $\Gamma_{\mathrm{corr}}$ from (10.6), the tensorial transformation follows because $T^{\mathrm{eff}}$ is the metric variation of a diffeomorphism-invariant functional (Remark 10.6), and the conservation law is Theorem 10.5. That is the difference between a rearrangement and a derivation.

Proposition 10.4 (Effective Einstein equations).

With the conventions (10.7), stationarity of $\Gamma_{\mathrm{eff}}$ with respect to $g^{\mu\nu}$ , i.e. $\delta\Gamma_{\mathrm{eff}}/\delta g^{\mu\nu}=0$ , is equivalent to $G_{\mu\nu}+\Lambda g_{\mu\nu} =8\pi G_{N}\Bigl(T^{\mathrm{visible}}_{\mu\nu}+T^{\mathrm{eff}}_{\mu\nu}\Bigr),\tag{10.9}$ where $G_{\mu\nu}=R_{\mu\nu}-\tfrac12 R\,g_{\mu\nu}$ .

Proof. Vary (10.6). The Einstein–Hilbert term in (10.7) has the standard variation $\frac{\delta}{\delta g^{\mu\nu}}\frac{1}{16\pi G_{N}}\int(R-2\Lambda)\sqrt{-g}\,d^{4}x =\frac{\sqrt{-g}}{16\pi G_{N}}\Bigl(R_{\mu\nu}-\tfrac12 Rg_{\mu\nu}+\Lambda g_{\mu\nu}\Bigr) =\frac{\sqrt{-g}}{16\pi G_{N}}\bigl(G_{\mu\nu}+\Lambda g_{\mu\nu}\bigr),$ using $\delta\sqrt{-g}=-\tfrac12\sqrt{-g}\,g_{\mu\nu}\delta g^{\mu\nu}$ and discarding the total-derivative term by the boundary hypothesis [16, §4.3]. By (10.7) and (10.8) the matter and correction terms contribute $-\tfrac{\sqrt{-g}}{2}T^{\mathrm{visible}}_{\mu\nu}$ and $-\tfrac{\sqrt{-g}}{2}T^{\mathrm{eff}}_{\mu\nu}$ respectively. Setting the sum to zero and multiplying by $16\pi G_{N}/\sqrt{-g}$ gives (10.9). ◻

Theorem 10.5 (Total covariant conservation).

Suppose in addition to Assumption 10.1 that the branch $\chi_{*}$ is diffeomorphism-equivariant, so that $\Gamma_{\mathrm{eff}}$ is a diffeomorphism-invariant functional of $(g,\varphi)$ . Then, on solutions of the effective visible-matter field equation $\delta\Gamma_{\mathrm{eff}}/\delta\varphi=0$ , $\nabla^{\mu}\Bigl(T^{\mathrm{visible}}_{\mu\nu}+T^{\mathrm{eff}}_{\mu\nu}\Bigr)=0 .\tag{10.10}$ In general, neither $\nabla^{\mu}T^{\mathrm{visible}}_{\mu\nu}=0$ nor $\nabla^{\mu}T^{\mathrm{eff}}_{\mu\nu}=0$ holds separately.

Proof. Let $\xi$ be a compactly supported smooth vector field on $M$ and let $\phi_{\varepsilon}$ be its flow. Diffeomorphism invariance of $\Gamma_{\mathrm{eff}}$ gives $\Gamma_{\mathrm{eff}}[\phi_{\varepsilon}^{*}g,\phi_{\varepsilon}^{*}\varphi]=\Gamma_{\mathrm{eff}}[g,\varphi]$ for all $\varepsilon$ ; differentiating at $\varepsilon=0$ and using $\delta_{\xi}g^{\mu\nu}=\mathcal{L}_{\xi}g^{\mu\nu}=-(\nabla^{\mu}\xi^{\nu}+\nabla^{\nu}\xi^{\mu})$ and $\delta_{\xi}\varphi=\mathcal{L}_{\xi}\varphi$ , we obtain the Noether identity $0=\int_{M}\Bigl( \frac{\delta\Gamma_{\mathrm{eff}}}{\delta g^{\mu\nu}}\,\mathcal{L}_{\xi}g^{\mu\nu} +\frac{\delta\Gamma_{\mathrm{eff}}}{\delta\varphi}\,\mathcal{L}_{\xi}\varphi \Bigr)d^{4}x .\tag{10.11}$ Restrict now to field configurations satisfying the visible-matter equation of the reduced theory, $\delta\Gamma_{\mathrm{eff}}/\delta\varphi=0$ ; the second term of (10.11) then drops. Split $\delta\Gamma_{\mathrm{eff}}/\delta g^{\mu\nu}$ using (10.6), (10.7), (10.8): writing $\frac{\delta\Gamma_{\mathrm{eff}}}{\delta g^{\mu\nu}} =\frac{\sqrt{-g}}{16\pi G_{N}}\bigl(G_{\mu\nu}+\Lambda g_{\mu\nu}\bigr) -\frac{\sqrt{-g}}{2}\Bigl(T^{\mathrm{visible}}_{\mu\nu}+T^{\mathrm{eff}}_{\mu\nu}\Bigr) =:\frac{\sqrt{-g}}{16\pi G_{N}}\,\mathcal{E}_{\mu\nu},$ and substituting into (10.11) with $\mathcal{L}_{\xi}g^{\mu\nu}=-2\nabla^{(\mu}\xi^{\nu)}$ gives $0=-\frac{2}{16\pi G_{N}}\int_{M}\sqrt{-g}\;\mathcal{E}_{\mu\nu}\,\nabla^{\mu}\xi^{\nu}\,d^{4}x =\frac{2}{16\pi G_{N}}\int_{M}\sqrt{-g}\;\bigl(\nabla^{\mu}\mathcal{E}_{\mu\nu}\bigr)\,\xi^{\nu}\,d^{4}x ,$ after one covariant integration by parts, the boundary term vanishing because $\xi$ has compact support. As $\xi$ is arbitrary, $\nabla^{\mu}\mathcal{E}_{\mu\nu}=0$ . Finally, the contracted Bianchi identity gives $\nabla^{\mu}(G_{\mu\nu}+\Lambda g_{\mu\nu})=0$ identically (metric-compatibility making $\nabla^{\mu}g_{\mu\nu}=0$ ), so $\nabla^{\mu}\bigl(T^{\mathrm{visible}}_{\mu\nu}+T^{\mathrm{eff}}_{\mu\nu}\bigr)=0$ , which is (10.10).

For the final sentence, the situation must be described with care, because $S_{\mathrm{visible}}$ and $\Gamma_{\mathrm{corr}}$ are separately diffeomorphism invariant (Definition 10.3), and each therefore does possess its own off-shell Noether identity, obtained by running the argument above on that functional alone. To state these without committing to the tensor type of $\varphi$ , introduce the Ward operator $\mathcal{W}_{\nu}$ : for any diffeomorphism-invariant functional $F[g,\varphi]$ with associated stress tensor $T^{F}_{\mu\nu}:=-\tfrac{2}{\sqrt{-g}}\,\delta F/\delta g^{\mu\nu}$ and Euler operator $E^{F}_{\varphi}:=\delta F/\delta\varphi$ , the same computation as above, with $\delta_{\xi}\varphi=\mathcal{L}_{\xi}\varphi$ and one integration by parts, yields the off-shell identity $\nabla^{\mu}T^{F}_{\mu\nu}=\mathcal{W}_{\nu}\bigl(E^{F}_{\varphi}\bigr),\tag{10.12}$ where $\mathcal{W}_{\nu}$ is the formal adjoint of the map $\xi^{\nu}\mapsto\mathcal{L}_{\xi}\varphi$ , contracted appropriately; it is linear in its argument, and its precise form depends on the tensor type of $\varphi$ but not on $F$ . For a scalar field $\varphi$ it reduces to $\mathcal{W}_{\nu}(E_{\varphi}) =\frac{1}{\sqrt{-g}}\,E_{\varphi}\,\nabla_{\nu}\varphi , \qquad\text{so that}\qquad \nabla^{\mu}T_{\mu\nu} =\frac{1}{\sqrt{-g}}\,\frac{\delta F}{\delta\varphi}\,\nabla_{\nu}\varphi ,\tag{10.13}$ in the conventions $T_{\mu\nu}=-\tfrac{2}{\sqrt{-g}}\,\delta F/\delta g^{\mu\nu}$ and $\delta_{\xi}\varphi=\mathcal{L}_{\xi}\varphi$ fixed in (10.7) and above. We stress that (10.13) is the scalar case; (10.12) is the statement for general $\varphi$ , and the scalar formula must not be read as an identity for tensor-valued $\varphi$ .

Applying (10.12) to $S_{\mathrm{visible}}$ and to $\Gamma_{\mathrm{corr}}$ , each invariant, hence each entitled to its own identity, and using the contracted Bianchi identity on the gravitational part of $S_{\mathrm{visible}}$ , we obtain, as identities holding for all field configurations, $\begin{aligned} \nabla^{\mu}T^{\mathrm{visible}}_{\mu\nu} &=\mathcal{W}_{\nu}\Bigl(\frac{\delta S_{\mathrm{visible}}}{\delta\varphi}\Bigr) \;=:\;J^{\mathrm{vis}}_{\nu}, \qquad\text{(10.14)}\\ \nabla^{\mu}T^{\mathrm{eff}}_{\mu\nu} &=\mathcal{W}_{\nu}\Bigl(\frac{\delta\Gamma_{\mathrm{corr}}}{\delta\varphi}\Bigr) \;=:\;J^{\mathrm{corr}}_{\nu} . \qquad\text{(10.15)} \end{aligned}$

So the two sectors do have Ward identities, and the reason separate conservation nevertheless fails is not an absence of identities but the on-shell condition. The physical configuration satisfies the field equation of the combined reduced theory, $\frac{\delta\Gamma_{\mathrm{eff}}}{\delta\varphi} =\frac{\delta S_{\mathrm{visible}}}{\delta\varphi}+\frac{\delta\Gamma_{\mathrm{corr}}}{\delta\varphi}=0 ,\tag{10.16}$ which does not imply $\delta S_{\mathrm{visible}}/\delta\varphi=0$ or $\delta\Gamma_{\mathrm{corr}}/\delta\varphi=0$ separately, it implies only that the two Euler operators are equal and opposite. Since $\mathcal{W}_{\nu}$ is linear, applying it to (10.16) gives $J^{\mathrm{vis}}_{\nu}+J^{\mathrm{corr}}_{\nu} =\mathcal{W}_{\nu}\Bigl(\frac{\delta S_{\mathrm{visible}}}{\delta\varphi} +\frac{\delta\Gamma_{\mathrm{corr}}}{\delta\varphi}\Bigr) =\mathcal{W}_{\nu}(0)=0 ,$ so the two sector Ward terms are equal and opposite and cancel in the sum of (10.14) and (10.15), which is exactly (10.10), while neither need vanish on its own. The common quantity $J^{\mathrm{vis}}_{\nu}=-J^{\mathrm{corr}}_{\nu}$ is the exchange current between the visible and eliminated sectors: it is the rate at which the eliminated sector feeds energy–momentum to the visible one.

The logical structure is therefore this. Separate diffeomorphism invariance supplies separate off-shell identities, (10.14) and (10.15), and nothing more. Separate conservation of a sector’s stress tensor requires, in addition, that that sector’s Ward term vanish on the combined solution, i.e. that $\mathcal{W}_{\nu}(\delta S_{\mathrm{visible}}/\delta\varphi)=0$ there, equivalently (by (10.16) and linearity) that $\mathcal{W}_{\nu}(\delta\Gamma_{\mathrm{corr}}/\delta\varphi)=0$ there. This holds, for instance, when $\Gamma_{\mathrm{corr}}$ does not depend on $\varphi$ at all, or when the visible field happens to satisfy its own equation $\delta S_{\mathrm{visible}}/\delta\varphi=0$ on the combined solution, or when $\mathcal{W}_{\nu}$ annihilates the relevant Euler operator. It is not supplied by invariance: diffeomorphism invariance of a sector gives that sector its Ward identity, but it does not put the sector on shell, and it therefore cannot by itself force the exchange current to vanish. Absent such a structure, the eliminated sector exchanges energy–momentum with the visible one, and only the sum is conserved. ◻

Remark 49 (Invariance versus covariance).

Three statements must be kept apart, and they are frequently run together.

$\Gamma_{\mathrm{eff}}$ is an invariant scalar functional: under the hypotheses of Theorem 10.5 we have $\Gamma_{\mathrm{eff}}[\phi^{*}g,\phi^{*}\varphi]=\Gamma_{\mathrm{eff}}[g,\varphi]$ . This is a statement about a single number assigned to a whole field configuration.
$T^{\mathrm{eff}}_{\mu\nu}$ is a covariant tensorial response: it transforms equivariantly as a $(0,2)$ -tensor field under diffeomorphisms, because it is the metric variation of an invariant functional. Geometrically it is a section of the associated tensor bundle $\mathrm{Sym}^{2}T^{*}M$ , evaluated on the given configuration, and the diffeomorphism group acts on it by pullback rather than fixing it: $T^{\mathrm{eff}}\bigl[\phi^{*}g,\phi^{*}\varphi\bigr] =\phi^{*}\,T^{\mathrm{eff}}\bigl[g,\varphi\bigr],\tag{10.17}$ in the same pullback convention used for the metric and the fields throughout. Consequently it is not an ordinary pointwise scalar function on the naive quotient of field space by diffeomorphisms: the value “ $T^{\mathrm{eff}}_{\mu\nu}(x)$ ” is not by itself a diffeomorphism-invariant number, because the label $x$ is not. Three levels must be kept apart here, and the second is routinely mistaken for the third.
- Tensorial response. $T^{\mathrm{eff}}_{\mu\nu}$ itself: an equivariant section of $\mathrm{Sym}^{2}T^{*}M$ , transforming by pullback as in (10.17).
- Covariant scalar fields obtained by contraction. Objects such as the trace $T^{\mathrm{eff}}:=g^{\mu\nu}T^{\mathrm{eff}}_{\mu\nu}$ , or $T^{\mathrm{eff}}_{\mu\nu}T^{\mathrm{eff}\,\mu\nu}$ . These are scalar fields: they carry no free indices and transform as scalars. But a scalar field is still a function on $M$ , and it is still not a diffeomorphism-invariant observable when evaluated at a coordinate-labelled point: under $\phi$ it goes to $\phi^{*}T^{\mathrm{eff}}$ , so the number $T^{\mathrm{eff}}(x)$ at a fixed coordinate label $x$ is not invariant. Contraction removes the indices; it does not remove the dependence on the labelling.
- Genuine diffeomorphism-invariant observables. These require a further step: invariant integration (e.g. $\int_{M}T^{\mathrm{eff}}\sqrt{-g}\,d^{4}x$ over an invariantly specified region, or an integral against an invariantly constructed density), smearing against equivariantly chosen test tensors, or relational evaluation, evaluating one field at a point located by the values of other fields, so that the point is specified by the configuration rather than by a coordinate.
The equivariant response is the primary object; covariant scalar fields are contractions of it; invariant observables are constructed from those by one of the three routes just named.
The effective field equations (10.9) are geometrically covariant: both sides are $(0,2)$ -tensors and the equation is preserved under diffeomorphisms. Covariance of the equations is a consequence of invariance of the functional; it is not the same statement, and it does not license treating $T^{\mathrm{eff}}$ as an invariant.