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Linearized field equations as the first-order consequence of the adopted field law, with the harmonic-gauge wave reduction and the quadratic residual control

Physical scope

Conditional deduction inside the classical gravitational framework: a one-parameter family of models of the adopted field law on an open set of the flat model, with smooth metric and stress-energy families, expanded to first order about the flat metric. The conclusion is the linearized equation with the first-order stress coefficient as its source, its vacuum and harmonic-gauge reductions, and, on the zero-source Lambda = 0 branch, a compact-set bound on the exact vacuum residual of a small-amplitude metric. The all-space harmonic-gauge existence clause additionally assumes the Axiom of Countable Choice. Excludes nonlinear back-reaction, non-smooth or distributional sources, curved backgrounds, boundaries, quantum fields and any comparison with observation; the stress-energy family is model input, not derived from the metric perturbation.

Statement

Adopt the field-law postulate Adopted Einstein field equation in the direct-equation presentation with its constants $Λ \in R$ , $G_{N} > 0$ , $c > 0$ , together with Adopted spacetime model and classical domain of the gravitational framework and Adopted matter-content assumptions: fields, equations, stress-energy and energy conditions (SI unit and coordinate conventions of the classical gravitational framework). Let $(M, g, N) = (R^{4}, η, \partial_{0})$ be the flat model with its inertial chart, let $Ω \subseteq R^{4}$ be a nonempty open set, let $(ε_{1}, h, q)$ be a flat linearized family over $Ω$ with $ε_{1} \leq ε$ , and let $(T (λ))_{λ \in (- ε, ε)}$ be a family of smooth symmetric covariant $2$ -tensor fields on $Ω$ whose component families $T_{a b} (λ, p)$ are smooth in $(λ, p)$ and such that for every $λ \in (- ε_{1}, ε_{1})$ the pair $(g (λ), T (λ))$ satisfies the adopted field law

$G_{a b} (g (λ)) + Λ g_{a b} (λ) = κ T_{a b} (λ) on Ω, κ : = \frac{8 π G_{N}}{c^{4}},$

$G$ being the Einstein tensor of the metric-selected connection (The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors, Metric compatibility, torsion freedom and the metric-selected connection class). Define $T_{a b}^{(1)} : = \partial_{λ} T_{a b} (λ) ∣_{λ = 0}$ . Then:

(a) Linearized field equation with source. On $Ω$ ,

$G_{a b}^{(1)} (h) + Λ h_{a b} = κ T_{a b}^{(1)},$

where $G^{(1)}$ is the first-order Einstein coefficient of Flat linearized perturbations: families, trace reversal, gauge transformation and harmonic gauge. In particular the linearized equation of the model is not an independent postulate: it is the first-order consequence of the adopted field law along the family.

(b) Vacuum and harmonic-gauge reduction. If $T (λ) = 0$ for every $λ$ and $Λ = 0$ , then $G_{a b}^{(1)} (h) = 0$ ; if in addition $h$ satisfies the harmonic-gauge condition $\partial^{a} {\overset{ˉ}{h}}_{a b} = 0$ (Flat linearized perturbations: families, trace reversal, gauge transformation and harmonic gauge clause (d)), then $□ {\overset{ˉ}{h}}_{a b} = 0$ componentwise on $Ω$ .

(c) Sourced wave equation in harmonic gauge. If $h$ is in harmonic gauge and solves the linearized equation with source $S$ and parameter $Λ$ in the sense of Plane-wave perturbations of the flat linearization: planar profiles, wave covectors, amplitudes, TT form and the linearized field equations clause (e), then $□ {\overset{ˉ}{h}}_{a b} = 2 Λ h_{a b} - 2 S_{a b}$ on $Ω$ .

(d) Quadratic residual of a linearized vacuum wave. Let $K \subseteq Ω$ be a nonempty compact set and let $L > 0$ be a length scale. In the fixed SI numerical chart define the real control functional

$B_{K} (A) : = \sup_{K} \max_{a, b} ∣ A_{a b} ∣ + L^{- 1} \sup_{K} \max_{a, b, c} ∣ \partial_{c} A_{a b} ∣ + L^{- 2} \sup_{K} \max_{a, b, c, d} ∣ \partial_{c} \partial_{d} A_{a b} ∣ .$

Take the fixed dimensionless numerical threshold $δ_{0} : = 1 / 16$ . If $h$ is a smooth symmetric perturbation on a neighborhood of $K$ with $G_{a b}^{(1)} (h) = 0$ on $K$ , then there is a finite dimensionless numerical constant $C_{K, L}$ such that, for every real $λ$ with $∣ λ ∣ B_{K} (h) \leq δ_{0}$ ,

$∣ G_{a b} (η + λ h) (p) ∣ \leq C_{K, L} λ^{2} B_{K} (h)^{2} / L^{2} (p \in K) .$

The functional $B_{K}$ and threshold are numerical chart controls, not physical observables; each component and derivative is first evaluated in the fixed SI length-coordinate convention. With $L$ in metres, the displayed residual has units $m^{- 2}$ . For the zero-source branch $T (λ) = 0$ with $Λ = 0$ , this bounds the residual of the exact vacuum field law on $K$ .

(e) All-space harmonic gauge (assuming ${A C}_{ω}$ ). If $Ω = R^{4}$ , $h$ solves the linearized vacuum equations with $Λ = 0$ , and the Axiom of Countable Choice is assumed (The Axiom of Countable Choice ( ${A C}_{ω}$ )), then there is a smooth covector field $ξ$ on $R^{4}$ such that $h_{a b}^{'} : = h_{a b} - \partial_{a} ξ_{b} - \partial_{b} ξ_{a}$ is in harmonic gauge, $G_{a b}^{(1)} (h^{'}) = 0$ and $□ {\overset{ˉ}{h}}_{a b}^{'} = 0$ componentwise.

Status. Clauses (a)-(e) are conditional deductions from the adopted field law and the cited mathematics for the stated families; they assert no observation, no measured constant and no fact about nature. The item contains no empirical premise.

Facts & Assumptions

Given: The adopted postulates, a flat linearized family $(ε_{1}, h, q)$ over $Ω$ , a smooth family $T (λ)$ of symmetric fields satisfying the field law for every $λ$ , and a nonempty compact set $K \subseteq Ω$ with $L > 0$ ; for clause (e), also assume ${A C}_{ω}$ .

[A1]

The model satisfies the adopted field law $G_{a b} + Λ g_{a b} = κ T_{a b}$ on its matter domain, with $κ = 8 π G_{N} / c^{4}$ ; the equation is model input, and the constants, the matter content and the domain are those of the cited postulates (Adopted Einstein field equation in the direct-equation presentation clauses (i)-(ii), Adopted spacetime model and classical domain of the gravitational framework, Adopted matter-content assumptions: fields, equations, stress-energy and energy conditions, SI unit and coordinate conventions of the classical gravitational framework).

[L1]

The family $g (λ)$ has the expansion $g_{a b} (λ, p) = η_{a b} + λ h_{a b} (p) + λ^{2} q_{a b} (λ, p)$ with smooth $h, q$ ; the component families of the metric-selected connection, curvature and Einstein tensor are smooth functions of $(λ, p)$ ; and the first-order coefficients of Flat linearized perturbations: families, trace reversal, gauge transformation and harmonic gauge are the $λ$ -derivatives at $0$ of these component families (Flat linearized perturbations: families, trace reversal, gauge transformation and harmonic gauge clauses (a)-(c), Metric compatibility, torsion freedom and the metric-selected connection class, The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors, $C^{r}$ and smooth maps between smooth manifolds).

[L2]

In harmonic gauge $\partial^{a} {\overset{ˉ}{h}}_{a b} = 0$ one has $G_{a b}^{(1)} (h) = - \frac{1}{2} □ {\overset{ˉ}{h}}_{a b}$ , and $G^{(1)}$ is unchanged under $h \mapsto h - \partial ξ - \partial ξ$ (First-order curvature of a flat linearized family, gauge transformations and the quadratic remainder clauses (c)-(e), Flat linearized perturbations: families, trace reversal, gauge transformation and harmonic gauge clauses (c)-(d)). For the all-space existence assertion, the nonnegative-radius spherical-mean kernel and zero-data Duhamel construction are those of All-space smooth wave equation with zero data, and harmonic gauge on flat space clauses (a)-(b); $H_{f} (x, 0) = 0$ and its right derivative at $s = 0$ is $f (x)$ . The signed extension used for negative time differences is smooth as shown in proof step 4.1 below, where The polar surface set function on the unit sphere and A linear map $T$ of $R^{n}$ sends Lebesgue measurable sets to Lebesgue measurable sets, with $λ_{n} (T [E]) = ∣ \det T ∣ λ_{n} (E)$ when $T$ is invertible and $T [E]$ Lebesgue null when it is not justify antipodal symmetry and hence preserves these initial data.

[L3]

In the inertial chart the Einstein tensor of $η + A$ is a smooth function of the finite component list $(A_{a b}, \partial_{c} A_{a b}, \partial_{c} \partial_{d} A_{a b})$ whenever $η + A$ is nondegenerate; at the zero list it vanishes, and its derivative there is $G^{(1)} (A)$ , which is linear in $A$ (Metric compatibility, torsion freedom and the metric-selected connection class, The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors, First-order curvature of a flat linearized family, gauge transformations and the quadratic remainder clauses (a), (e), Flat linearized perturbations: families, trace reversal, gauge transformation and harmonic gauge clause (b)). The Taylor estimate of The Lagrange and Cauchy forms of Taylor's remainder applies on compact subsets of this finite-dimensional jet domain, and matrices within maximum-entry distance $1 / 8$ of $η$ are Lorentzian (Lorentzian metrics are stable under small compactly supported symmetric perturbations clause (a)).

Proof

technique · direct; differentiate the field-law identity along the family at $\lambda=0$, then apply the harmonic-gauge reduction and the quadratic-remainder estimate

1.1A1L1given

(The residual family.) Define $Φ_{a b} (λ, p) : = G_{a b} (g (λ)) (p) + Λ g_{a b} (λ) (p) - κ T_{a b} (λ, p)$ for $(λ, p) \in (- ε_{1}, ε_{1}) \times Ω$ . By [L1] and the smoothness of $T$ , each component $Φ_{a b}$ is differentiable in $λ$ at every $λ$ . By hypothesis $Φ_{a b} (λ, p) = 0$ for all $(λ, p)$ , so $\partial_{λ} Φ_{a b} (0, p) = 0$ for every $p$ (the difference quotients of the identically zero function vanish).

1.2L2algebra

(Sourced wave equation.) If $h$ is in harmonic gauge and $G_{a b}^{(1)} (h) + Λ h_{a b} = S_{a b}$ , then substituting [L2] gives $- \frac{1}{2} □ {\overset{ˉ}{h}}_{a b} = S_{a b} - Λ h_{a b}$ , that is $□ {\overset{ˉ}{h}}_{a b} = 2 Λ h_{a b} - 2 S_{a b}$ . This proves (c).

1.3L3algebra

(Quadratic residual.) Let $h$ satisfy $G_{a b}^{(1)} (h) = 0$ on $K$ and choose $λ \in R$ with $∣ λ ∣ B_{K} (h) \leq δ_{0} = 1 / 16$ . Put $A : = λ h$ and at each $p \in K$ let $V_{A} (p)$ be the finite list $(A_{a b}, \partial_{c} A_{a b}, \partial_{c} \partial_{d} A_{a b})$ . By the displayed definition of $B_{K}$ , $∥ V_{A} (p) ∥_{\max} \leq m_{L} B_{K} (A)$ , where $m_{L} : = \max {1, L, L^{2}}$ ; in particular $∣ A_{a b} (p) ∣ \leq 1 / 16 < 1 / 8$ , so [L3] makes $η + A$ Lorentzian. The jet lists along all segments $t V_{A} (p)$ , $p \in K$ , $0 \leq t \leq 1$ , lie in the compact set given by $\max ∣ A_{a b} ∣ \leq 1 / 16$ , $\max ∣ \partial_{c} A_{a b} ∣ \leq L δ_{0}$ and $\max ∣ \partial_{c} \partial_{d} A_{a b} ∣ \leq L^{2} δ_{0}$ . On this set the smooth Einstein-tensor map of [L3] has Hessian bounded by a finite $H_{K, L}$ . Taylor's estimate along $t V_{A} (p)$ gives $∣ G_{a b} (η + A) (p) - G_{a b}^{(1)} (A) (p) ∣ \leq \frac{1}{2} H_{K, L} m_{L}^{2} B_{K} (A)^{2} = C_{K, L} B_{K} (A)^{2} / L^{2}$ , with $C_{K, L} : = \frac{1}{2} H_{K, L} m_{L}^{2} L^{2}$ . Since $B_{K} (A) = ∣ λ ∣ B_{K} (h)$ and $G_{a b}^{(1)} (A) = λ G_{a b}^{(1)} (h) = 0$ , this is (d).

2.1L1step 1.1algebra

(First-order equation.) Differentiating the definition of $Φ$ at $λ = 0$ and evaluating at $p$ , $0 = \partial_{λ} G_{a b} (g (λ)) ∣_{0} + Λ \partial_{λ} g_{a b} (λ) ∣_{0} - κ {\partial_{λ} T_{a b} (λ, p) ∣}_{λ = 0} .$ By [L1] the first term is the first-order Einstein coefficient $G_{a b}^{(1)} (h) (p)$ , the second is $Λ h_{a b} (p)$ , and the third is $κ T_{a b}^{(1)} (p)$ . Hence $G_{a b}^{(1)} (h) + Λ h_{a b} = κ T_{a b}^{(1)}$ on $Ω$ , which is (a).

3.1L2step 2.1algebra

(Vacuum branch and wave form.) If $T (λ) = 0$ for every $λ$ , then $T^{(1)} = 0$ , and with $Λ = 0$ step 2.1 gives $G_{a b}^{(1)} (h) = 0$ . If in addition $\partial^{a} {\overset{ˉ}{h}}_{a b} = 0$ , then [L2] gives $- \frac{1}{2} □ {\overset{ˉ}{h}}_{a b} = G_{a b}^{(1)} (h) = 0$ for every $a, b$ , so $□ {\overset{ˉ}{h}}_{a b} = 0$ . This proves (b).

4.1givenL2step 3.1algebra

(All-space harmonic gauge.) Let $Ω = R^{4}$ , let $h$ solve the linearized vacuum equations with $Λ = 0$ , and set $F_{b} (x, t) : = - \partial^{a} {\overset{ˉ}{h}}_{a b} (x, t)$ in the coordinates $x^{0} = c t$ . For $f \in C^{\infty} (R^{3})$ define the signed-radius mean ${\tilde{M}}_{r} f (x) : = (4 π)^{- 1} \int_{S^{2}} f (x + r ω) d σ (ω)$ for $r \in R$ . The integral is smooth in $(x, r)$ by differentiation under the integral on compact parameter sets, as in the proof of All-space smooth wave equation with zero data, and harmonic gauge on flat space clause (a). The antipodal substitution $ω \mapsto - ω$ preserves $σ$ : by the cone-volume definition in The polar surface set function on the unit sphere, the cone of $- E$ is the image of the cone of $E$ under $y \mapsto - y$ , a linear map with determinant $- 1$ in $R^{3}$ and hence preserves Lebesgue measure by A linear map $T$ of $R^{n}$ sends Lebesgue measurable sets to Lebesgue measurable sets, with $λ_{n} (T [E]) = ∣ \det T ∣ λ_{n} (E)$ when $T$ is invertible and $T [E]$ Lebesgue null when it is not. Thus ${\tilde{M}}_{- r} f = {\tilde{M}}_{r} f$ , and for $r \geq 0$ it agrees with $M_{r} f$ . The kernel ${\tilde{H}}_{f} (x, s) : = s {\tilde{M}}_{c s} f (x)$ is therefore smooth for signed $s$ , odd in $s$ , agrees with the kernel of [L2] for $s \geq 0$ , and satisfies $(c^{- 2} \partial_{s}^{2} - Δ_{x}) {\tilde{H}}_{f} = 0$ : for $s > 0$ this follows from All-space smooth wave equation with zero data, and harmonic gauge on flat space clause (a), for $s < 0$ by oddness, and at $s = 0$ by smoothness and continuity. The same moving-endpoint differentiation as in that theorem's proof (b), now with this smooth two-sided kernel, shows that $ξ_{b} (x, t) : = c^{2} \int_{0}^{t} {\tilde{H}}_{F_{b} (\cdot, v)} (x, t - v) d v$ is smooth on $R^{3} \times R$ and satisfies $(c^{- 2} \partial_{t}^{2} - Δ_{x}) ξ_{b} = F_{b}$ for every signed $t$ ; hence $- □ ξ_{b} = F_{b} = - \partial^{a} {\overset{ˉ}{h}}_{a b}$ and $□ ξ_{b} = \partial^{a} {\overset{ˉ}{h}}_{a b}$ . The gauge identity of [L2] then gives $\partial^{a} {\overset{ˉ}{h}}_{a b}^{'} = 0$ for $h_{a b}^{'} = h_{a b} - \partial_{a} ξ_{b} - \partial_{b} ξ_{a}$ . By gauge invariance in [L2] and the vacuum conclusion of step 3.1 applied only to $h$ , $G_{a b}^{(1)} (h^{'}) = G_{a b}^{(1)} (h) = 0$ ; applying the harmonic-gauge identity of [L2] to $h^{'}$ yields $□ {\overset{ˉ}{h}}_{a b}^{'} = 0$ . This proves (e).

5.1step 1.1step 1.2step 1.3step 2.1step 3.1step 4.1A1∎

(Status and conclusion.) Steps 1.1-4.1 use only the field-law hypothesis for the family, the differentiability of the component families and the cited mathematical reductions. No observation, apparatus, measured constant or statistical model is used, and the family hypothesis is precisely the adopted field law evaluated on the given families; therefore the conclusions are conditional model deductions.

Remarks

The source is model input. Clause (a) produces the source $κ T^{(1)}$ from the matter sector of the model; the theorem does not construct a metric family solving the exact field law for a prescribed matter content, and (d) shows only that a solution of the linearized vacuum equation has an exact vacuum residual of order $λ^{2}$ . The existence theory for the exact equations is a different module.
Presentations and the action formulation. A model may instead adopt the action formulation Adopted classical gravitational action and variational field law; by Stationarity of the adopted total action is equivalent to the Einstein equation with the framework's coupling its solutions satisfy the same field law, so the linearization above applies without change. The two presentations are not combined as independent premises.
What is not covered. The linearization about a non-flat background, sources that are not smooth on $Ω$ , boundary and initial-value problems for the linearized equation on bounded domains, and the back-reaction of the perturbation on the matter are outside the stated scope.

Linearized field equations as the first-order consequence of the adopted field law, with the harmonic-gauge wave reduction and the quadratic residual control

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources