Scope, status and deferred material of the linearized-waves page: what the model predicts, what is idealization and what is not covered

This note fixes what the physics items of the linearized-waves page do and do not claim; it proves nothing by itself.

• What is proved. Linearized field equations as the first-order consequence of the adopted field law, with the harmonic-gauge wave reduction and the quadratic residual control derives the linearized field equation and its vacuum branch by differentiating the adopted field law Adopted Einstein field equation in the direct-equation presentation along a smooth family of exact solutions. Its separate quadratic residual bound follows from the mathematical Taylor estimate for the Einstein tensor once $G^{(1)}(h)=0$; it does not require $\eta +\lambda h$ to belong to an exact solution family. The all-space harmonic-gauge existence clause additionally assumes ${\mathrm{A}\mathrm{C}}_{\omega }$. The mathematical existence statement for the vacuum plane waves is TT plane waves with arbitrary bounded profiles solve the linearized vacuum equations, are in harmonic gauge, and give Lorentzian small-amplitude metrics with a quadratic Einstein-tensor residual: for every pair of bounded smooth profiles there is an exact solution of the linearized vacuum equations in harmonic gauge, whose exact metric in the zero-source, $\mathrm{\Lambda }=0$ branch has vacuum field-law residual bounded by a constant times the square of the amplitude on nonempty compact sets. Plane-wave polarizations of the linearized vacuum model: two phase coefficients for each of two polarization types reads the amplitude classification of Harmonic-gauge plane-wave amplitudes split uniquely into a pure plane-wave gauge part and a transverse-traceless representative, with a two-dimensional space of polarizations as the statement that the model has two polarization types at fixed null wave covector, with the gauge-invariant first-order curvature coefficients as their geometric content; no operational observable is defined on this page. Response of ideal test bodies and of the ideal two-body detector to a TT plane wave: free-fall worldlines, first-order relative acceleration and strain derives the model's infinitesimal relative-acceleration law for a geodesic variation and the finite path-length strain of the ideal two-body detector of The linearized-wave approximation regime and an ideal two-body detector with a prescribed finite arm path.

• What is conditional. Every physical statement on the page is a deduction from the adopted postulates of Adopted spacetime model and classical domain of the gravitational framework, Adopted matter-content assumptions: fields, equations, stress-energy and energy conditions, Adopted Einstein field equation in the direct-equation presentation and Adopted clock, free-fall and light-ray assumptions of the classical gravitational framework, together with the explicit hypotheses of the mathematical items. The postulates are adopted model assumptions, not derived results; no item on the page establishes the postulates, and no item compares a prediction with a measurement. For a TT perturbation the metric family $\eta +\lambda h$ satisfies the zero-source, $\mathrm{\Lambda }=0$ equation through first order; its residual is bounded by $O({\lambda }^2)$ on nonempty compact sets for sufficiently small amplitudes. This estimate does not establish an exact vacuum solution for nonzero $\lambda$. The admitted case $h=0$ is exactly flat and solves that equation for every $\lambda$. The polarization and response statements retain their stated linearized-model interpretation.

• Idealizations of the detector. The "detector" of The linearized-wave approximation regime and an ideal two-body detector with a prescribed finite arm path is two freely falling test bodies together with the metric length of their prescribed chart-straight spatial segment at equal inertial-chart time. This is a path-dependent arm length, not an intrinsic shortest distance between the endpoints. Its relative-acceleration claim is the infinitesimal deviation law, not a finite-pair acceleration formula. The detector has no rigidity, elasticity, internal resonance, readout, noise, light propagation or calibration. The strain function is a model quantity; identifying it with a recorded signal of a real instrument would require an apparatus model with its own assumptions, empirical premises with uncertainties and a statistical analysis, none of which occurs on this page. In particular no numerical value of an amplitude, frequency, strain or event rate is asserted anywhere in the batch.

• Frame and gauge caveats. The transverse-traceless form and the amplitudes $A_{+},A_{\times }$ are defined with respect to an adapted orthonormal frame (The wave screen of an observer and propagation direction, and the transverse-traceless projection of a symmetric two-tensor, Harmonic-gauge plane-wave amplitudes split uniquely into a pure plane-wave gauge part and a transverse-traceless representative, with a two-dimensional space of polarizations); the pair $(A_{+},A_{\times })$ transforms under screen rotations as in The screen and spatial projectors are orthogonal projections of rank two and three, and the TT projection is the orthogonal projection onto the two-dimensional transverse-traceless space clause (d), and the space of classes, not the particular representative, is the frame-independent content. The first-order curvature is gauge invariant (First-order curvature of the flat linearization and of TT plane waves, its gauge invariance and its contraction to the linearized Ricci tensor clause (b)). The projection to the physical screen is two-dimensional; a trace subtraction with the half weight of a three- or four-dimensional projector is not the transverse-traceless projection (The screen and spatial projectors are orthogonal projections of rank two and three, and the TT projection is the orthogonal projection onto the two-dimensional transverse-traceless space remark and the counterexample recorded there).

• What is deferred. The following are outside this page and are not proved, assumed or silently used: the construction of solutions of the exact vacuum equations with prescribed initial data or from specified sources; the linearization about a non-flat background; radiation from localized slow sources, the quadrupole formula, the energy and angular momentum carried by waves, asymptotic and ADM-type charges, and the far-zone scale separation; the propagation of light in a wave background and the optical response of an interferometer; post-Newtonian and nonlinear corrections; the existence and properties of exact strong-field solutions; quantum aspects of gravitational waves and any gravitational quantum theory. Sources and detectors are experiment-level material and require retrieved primary reports, apparatus and sampling models, uncertainties and a stated statistical inference before any reported observation can be recorded in the library.