Plane-wave polarizations of the linearized vacuum model: two phase coefficients for each of two polarization types

Adopt the field-law postulate Adopted Einstein field equation in the direct-equation presentation together with Adopted spacetime model and classical domain of the gravitational framework and Adopted matter-content assumptions: fields, equations, stress-energy and energy conditions, and consider the vacuum branch $T=0$, $\mathrm{\Lambda }=0$ (SI unit and coordinate conventions of the classical gravitational framework). Let $(M,g,N)=({\mathbb{R}}^4,\eta ,{\mathrm{\partial }}_0)$ be the flat model with its inertial chart and let $\mathrm{\Omega }\subseteq {\mathbb{R}}^4$ be a nonempty open set. Then:

(a) First-order perturbations of vacuum families. If $(g(\lambda ),T(\lambda ){)}_{\lambda \in (-\epsilon ,\epsilon )}$ is a family of solutions of the adopted field law on $\mathrm{\Omega }$ with $T(\lambda )=0$ for every $\lambda$, realised as a flat linearized family with first-order perturbation $h$, then $G_{ab}^{(1)}(h)=0$ on $\mathrm{\Omega }$: the first-order perturbation of a family of vacuum model spacetimes solves the linearized vacuum equations of Plane-wave perturbations of the flat linearization: planar profiles, wave covectors, amplitudes, TT form and the linearized field equations clause (e).

(b) TT plane waves are vacuum waves. Let $A_{+},A_{\times }\in C^{\mathrm{\infty }}(\mathbb{R})$ be bounded and let $h$ be the TT planar perturbation of Plane-wave perturbations of the flat linearization: planar profiles, wave covectors, amplitudes, TT form and the linearized field equations clause (b) with these amplitudes. Then $h$ is in harmonic gauge and $G_{ab}^{(1)}(h)=0$. Moreover, for every nonempty compact set $K$ and length scale $L>0$, the exact metric $\eta +\lambda h$ satisfies the vacuum field law on $K$ up to a residual bounded by $C_{K,L}{\lambda }^2{B}_K(h{)}^2\mathrm{/}{L}^2$ whenever $\mathrm{\mid }\lambda \mathrm{\mid }{B}_K(h)\le 1\mathrm{/}16$, where $B_K$ is the numerical chart control of Linearized field equations as the first-order consequence of the adopted field law, with the harmonic-gauge wave reduction and the quadratic residual control clause (d). Since $h$ is smooth, this condition holds for sufficiently small $\mathrm{\mid }\lambda \mathrm{\mid }$. The model therefore contains these transverse-traceless plane waves as linearized vacuum waves in the stated approximation. In the adopted SI length coordinates $x^0=ct$, a constant phase $u=x^0-x^3$ moves in the $+x^3$ direction at coordinate speed $c$. The residual estimate is read in SI numerical chart components, with $L$ in metres and its chart-control constants as real numerical values; the Einstein-tensor residual then has units ${\mathrm{m}}^{-2}$.

(c) Two polarization types and phase coefficients. Fix a nonzero null covector $k$ and an adapted orthonormal frame as in 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 clause (a), so that $k_A=k(e_A)=\omega (-1,0,0,1)$ and $k^A=\omega (1,0,0,1)$ in that frame, and let $V_0$, $G$ and $T$ be the amplitude spaces of that lemma. In the adapted-frame SI length coordinates the phase is $k_A{x}^A=-\omega (x^0-x^3)$, so $\omega$ has units ${\mathrm{m}}^{-1}$ and the corresponding physical angular frequency at fixed spatial position is $\varpi :=\mathrm{\mid }{\mathrm{\partial }}_t(k_A{x}^A)\mathrm{\mid }=c\mathrm{\mid }\omega \mathrm{\mid }$ in ${\mathrm{s}}^{-1}$.

For one real cosine coefficient, $h_X(x):=X\cos (k\cdot x)$, the quotient $V_0\mathrm{/}G$ is two-dimensional, and the maps

$V_0\mathrm{/}G⟶T⟶{\mathbb{R}}^2,[X]⟼\pi (X)⟼(A_{+},A_{\times }):=(X_{11}^{\mathrm{\prime }},X_{12}^{\mathrm{\prime }}),$

identify it with ${\mathbb{R}}^2$; here $\pi (X)$ is the unique TT representative of clause (c) of that lemma and $(A_{+},A_{\times })$ its amplitudes in the screen basis. The same quotient applies independently to one real sine coefficient. Thus, for the real plane-wave form $h_{ab}=X_{ab}\cos (k\cdot x)+Y_{ab}\sin (k\cdot x)$ of Plane-wave perturbations of the flat linearization: planar profiles, wave covectors, amplitudes, TT form and the linearized field equations clause (c), the full harmonic-gauge amplitude quotient is $(V_0\mathrm{/}G)\oplus (V_0\mathrm{/}G)\cong {\mathbb{R}}^4$: it has two polarization types, each with independent cosine and sine coefficients. The gauge transformations act independently on the two coefficients by choosing ${\xi }_b=Z_b^X\sin (k\cdot x)-Z_b^Y\cos (k\cdot x)$.

Let $\mathcal{F}:=C^{\mathrm{\infty }}({\mathbb{R}}^4,{\mathbb{R}}^{4\times 4\times 4\times 4})$ be the real vector space of smooth arrays of curvature components in the fixed inertial chart. For the cosine coefficient, the first-order curvature map $V_0\mathrm{/}G\to \mathcal{F}$ is $[X]\mapsto (x\mapsto R_{abcd}^{(1)}(h_X)(x))$. It is well defined because $R^{(1)}$ is gauge invariant by 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), and is injective: for the plane wave $h_{ab}=X_{ab}\cos (k\cdot x)$ with TT representative $X^{\mathrm{\prime }}$ one has

$R_{0101}^{(1)}=\frac{1}{2}{\omega }^2{A}_{+}\cos (k\cdot x),R_{0102}^{(1)}=\frac{1}{2}{\omega }^2{A}_{\times }\cos (k\cdot x),$

so evaluating these components at $x=0$ recovers $(A_{+},A_{\times })$, since $\cos (k\cdot 0)=1$ and $\omega \mathrm{\ne }0$. The sine coefficient has the same curvature formula with $\sin (k\cdot x)$ in place of $\cos (k\cdot x)$; because the nonzero linear phase $k\cdot x$ attains $\pi \mathrm{/}2$, its screen coefficients are recovered by evaluating at a point with $k\cdot x=\pi \mathrm{/}2$. At phase $0$ the sine coefficient vanishes, and at phase $\pi \mathrm{/}2$ the cosine coefficient vanishes while the sine coefficient is nonzero (Pi as twice the smallest positive zero of cosine, Parity and the Pythagorean identity for sine and cosine). Thus the curvature map is injective on the full real coefficient-pair quotient as well. A different screen basis rotates each phase-coefficient pair 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); the quotient classes and their curvature tensors, rather than their coordinate pairs, are the frame-independent content. In this precise sense the linearized vacuum model has two polarization types at fixed $k$.

Status. Clauses (a)-(c) are conditional deductions from the adopted postulates and the cited mathematics. No source distribution, energy flux, observed radiation, detector, measured amplitude or statistical statement is made; the item contains no empirical premise.

• Counting is not the statement. The familiar count "10 components minus 4 gauge conditions minus 4 residual gauges gives 2" is replaced here by the exact decomposition $V_0=T\oplus G$ of amplitude space: every claim about the number of polarizations is a statement about this finite-dimensional space at a fixed null $k$, and the profile-level solutions of clause (b) are not classified by it.

• What is not included. Radiation from a source, the quadrupole formula, energy and angular momentum carried by waves, the propagation of waves on a curved background and the response of a detector are not part of this item; the detector response is the subject of 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, and source and energy questions are outside the scope of this page.