Plane-wave perturbations of the flat linearization: planar profiles, wave covectors, amplitudes, TT form and the linearized field equations

Let $(M,g,N)=({\mathbb{R}}^4,\eta ,{\mathrm{\partial }}_0)$ be the flat model with its inertial chart $x$ and constant metric components ${\eta }_{ab}=\mathrm{diag}(-1,1,1,1)$ (The flat model spacetime: the Minkowski metric on R^4 and its inertial chart, Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model), and let $\mathrm{\Omega }\subseteq {\mathbb{R}}^4$ be a nonempty open set, identified with its image under the chart; all components and all partial derivatives below are taken in this chart. Let $c>0$ be a real parameter. Throughout, a symmetric field means a smooth symmetric covariant $2$-tensor field on $\mathrm{\Omega }$ (Smooth covariant 2-tensor fields on a smooth manifold) and indices are raised and lowered with $\eta$ and its inverse ${\eta }^{-1}$ (The inverse metric, musical isomorphisms and index raising).

(a) Phase coordinate and wave covector. The phase coordinate of the planar waves of this page is $u(x):=x^0-x^3$. Its differential $n:=du$ has components $n_a={\delta }_a^0-{\delta }_a^3$, so that $n_0=1$, $n_3=-1$, $n_1=n_2=0$ and ${\eta }^{ab}{n}_a{n}_b=-1+1=0$: $n$ is a nonzero null covector, and the vector with spatial components $(0,0,-1)$ has Euclidean norm $1$ in the sense of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$. The vector $T:={\mathrm{\partial }}_0+{\mathrm{\partial }}_3$ is null and future-directed, and $\eta (T,\cdot )=-n$, so that the phase covector is the metric dual of $-T$. For $\omega \in \mathbb{R}$ we call $k_a:=\omega {\eta }_{ab}{T}^b=\omega (-1,0,0,1)$ a wave covector; $\omega$ is its real phase coefficient and $\mathrm{\mid }\omega \mathrm{\mid }$ its absolute value. Its raised components are $k^a=\omega T^a=\omega (1,0,0,1)$, so that $k\mathrm{\ne }0$ for $\omega \mathrm{\ne }0$ and $k^a{k}_a=0$. For a general null covector $k\mathrm{\ne }0$ of the flat model the adapted orthonormal frames are treated in the gauge-decomposition lemma of this page; clauses (a)-(d) fix the standard propagation direction $x^3$ used on this page.

(b) Planar perturbations and TT profiles. A symmetric field $h$ on $\mathrm{\Omega }$ is a planar perturbation when there is a smooth map $F=(F_{ab}):\mathbb{R}\to {\mathbb{R}}^{4\times 4}$ with values in the symmetric matrices such that $h_{ab}(x)=F_{ab}(u(x))$ for all $x\in \mathrm{\Omega }$; such an $F$ is a planar profile. The image $u(\mathrm{\Omega })$ is open: near each $x\in \mathrm{\Omega }$, varying $x^0$ through a sufficiently small interval varies $u(x)$ through an interval while remaining in $\mathrm{\Omega }$. The restriction $F{\mathrm{\mid }}_{u(\mathrm{\Omega })}$ is uniquely determined by $h$, but an extension of it outside $u(\mathrm{\Omega })$ is not determined unless $u(\mathrm{\Omega })=\mathbb{R}$. A planar perturbation is transverse traceless (TT) with respect to the inertial frame when $h_{a0}=h_{a3}=0$ and ${\eta }^{ab}{h}_{ab}=0$ on $\mathrm{\Omega }$; equivalently, a chosen profile satisfies $F_{a0}=F_{a3}=0$ and $F_{11}+F_{22}=0$ on $u(\mathrm{\Omega })$. In that case its restriction to $u(\mathrm{\Omega })$ has the form

$F_{ab}(u)=A_{+}(u)({\delta }_a^1{\delta }_b^1-{\delta }_a^2{\delta }_b^2)+A_{\times }(u)({\delta }_a^1{\delta }_b^2+{\delta }_a^2{\delta }_b^1)$

for smooth functions $A_{+},A_{\times }:u(\mathrm{\Omega })\to \mathbb{R}$, uniquely given by $A_{+}=F_{11}{\mathrm{\mid }}_{u(\mathrm{\Omega })}$ and $A_{\times }=F_{12}{\mathrm{\mid }}_{u(\mathrm{\Omega })}$; these are the plus and cross amplitudes on the phase image. In particular a TT planar perturbation is spatial ($h_{a0}=0$), traceless (${\eta }^{ab}{h}_{ab}=0$) and transverse (its contraction with the wave covector of (a) vanishes).

(c) Single-frequency plane waves. Let $k\mathrm{\ne }0$ be a null covector of the flat model and let $X=(X_{ab})$ and $Y=(Y_{ab})$ be real symmetric matrices. The symmetric field

$h_{ab}(x):=X_{ab}\cos (k\cdot x)+Y_{ab}\sin (k\cdot x),k\cdot x:=k_a{x}^a,$

where $\cos$ and $\sin$ are the power-series functions (Sine and cosine defined by their real power series), is the plane-wave perturbation with wave covector $k$ and amplitudes $(X,Y)$. Its trace-reversed amplitudes are ${\bar{X}}_{ab}:=X_{ab}-\frac{1}{2}{\eta }_{ab}{\eta }^{cd}{X}_{cd}$ and ${\bar{Y}}_{ab}:=Y_{ab}-\frac{1}{2}{\eta }_{ab}{\eta }^{cd}{Y}_{cd}$, in agreement with the trace reversal of Flat linearized perturbations: families, trace reversal, gauge transformation and harmonic gauge clause (c). The plane wave is in harmonic gauge when $k^a{\bar{X}}_{ab}=0$ and $k^a{\bar{Y}}_{ab}=0$. The plane wave is TT with respect to the wave covector $k$ when, in some $\eta$-orthonormal frame $(e_0,e_1,e_2,e_3)$ with $e_0$ future timelike, $e_1,e_2$ spanning the screen of (d) and $k_A=k(e_A)=\omega (-1,0,0,1)$ for some $\omega \mathrm{\ne }0$ (existence of such a frame is proved in the gauge-decomposition lemma of this page, where also $k^A=\omega (1,0,0,1)$), both amplitude matrices have components $X_{a0}=X_{a3}=Y_{a0}=Y_{a3}=0$ and $X_{11}+X_{22}=Y_{11}+Y_{22}=0$ in that frame.

(d) Screen amplitudes. In the situation of (c) with $k_A=\omega (-1,0,0,1)$ in the adapted frame, the screen of $k$ is the two-dimensional linear subspace ${\mathrm{\Sigma }}_k:=\{v\in T_p{\mathbb{R}}^4:\eta (v,e_0)=\eta (v,e_3)=0\}$, which is independent of the point $p$ and is the orthogonal complement of $\mathrm{span}\{e_0,e_3\}$ in the sense of eta-orthogonality and orthogonal complements in a Lorentz vector space. The plus and cross amplitudes of a screen tensor $A$ (a symmetric bilinear form on ${\mathrm{\Sigma }}_k$) with respect to an ordered $\eta$-orthonormal basis $(e_1,e_2)$ of ${\mathrm{\Sigma }}_k$ are $A_{+}:=A(e_1,e_1)$ and $A_{\times }:=A(e_1,e_2)$.

(e) Linearized field equations. Let $\mathrm{\Lambda }\in \mathbb{R}$ be a real parameter and let $S=(S_{ab})$ be a symmetric field on $\mathrm{\Omega }$. A symmetric field $h$ solves the linearized field equation with source $S$ and parameter $\mathrm{\Lambda }$ when $G_{ab}^{(1)}(h)+\mathrm{\Lambda }{h}_{ab}=S_{ab}$ on $\mathrm{\Omega }$, with $G^{(1)}$ the first-order Einstein coefficient of Flat linearized perturbations: families, trace reversal, gauge transformation and harmonic gauge clause (b). It solves the linearized vacuum equations when $S=0$ and $\mathrm{\Lambda }=0$; and it solves the linearized vacuum equations with parameter $\mathrm{\Lambda }$ when $S=0$ and $\mathrm{\Lambda }$ is arbitrary. A planar perturbation (b) or a plane wave (c) is a linearized vacuum wave when it solves the linearized vacuum equations with $\mathrm{\Lambda }=0$.

(f) Scope. This item fixes objects and terminology only. It asserts no existence of a profile or amplitude solving any equation, no relation between the single-frequency and planar descriptions, and no physical interpretation; in particular the harmonic-gauge condition in (c) is a named algebraic condition whose equivalence with the differential condition of Flat linearized perturbations: families, trace reversal, gauge transformation and harmonic gauge clause (d) is proved in the gauge-decomposition lemma of this page, and the terminology "TT" in (b)-(c) is defined relative to a chosen orthonormal frame and not claimed to be geometrically canonical.