Linearized Waves

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.

Let $(M,g,N)=({\mathbb{R}}^4,\eta ,{\mathrm{\partial }}_0)$ be the flat model with its inertial chart $x$ and 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). Since the chart identifies each $T_p{\mathbb{R}}^4$ with ${\mathbb{R}}^4$, all objects below are defined by their components in this chart; they are the components at a point, and a subscript $p$ is suppressed. Indices are raised and lowered with $\eta$ and ${\eta }^{-1}$ (The inverse metric, musical isomorphisms and index raising), and orthogonality is in the sense of eta-orthogonality and orthogonal complements in a Lorentz vector space.

(a) Observer, propagation direction, screen. Let $n=n^a{\mathrm{\partial }}_a$ be an observer of the flat model: $n$ is future-directed, $\eta (n,n)=-1$ (Timelike, null, spacelike and causal vectors of a Lorentz vector space); and let $s=s^a{\mathrm{\partial }}_a$ be a propagation direction: $\eta (s,s)=1$ and $\eta (n,s)=0$. The pair defines the two-dimensional screen $\mathrm{\Sigma }(n,s):=\{v\in {\mathbb{R}}^4:\eta (v,n)=0,\eta (v,s)=0\}$, on which the restriction of $\eta$ is positive definite by The observer rest-space decomposition and the spatial projection. The screen projector and the spatial projector of the pair are the $(1,1)$-tensors with components

$p^a{}_b:={\delta }^a{}_b+n^a{n}_b-s^a{s}_b,h^a{}_b:={\delta }^a{}_b+n^a{n}_b,$

and the covariant form of the screen projector is $p_{ab}:={\eta }_{ac}{p}^c{}_b={\eta }_{ab}+n_a{n}_b-s_a{s}_b$. The properties $p^2=p$, $h^2=h$, $\mathrm{im}p=\mathrm{\Sigma }(n,s)$ and $\mathrm{im}h=n^{\perp }$ are proved in the screen-projector lemma of this page.

(b) TT projection. Let $A=(A_{ab})$ be a symmetric covariant $2$-tensor at a point. Its transverse-traceless (TT) projection with respect to $(n,s)$ is the symmetric covariant $2$-tensor

${\mathrm{\Pi }}_{n,s}(A{)}_{ab}:=p_a{}^c{p}_b{}^d{A}_{cd}-\frac{1}{2}{p}_{ab}{p}^{cd}{A}_{cd},p^{cd}:={\eta }^{ce}{\eta }^{df}{p}_{ef}\mathrm{.}$

(c) Screen amplitudes. Let $(e_1,e_2)$ be an ordered $\eta$-orthonormal basis of $\mathrm{\Sigma }(n,s)$ (which exists by Every finite-dimensional real or complex inner product space has an orthonormal basis applied to the positive-definite restriction of $\eta$). For a symmetric covariant $2$-tensor $A$ the plus and cross amplitudes of $A$ with respect to $(e_1,e_2)$ are $A_{+}:=A(e_1,e_1)$ and $A_{\times }:=A(e_1,e_2)$; if $A$ is TT with respect to $(n,s)$ in the sense of clause (d) below, then, extending the screen dual basis to covectors on ${\mathbb{R}}^4$ by $e^i(v):=\eta (e_i,v)$ for $i=1,2$ (so both vanish on $n,s$),

$A=A_{+}(e^1\otimes e^1-e^2\otimes e^2)+A_{\times }(e^1\otimes e^2+e^2\otimes e^1)\mathrm{.}$

For a TT screen tensor $A$, under the rotation $e_1^{\mathrm{\prime }}=\cos \theta e_1+\sin \theta e_2$, $e_2^{\mathrm{\prime }}=-\sin \theta e_1+\cos \theta e_2$ of the screen basis the amplitudes transform as $A_{+}^{\mathrm{\prime }}=A_{+}\cos 2\theta +A_{\times }\sin 2\theta$ and $A_{\times }^{\mathrm{\prime }}=-A_{+}\sin 2\theta +A_{\times }\cos 2\theta$. Here $\sin$ and $\cos$ are the power-series functions of Sine and cosine defined by their real power series; expanding $A(e_1^{\mathrm{\prime }},e_1^{\mathrm{\prime }})$ and $A(e_1^{\mathrm{\prime }},e_2^{\mathrm{\prime }})$ by bilinearity and applying the addition formulas of The addition formulas for sine and cosine gives the two displayed transformations.

(d) TT tensors and TT perturbations. A symmetric covariant $2$-tensor $A$ is transverse-traceless (TT) with respect to $(n,s)$ when $A(n,\cdot )=0$, $A(s,\cdot )=0$ and ${\eta }^{ab}{A}_{ab}=0$. For a symmetric field $h$ on a nonempty open set $\mathrm{\Omega }\subseteq {\mathbb{R}}^4$, use the constant smooth extensions of the fixed component vectors: $n_q:=n^a{\mathrm{\partial }}_a{\mathrm{\mid }}_q$ and $s_q:=s^a{\mathrm{\partial }}_a{\mathrm{\mid }}_q$ for $q\in \mathrm{\Omega }$. Here a symmetric field is a smooth covariant $2$-tensor field with symmetric components in the sense of Tensor fields of type (r,s) on a Lorentzian four-manifold and their components. Then "TT with respect to $(n,s)$" means $h_q(n_q,\cdot )=h_q(s_q,\cdot )=0$ and ${\eta }^{ab}{h}_{ab}(q)=0$ for every $q\in \mathrm{\Omega }$. It is spatial with respect to $n$ when $h_q(n_q,\cdot )=0$ for every $q\in \mathrm{\Omega }$.

(e) Scope. This item fixes the screen, the two projections and the amplitude terminology, and states the rotation law of clause (c) as part of the naming. It asserts no property of ${\mathrm{\Pi }}_{n,s}$ (idempotence, rank, image and trace are proved in the screen-projector lemma of this page), no relation to the planar perturbations of the plane-wave definition of this page (established in the gauge-decomposition lemma of this page) and no physical interpretation of the amplitudes.

Let $(M,g,N)=({\mathbb{R}}^4,\eta ,{\mathrm{\partial }}_0)$ be the flat model with its inertial chart $x$ (The flat model spacetime: the Minkowski metric on R^4 and its inertial chart) and let $u(x):=x^0-x^3$, $n:=du$, be the phase coordinate and its differential of Plane-wave perturbations of the flat linearization: planar profiles, wave covectors, amplitudes, TT form and the linearized field equations clause (a); let $\mathrm{\square }$ be the flat d'Alembertian of Flat linearized perturbations: families, trace reversal, gauge transformation and harmonic gauge clause (c). Then:

(a) The covector $n$ is nonzero and null, ${\eta }^{ab}{n}_a{n}_b=0$, and its spatial part has Euclidean norm $1$; the vector field $T:={\mathrm{\partial }}_0+{\mathrm{\partial }}_3$ is nonzero, null and future-directed, and $\eta (T,\cdot )=-n$, so $n(T)=0$ and the wave covector $k=\omega \eta (T,\cdot )$ of Plane-wave perturbations of the flat linearization: planar profiles, wave covectors, amplitudes, TT form and the linearized field equations clause (a) is the metric dual of $\omega T$, with components $k_a=\omega (-1,0,0,1)$ and $k^a=\omega (1,0,0,1)$. For $\omega >0$ this vector is future null, for $\omega <0$ it is past null, and for $\omega =0$ it is zero and hence not null.

(b) For every $f\in C^{\mathrm{\infty }}(\mathbb{R})$ the function $\phi :=f\circ u$ is smooth on ${\mathbb{R}}^4$, and $T\phi =0$ and $\mathrm{\square }\phi =0$ at every point.

(c) For every smooth map $F=(F_{ab}):\mathbb{R}\to {\mathbb{R}}^{4\times 4}$ with values in the symmetric matrices, the planar perturbation $h_{ab}(x):=F_{ab}(u(x))$ of Plane-wave perturbations of the flat linearization: planar profiles, wave covectors, amplitudes, TT form and the linearized field equations clause (b) is a smooth symmetric field with ${\mathrm{\partial }}_c{h}_{ab}=F_{ab}^{\mathrm{\prime }}(u)n_c$ and $\mathrm{\square }{h}_{ab}=0$ componentwise on ${\mathbb{R}}^4$; moreover $({\mathrm{\partial }}_0+{\mathrm{\partial }}_3)h_{ab}=0$. If in addition the profile is TT, then ${\eta }^{ab}{h}_{ab}=0$ and the divergence ${\eta }^{bc}{\mathrm{\partial }}_b{h}_{ac}$ vanishes componentwise.

(d) The profile is constant along the integral curves of $T$: for every $x$ and every $\sigma \in \mathbb{R}$, $u(x+\sigma T)=u(x)$ and $h_{ab}(x+\sigma T)=h_{ab}(x)$. Under the coordinate reparametrization $t_c:=x^0\mathrm{/}c$, these curves satisfy $\mathrm{\mid }d{x}^3\mathrm{/}d{t}_c\mathrm{\mid }=c$. Here $c>0$ is only the real parameter used in this mathematical coordinate change; no physical unit or physical-time interpretation is asserted in this lemma.

• Terminology. The wave covector is $k=-\omega n$, and its metric-dual vector is $\omega T$. For $\omega \mathrm{\ne }0$ this vector spans the transport line; the covector annihilates $T$ because that line is null. A covector and a vector have different types, so no claim of distinct vector directions follows from their different component signs.

• Scope. The lemma is a statement about the flat model and the fixed inertial chart. A change of inertial chart mixes the roles of the coordinate directions, and a general null wave covector is treated in the gauge-decomposition lemma of this page; no claim about curved backgrounds is made.

Let $(M,g,N)=({\mathbb{R}}^4,\eta ,{\mathrm{\partial }}_0)$ be the flat model with its inertial chart (The flat model spacetime: the Minkowski metric on R^4 and its inertial chart), let $(n,s)$ be an observer and propagation direction and let $p$, $h$, $\mathrm{\Sigma }(n,s)$, ${\mathrm{\Pi }}_{n,s}$ and the amplitude terminology be as in The wave screen of an observer and propagation direction, and the transverse-traceless projection of a symmetric two-tensor. Write $V:={\mathbb{R}}^4$ and let $\mathcal{S}$ be the space of symmetric covariant $2$-tensors on $V$. Then:

(a) Projections. $p$ and $h$ are linear endomorphisms of $V$ with $p^2=p$, $h^2=h$, $\ker p=\mathrm{span}\{n,s\}$, $\mathrm{im}p=\mathrm{\Sigma }(n,s)$, $\ker h=\mathbb{R}n$, $\mathrm{im}h=n^{\perp }$; both are $\eta$-self-adjoint, $\eta (pu,v)=\eta (u,pv)$ and $\eta (hu,v)=\eta (u,hv)$ for all $u,v$. Moreover $h-p=s\otimes s^{\mathrm{♭}}$, that is $h^a{}_b-p^a{}_b=s^a{s}_b$.

(b) Ranks and traces. $\mathrm{rank}p=\dim \mathrm{\Sigma }(n,s)=2$, $p^a{}_a=2$, $\mathrm{rank}h=3$, $h^a{}_a=3$.

(c) TT projection. ${\mathrm{\Pi }}_{n,s}$ is a linear endomorphism of $\mathcal{S}$ with ${\mathrm{\Pi }}_{n,s}^2={\mathrm{\Pi }}_{n,s}$; for every $A\in \mathcal{S}$ the tensor ${\mathrm{\Pi }}_{n,s}(A)$ is TT with respect to $(n,s)$ in the sense of The wave screen of an observer and propagation direction, and the transverse-traceless projection of a symmetric two-tensor clause (d); and ${\mathrm{\Pi }}_{n,s}(A)=A$ exactly when $A$ is TT. The TT tensors form the two-dimensional linear subspace $T_{n,s}=\{B\in \mathcal{S}:B(n,\cdot )=B(s,\cdot )=0,{\eta }^{ab}{B}_{ab}=0\}$, so $\dim T_{n,s}=2$.

(d) Orthogonal projection and amplitudes. Let $⟨A,B⟩:=h^{ac}{h}^{bd}{A}_{ab}{B}_{cd}$ on the subspace of tensors spatial with respect to $n$. This is a positive definite inner product, and for every spatial $A$ and every TT $B$, $⟨A-{\mathrm{\Pi }}_{n,s}(A),B⟩=0$; so ${\mathrm{\Pi }}_{n,s}$ restricted to the spatial tensors is the orthogonal projection onto $T_{n,s}$ (The orthogonal projection $P_{W}v$ is the $W$-component in $V=W\oplus W^{\perp }$, For a subspace $W$ of a finite-dimensional inner product space, $V=W\oplus W^{\perp }$). With respect to an ordered $\eta$-orthonormal screen basis $(e_1,e_2)$ a TT tensor is

$B=B_{11}(e^1\otimes e^1-e^2\otimes e^2)+B_{12}(e^1\otimes e^2+e^2\otimes e^1),$

so $B$ is determined by the pair $(A_{+},A_{\times })=(B_{11},B_{12})$, and under a screen rotation by $\theta$ this pair transforms as in The wave screen of an observer and propagation direction, and the transverse-traceless projection of a symmetric two-tensor clause (c); in particular the map $T_{n,s}\to {\mathbb{R}}^2$, $B\mapsto (A_{+},A_{\times })$, is a linear isomorphism.