The wave screen of an observer and propagation direction, and the transverse-traceless projection of a symmetric two-tensor

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.