Linearized Waves — Examples

Use the SI length coordinates $x^0=ct,x^1,x^2,x^3$ of SI unit and coordinate conventions of the classical gravitational framework. Fix dimensionless real amplitudes $a,b$ and a positive wave-number parameter $\omega$ with units ${\mathrm{m}}^{-1}$. The phase coordinate $u=x^0-x^3$ has units $\mathrm{m}$, so $\omega u$ is dimensionless. At a fixed spatial point the model angular frequency is $c\mathrm{\mid }\omega \mathrm{\mid }$ in ${\mathrm{s}}^{-1}$. Put

$A_{+}(u):=a\cos (\omega u),A_{\times }(u):=b\sin (\omega u),u=x^0-x^3,$

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: $h_{11}=A_{+}$, $h_{22}=-A_{+}$, $h_{12}=h_{21}=A_{\times }$ and all other components zero. Let $M:=\max \{\mathrm{\mid }a\mathrm{\mid },\mathrm{\mid }b\mathrm{\mid }\}$ and choose $\mathrm{\mid }\lambda \mathrm{\mid }<1\mathrm{/}(2M)$ when $M>0$. Then:

1. $h$ is traceless and in harmonic gauge, $\mathrm{\square }{h}_{ab}=0$, and $G_{ab}^{(1)}(h)=0$: the perturbation solves the linearized vacuum equations; the metric $g(\lambda )=\eta +\lambda h$ is a smooth Lorentzian metric of signature $(-,+,+,+)$ with time orientation ${\mathrm{\partial }}_0$, and in the zero-source, $\mathrm{\Lambda }=0$ branch, for every nonempty compact $K$ and length scale $L$, the exact vacuum residual is bounded by a constant times ${\lambda }^2$ whenever $\mathrm{\mid }\lambda \mathrm{\mid }{B}_K(h)\le 1\mathrm{/}16$; since the profiles are smooth, this holds for sufficiently small $\mathrm{\mid }\lambda \mathrm{\mid }$ on each fixed compact set.
2. The first-order curvature components are $R_{0101}^{(1)}=\frac{1}{2}a{\omega }^2\cos (\omega u),R_{0102}^{(1)}=\frac{1}{2}b{\omega }^2\sin (\omega u),R_{0202}^{(1)}=-\frac{1}{2}a{\omega }^2\cos (\omega u),$ and $R_{0i0j}^{(1)}=0$ whenever $i=3$ or $j=3$.
3. At a point where $u=0$ the curvature values are $R_{0101}^{(1)}=\frac{1}{2}a{\omega }^2$ and $R_{0102}^{(1)}=0$. For the ideal detector of The linearized-wave approximation regime and an ideal two-body detector with a prescribed finite arm path, the arm along $e_1$ has strain $\sqrt{1+\lambda a\cos (\omega u)}-1$, which equals $\sqrt{1+\lambda a}-1$ at $u=0$; the arm along $(e_1+e_2)\mathrm{/}\sqrt{2}$ has strain $\sqrt{1+\lambda b\sin (\omega u)}-1$ as a function of general $u$, whose value at $u=0$ is $0$.

All three clauses are conditional statements of the model: clause 3 is the strain function of the ideal detector defined in The linearized-wave approximation regime and an ideal two-body detector with a prescribed finite arm path evaluated on the stated family; no observation, apparatus or statistical statement is made, and no numerical amplitude, frequency or strain of a real wave is asserted.

• Sign and phase conventions. The signs follow the curvature convention of First-order curvature of the flat linearization and of TT plane waves, its gauge invariance and its contraction to the linearized Ricci tensor; with $a>0$ the component $R_{0101}^{(1)}$ is positive at $u=0$. The example is a mathematical computation in the flat model and asserts no physical amplitude, frequency or measured strain.

• Superposition. Since $\mathrm{\square }$ and $G^{(1)}$ are linear, a finite sum of TT plane waves with different wave covectors is again a solution of the linearized vacuum equations, by the same computation. No completeness of such sums is claimed here.

Work in the inertial chart of the flat model (The flat model spacetime: the Minkowski metric on R^4 and its inertial chart) with the observer $n={\mathrm{\partial }}_0$ and propagation direction $s={\mathrm{\partial }}_3$, so that the screen is $\mathrm{\Sigma }=\mathbb{R}{e}_1\oplus \mathbb{R}{e}_2$ and, in the basis $(e_0,e_1,e_2,e_3)$, the screen and spatial projectors of The wave screen of an observer and propagation direction, and the transverse-traceless projection of a symmetric two-tensor have the component matrices

$p=\mathrm{diag}(0,1,1,0),h=\mathrm{diag}(0,1,1,1)\mathrm{.}$

Let $A$ be the symmetric spatial tensor with component matrix

$(A_{ab})=\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 3& 1& 2\\ 0& 1& -1& 0\\ 0& 2& 0& 5\end{array}\right),$

so $A(n,\cdot )=0$. Then:

1. The screen projection and the trace of $A$ on the screen are $p_a{}^c{p}_b{}^d{A}_{cd}=\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 3& 1& 0\\ 0& 1& -1& 0\\ 0& 0& 0& 0\end{array}\right),p^{cd}{A}_{cd}=A_{11}+A_{22}=2.$
2. The TT projection is ${\mathrm{\Pi }}_{n,s}(A)=\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 2& 1& 0\\ 0& 1& -2& 0\\ 0& 0& 0& 0\end{array}\right),$ which is transverse (${\mathrm{\Pi }}_{n,s}(A)(s,\cdot )=0$), traceless (${\eta }^{ab}{\mathrm{\Pi }}_{n,s}(A{)}_{ab}=2-2=0$) and equals its own TT projection: ${\mathrm{\Pi }}_{n,s}({\mathrm{\Pi }}_{n,s}(A))={\mathrm{\Pi }}_{n,s}(A)$.
3. Its amplitudes are $A_{+}=2$ and $A_{\times }=1$, and it is the tensor $2(e^1\otimes e^1-e^2\otimes e^2)+(e^1\otimes e^2+e^2\otimes e^1)$.

• The projection kills the $3$-components. The longitudinal entries $A_{13}=2$, $A_{23}=0$ and the trace component $A_{33}=5$ are removed by the screen projection; the screen trace $A_{11}+A_{22}=2$ is then removed by the half-trace subtraction, leaving the traceless screen block with amplitudes $(2,1)$.

• No physical reading. The example computes a projection of a chosen matrix in the flat model; the tensor $A$ is not a metric perturbation and no wave, source or observation is involved.

Work in the inertial chart of the flat model with the wave covector $k^a=\omega (1,0,0,1)$, $\omega =1$, and let $\mathcal{X}$ be the space of symmetric amplitudes of Plane-wave perturbations of the flat linearization: planar profiles, wave covectors, amplitudes, TT form and the linearized field equations clause (c). Consider the amplitude

$X=\left(\begin{array}{cccc}-2& -2& 0& 0\\ -2& 1& 0& 2\\ 0& 0& -1& 0\\ 0& 2& 0& 2\end{array}\right),$

whose components in the orthonormal basis $(e_0,e_1,e_2,e_3)$ are displayed in the order $(0,1,2,3)$. Then:

1. $X$ lies in $V_0=\{X:k^a{\bar{X}}_{ab}=0\}$ and is not TT: $X(e_0,e_0)=-2\mathrm{\ne }0$ and its trace is $\mathrm{tr}X=-X_{00}+X_{11}+X_{22}+X_{33}=2+1-1+2=4\mathrm{\ne }0$.
2. The unique $Z$ 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 clause (c) with $X_{ab}^{\mathrm{\prime }}:=X_{ab}-k_a{Z}_b-k_b{Z}_a$ TT is $Z=(1,2,0,1)$, and $X^{\mathrm{\prime }}=\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& 0\end{array}\right),$ so the amplitudes of the TT representative are $A_{+}=1$, $A_{\times }=0$.
3. With ${\xi }_b=Z_b\sin (k\cdot x)$ the perturbation $h_{ab}=X_{ab}\cos (k\cdot x)$ satisfies $h_{ab}^{\mathrm{\prime }}:=h_{ab}-{\mathrm{\partial }}_a{\xi }_b-{\mathrm{\partial }}_b{\xi }_a=X_{ab}^{\mathrm{\prime }}\cos (k\cdot x)$, which is a TT plane wave in harmonic gauge.

• Uniqueness. The example exhibits the unique solution of the linear system 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 clause (c): any other $Z$ would give a different tensor whose $0$-components would not vanish, and the lemma's uniqueness statement forbids two TT representatives in the same class.

• What the amplitudes are. The pair $(A_{+},A_{\times })=(1,0)$ is the gauge-invariant first-order content of this class up to the frame choice: a screen rotation mixes the 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), and the class, not the coordinate pair, is the intrinsic object.

Statement refuted: "For every real $v$ with $-1<v<1$ and every smooth $f:\mathbb{R}\to \mathbb{R}$, the symmetric perturbation $h$ with components

$h_{11}(x)=f(x^3-v{x}^0),h_{22}(x)=-f(x^3-v{x}^0),h_{12}=h_{21}=0,\text{all other components }0,$

solves the linearized vacuum equations of the flat linearization, $G_{ab}^{(1)}(h)=0$."

Here the phase parameter is defined by the level sets $x^3-v{x}^0=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}$; their coordinate ratio is $\mathrm{\mid }d{x}^3\mathrm{/}d{x}^0\mathrm{\mid }=\mathrm{\mid }v\mathrm{\mid }<1$, while the null phase has coordinate ratio $1$ in the same flat chart.

• Why the phase rate matters. For a profile $f(x^3-v{x}^0)$ one computes $\mathrm{\square }f=(1-v^2)f^{\mathrm{\prime }\mathrm{\prime }}$, which vanishes for every $f$ exactly when $v^2=1$. The distinguished null profile of the vacuum plane waves is $u=x^0-x^3$, for which the profiles with the same spatial structure do solve the equations (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 clause (b)); the counterexample uses the sub-null coordinate ratio $v=\frac{1}{2}$ and exhibits the failure quantitatively.

• The perturbation is not a solution of the exact field law either. Since $G_{11}^{(1)}\mathrm{\ne }0$, the perturbation is not even a first-order vacuum solution, so no exact vacuum metric is obtained by "correcting" it at higher order in the displayed family; the counterexample concerns the stated universal claim only.

Statement refuted: "Let $n$ be an observer and $s$ a propagation direction of the flat model, let $h^a{}_b={\delta }^a{}_b+n^a{n}_b$ be the spatial projector of clause (a) of The wave screen of an observer and propagation direction, and the transverse-traceless projection of a symmetric two-tensor, let $h_{ab}={\eta }_{ac}{h}^c{}_b$, and let $A$ be any symmetric covariant $2$-tensor with $A(n,\cdot )=0$. Then the tensor

$A^{\mathrm{T}\mathrm{T}}:=h_a{}^c{h}_b{}^d{A}_{cd}-\frac{1}{2}{h}_{ab}{h}^{cd}{A}_{cd}$

is transverse-traceless: $A^{\mathrm{T}\mathrm{T}}(n,\cdot )=0$, $A^{\mathrm{T}\mathrm{T}}(s,\cdot )=0$ and ${\eta }^{ab}{A}_{ab}^{\mathrm{T}\mathrm{T}}=0$."