An explicit sinusoidal TT plane wave: harmonic gauge, linearized vacuum equations, curvature components and detector strain

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.