Propagation, Scattering, Lensing, and Wave Optics

Gravitational radiation crosses several conceptually distinct propagation regimes between emission and detection. A short wavelength follows affinely parametrized null rays, its amplitude focuses with the ray bundle, and its tensor polarization is parallel transported. Curvature can also scatter radiation into a tail; a lens can create several stationary images; finite wavelength turns those images into a diffraction pattern; and cosmic expansion maps source-frame scales into detector-frame ones. This chapter develops those effects without importing instrument likelihoods or theory-specific modified-dispersion constraints.

Helpful background. Waves, Green Functions, and Geometric Optics supplies characteristics, Green functions, and stationary phase; Curvature supplies geodesics, focusing, and scattering potentials; Einstein Equations supplies the linearized tensor equation; Relativistic Cosmology supplies redshift and distance measures.

Enter the propagation problem

The first decision is not “which waveform model?” but “which propagation approximation is controlled?” Let $\lambda$ be the local wavelength and $L$ the shortest background scale sampled by the wave. If $\lambda/L\ll1$ away from special regions, geometric optics supplies the local wavefront and transport. That does not automatically validate geometric optics through a compact lens: the separate lens parameter $w$, built from frequency and lens delay scale, can still be order unity. Nor does a valid ray expansion eliminate curvature tails, which are nonlocal parts of a retarded Green function rather than corrections assigned to one ray.

The chapter treats the emitted wave as given. Source dynamics and multipolar generation determine the data placed on an outgoing wavefront. Detailed compact-object tail spectra and decay exponents are developed in Black-Hole Perturbation Theory, the background expansion history in Relativistic Cosmology, and population hypotheses and detector likelihoods in Gravitational-Wave Astronomy. Here those inputs meet at a source-independent propagation map.

Shared convention dictionary

All pages use signature $(-,+,+,+)$ and $c=1$, with $G$ displayed. A local positive-frequency phase is

$$e^{i\theta}=e^{-i\omega t+i\boldsymbol{k}\cdot\boldsymbol{x}}, \qquad k_\mu=\nabla_\mu\theta.$$

Raising the index makes $k^\mu$ future directed. At leading eikonal order,

$$k^\mu k_\mu=0, \qquad k^\nu\nabla_\nu k^\mu=0, \qquad \omega_{(u)}=-k_\mu u^\mu>0,$$

so its integral curves are affinely parametrized null rays. A transported polarization basis consists of two orthonormal spacelike screen vectors $s_A^\mu$ transverse to the ray; rotating that basis by $\chi$ sends the complex strain $h=h_+-ih_\times$ to $e^{-2i\chi}h$. This is basis covariance, not an additional polarization.

Frequency-domain lensing uses

$$h(t)=\int\tilde h(f)e^{-2\pi ift}\,df.$$

The dimensionless Fermat potential has its global minimum fixed to $T_{\min}=0$. For a point lens,

$$w=8\pi G M_L(1+z_L)f_{\mathrm{det}}, \qquad t_d=4GM_L(1+z_L)T.$$

Thus $2\pi f_{\mathrm{det}}t_d=wT$, with no hidden $2\pi$ or common-delay phase. The source and detector frames are comoving cosmological frames unless an endpoint peculiar velocity is declared, and

$$f_{\mathrm{det}}=\frac{f_{\mathrm{src}}}{1+z}, \qquad dt_{\mathrm{det}}=(1+z)dt_{\mathrm{src}}.$$

These choices align the eikonal transport of Flanagan and Hughes Flanagan and Hughes 2005, §5, pp. 23–29, PDF, the thin-lens phase of Takahashi and Nakamura Takahashi and Nakamura 2003, §2, pp. 3–8, PDF, and the FLRW tensor evolution of Caprini and Figueroa Caprini and Figueroa 2018, §2.4, pp. 19–22, PDF.

Six pages, one sequence

Read the pages in the sequence below; each changes exactly one part of the propagation description.

1. Geometric Optics for Gravitational Waves derives $k^2=0$, affine null rays, amplitude–area conservation, and parallel polarization transport, then recovers a flat plane wave.
2. Curvature Scattering and Tails decomposes a local retarded Green function into direct and interior-cone pieces, checks four-dimensional flat Huygens propagation, and separates propagation tails from source hereditary terms. The local support construction is given by Poisson, Pound, and Vega 2011, §§14.1–14.5.
3. Strong and Weak Gravitational Lensing solves the point-mass lens equation and tracks magnification, parity, delay ordering, and stationary-phase Morse factors.
4. Wave-Optics Diffraction normalizes the amplification integral, shows $F\to1$ at low $w$, and recovers the geometric image sum at high $w$.
5. Polarization Transport constructs the parallel screen dyad and distinguishes endpoint basis rotation from physical helicity mixing. Beyond-geometric-optics apparent components are observer dependent, as developed in Cusin and Lagos 2020, §IV, pp. 8–9, and Appendix A, pp. 11–12, PDF.
6. Cosmological Propagation and Redshift derives $h_k\propto1/a$ for free subhorizon modes, maps source to detector frequency, and obtains the luminosity-distance amplitude law.

Regimes and discriminants

The comparison axes are causal support, controlling scale, mathematical representation, and the physical effect on the waveform.

Regime	Controlling condition	Representation	Distinctive effect
Geometric transport	$\lambda/L\ll1$ along one bundle	Eikonal plus transport equations	Null phase, focusing, parallel polarization
Curvature tail	Nontrivial retarded interior support	Direct plus tail Green function	Signal after the direct null-cone arrival
Multiple imaging	Several stationary Fermat paths	Discrete image sum	Magnifications, delays, parities, Morse phases
Diffraction	$w\lesssim1$ or a caustic invalidates stationary phase	Lens-plane amplification integral	Smoothed focusing and frequency-dependent interference
Cosmological propagation	Subhorizon wave on evolving FLRW background	Rescaled mode $\mu_k=a h_k$	Redshift, time dilation, inverse-$D_L$ amplitude
Genuine extra polarization	Additional dynamical field or altered constraint	Theory-specific field equations	New independent radiative degree of freedom

These rows can coexist along one journey. A wave can propagate geometrically through most of an FLRW spacetime, diffract at one compact lens, and be compared in a rotated detector screen. Conversely, a frequency-dependent modulation is not by itself proof of diffraction: it might be interference between high-frequency images. A delayed weak signal is not automatically a second image: it might be a curvature tail. The causal and scaling tests in the table decide which model is justified.

Diagnostic exercise

A detector receives a wave at $f_{\mathrm{det}}=50\,\mathrm{Hz}$ from a comoving source at $z=2$. Along most of the path $\lambda/L=10^{-3}$. It crosses a thin lens whose redshifted gravitational time is $GM_L(1+z_L)=10^{-4}\,\mathrm{s}$. Calculate the source-frame frequency and $w$. Which approximation should describe the smooth segments and which should describe the lens? Would an apparent scalar projection at the lens, by itself, establish a genuine extra polarization?

Solution

The frame map gives

$$f_{\mathrm{src}}=(1+z)f_{\mathrm{det}} =3\times50\,\mathrm{Hz}=150\,\mathrm{Hz}.$$

The lens parameter is

$$w=8\pi\left(10^{-4}\,\mathrm{s}\right) \left(50\,\mathrm{s}^{-1}\right) =0.04\pi\simeq0.126.$$

Because $\lambda/L=10^{-3}$, geometric-optics phase, focusing, and screen transport are controlled on the smooth portions of the path. At the lens, however, $w\ll1$, so discrete stationary images are not a controlled description; the normalized wave-optics amplification integral is required and tends toward weak net lensing as $w\to0$. An apparent scalar projection does not establish a new degree of freedom, because diffraction smears the single screen direction and the decomposition can depend on the observer. Evidence for a genuine extra polarization would require a theory-specific independent radiative mode and an observer-invariant propagation prediction.

References

• Chiara Caprini and Daniel G. Figueroa, “Cosmological Backgrounds of Gravitational Waves,” Classical and Quantum Gravity 35 (2018) 163001, arXiv:1801.04268v3, DOI, Open PDF.
• Éanna É. Flanagan and Scott A. Hughes, “The Basics of Gravitational Wave Theory,” New Journal of Physics 7 (2005) 204, arXiv:gr-qc/0501041v3, DOI, Open PDF.
• Eric Poisson, Adam Pound, and Ian Vega, “The Motion of Point Particles in Curved Spacetime,” Living Reviews in Relativity 14 (2011) 7, arXiv:1102.0529v3, DOI, Open PDF.
• Giulia Cusin and Macarena Lagos, “Gravitational Wave Propagation Beyond Geometric Optics,” Physical Review D 101 (2020) 044041, arXiv:1910.13326v2, DOI, Open PDF.
• Ryuichi Takahashi and Takashi Nakamura, “Wave Effects in the Gravitational Lensing of Gravitational Waves from Chirping Binaries,” The Astrophysical Journal 595 (2003) 1039–1051, arXiv:astro-ph/0305055v2, DOI, Open PDF.