Geometric Optics for Gravitational Waves

When a gravitational wavelength $\lambda$ is much shorter than the background curvature scale $L$, phase varies rapidly while amplitude, polarization, and geometry vary slowly. Expanding in $\epsilon=\lambda/L$ turns the curved-spacetime wave equation into a hierarchy: the leading equation makes the wave covector null, its gradient character makes the rays affinely parametrized null geodesics, and the next equation transports amplitude and polarization. This is geometric optics for gravitational waves. It fails near caustics and whenever wavelength-scale curvature or lens structure makes diffraction important. Instrument likelihoods and theory-specific modified dispersion are not treated here.

Required background. Waves, Green Functions, and Geometric Optics supplies eikonal expansions, characteristics, and transport equations; Curvature supplies the Riemann coupling and geodesic-deviation scale.

Helpful background. Einstein Equations supplies the linearized vacuum equation; Relativistic Cosmology supplies expanding backgrounds and source-to-observer frames.

The eikonal hierarchy

Let $\bar g_{\mu\nu}$ be a slowly varying background and $h_{\mu\nu}$ a small perturbation. In traceless Lorenz gauge on a vacuum background, take the convention-explicit wave equation

$$\bar\square h_{\mu\nu} -2\bar R_{\alpha\mu\nu\beta}h^{\alpha\beta}=0, \qquad \bar\nabla^\mu h_{\mu\nu}=0, \qquad h^\mu{}_\mu=0.$$

Use the rapidly varying ansatz

$$h_{\mu\nu} =\operatorname{Re}\left\{ \left(A^{(0)}_{\mu\nu}+\epsilon A^{(1)}_{\mu\nu}+\cdots\right) e^{i\theta/\epsilon} \right\}, \qquad k_\mu:=\bar\nabla_\mu\theta.$$

The phase is chosen so a local positive-frequency wave has $\theta=-\omega_0 t+\boldsymbol{k}_0\!\cdot\boldsymbol{x}$. The physical phase is $\theta/\epsilon$, so its wave covector is $K_\mu=k_\mu/\epsilon$ and its local frequency is $\omega_0/\epsilon$. Raising the index makes both $k^\mu$ and $K^\mu$ future directed. The terms of order $\epsilon^{-2}$ give

$$k_\mu k^\mu=0.$$

Thus constant-phase hypersurfaces are null. The leading gauge condition also gives $k^\mu A^{(0)}_{\mu\nu}=0$, while the trace condition removes the scalar trace. The curvature term carries no rapid derivatives and first affects lower orders. Cusin and Lagos display this hierarchy and its sourced corrections beyond geometric optics Cusin and Lagos 2020, §§II–III, pp. 3–8, PDF.

Affine null rays

Because $k_\mu$ is a gradient, its covariant derivative is symmetric. Differentiate the eikonal equation:

$$k^\nu\bar\nabla_\nu k_\mu =k^\nu\bar\nabla_\mu k_\nu =\frac12\bar\nabla_\mu(k^\nu k_\nu) =0.$$

Integral curves $dx^\mu/d\lambda=k^\mu$ are therefore null geodesics with affine parameter $\lambda$. An arbitrary rescaling varying along a ray would destroy affine parametrization; the normalization is fixed by a source or observer frequency,

$$\omega_{(u)}=-K_\mu u^\mu=-\frac{k_\mu u^\mu}{\epsilon},$$

for a future-directed observer $u^\mu$. Bending, focusing, and redshift arise from the background geometry even though the local dispersion relation remains null.

Amplitude and polarization transport

Write the leading tensor as

$$A^{(0)}_{\mu\nu}=\mathcal A\,e_{\mu\nu}, \qquad e_{\mu\nu}\bar e^{\mu\nu}=2.$$

The next eikonal order is

$$2k^\alpha\bar\nabla_\alpha A^{(0)}_{\mu\nu} +(\bar\nabla_\alpha k^\alpha)A^{(0)}_{\mu\nu}=0.$$

Separate magnitude from direction to obtain

$$k^\alpha\bar\nabla_\alpha e_{\mu\nu}=0, \qquad k^\alpha\bar\nabla_\alpha\ln\mathcal A =-\frac12\bar\nabla_\alpha k^\alpha.$$

Polarization is parallel transported along the ray, and the current

$$J^\mu=\mathcal A^2k^\mu, \qquad \bar\nabla_\mu J^\mu=0,$$

expresses leading flux conservation. If a narrow ray bundle has screen area $\mathcal S$, then $d\ln\mathcal S/d\lambda=\bar\nabla_\mu k^\mu$ before a caustic, so $\mathcal A\sqrt{\mathcal S}$ is constant. Flanagan and Hughes explain why the short-wavelength split is locally meaningful and how curvature drives slow evolution Flanagan and Hughes 2005, §5, pp. 23–29, PDF.

Screen basis and validity

Choose two orthonormal spacelike vectors $s_1^\mu,s_2^\mu$ orthogonal to $k^\mu$ and to an auxiliary null direction. Parallel transport them along the ray. The tensors

$$e^+_{\mu\nu}=s_{1\mu}s_{1\nu}-s_{2\mu}s_{2\nu}, \qquad e^\times_{\mu\nu}=s_{1\mu}s_{2\nu}+s_{2\mu}s_{1\nu}$$

span the two transverse tensor polarizations. Their components stay constant in this transported screen, although a different observer screen may be rotated. Such basis rotation is not a new polarization degree of freedom.

The expansion requires $\lambda/L\ll1$ and slow variation of amplitude across a wavelength. At a caustic, geometric optics predicts $\mathcal S\to0$ and a divergent amplitude; the exact wave stays finite and requires diffraction or a uniform approximation. Curvature scattering and tails likewise involve information that cannot be assigned to one ray alone.

Flat-spacetime check

For this explicit solution, absorb the bookkeeping parameter $\epsilon$ into the physical frequency and denote the physical wave covector by $k_\mu$. In Minkowski spacetime choose a ray along $+z$,

$$k_\mu=(-\omega,0,0,\omega), \qquad \theta=-\omega(t-z),$$

and the constant screen vectors $s_1^\mu=(0,1,0,0)$ and $s_2^\mu=(0,0,1,0)$. Then $k^\mu=(\omega,0,0,\omega)$ is future directed, $k^\nu\partial_\nu k^\mu=0$, and the expansion $\partial_\mu k^\mu$ vanishes. Both $\mathcal A$ and the polarization tensor are therefore constant along the ray. A complex representative is

$$h_{\mu\nu} =\frac{\mathcal A}{\sqrt2}\left(e^+_{\mu\nu}-i e^\times_{\mu\nu}\right) e^{-i\omega(t-z)},$$

whose real part is a transverse plane wave. The factor $1/\sqrt2$ keeps $e_{\mu\nu}\bar e^{\mu\nu}=2$ as declared above. This elementary limit checks three conventions at once: the sign in $e^{-i\omega t}$, the future direction of the raised wave vector, and the absence of spurious amplitude or polarization evolution in a nonexpanding parallel beam.

The ray description should not be overinterpreted. It determines the local wavefront and leading transport, but it does not by itself describe interference between several rays, the interior-of-the-light-cone support of a curved-spacetime Green function, or wavelength-scale diffraction. Those are distinct propagation regimes developed on the following pages.

It also separates propagation from generation: the source fixes the initial amplitude and polarization on a wavefront, whereas the equations above evolve that data through the slowly varying exterior geometry. Any claim about emitted multipoles must therefore come from a source calculation, not from geometric optics alone.

Exercise

An outgoing bundle in flat spacetime subtends a fixed solid angle $\Delta\Omega$. Determine its amplitude scaling and show that the total power through the bundle is independent of radius.

Solution

The screen area is $\mathcal S=r^2\Delta\Omega$. Conservation of $\mathcal A^2\mathcal S$ gives $\mathcal A\propto r^{-1}$. The local energy flux is quadratic in a derivative of the wave, so at fixed frequency it scales as $\mathcal A^2\propto r^{-2}$. Multiplying by the bundle area $r^2\Delta\Omega$ removes the radial dependence, recovering constant radiated power and the familiar far-zone $1/r$ strain.

References

• 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.
• Giulia Cusin and Macarena Lagos, “Gravitational Wave Propagation Beyond Geometric Optics,” Physical Review D 101 (2020) 044041, arXiv:1910.13326v2, DOI, Open PDF.