The Eikonal Expansion

The eikonal expansion separates a rapidly varying phase from a slowly varying electromagnetic amplitude. Its leading equation fixes the characteristic cone; the next equation transports amplitude and polarization; lower orders contain curvature, diffraction, and other finite-wavelength corrections. The hierarchy is controlled only when the reduced wavelength is small compared with every background and envelope scale sampled by the wave.

Required background. Classical Fields and Their Actions supplies gauge fields and Lorenz gauge; The Wave Equation and Characteristics supplies WKB order counting; Affine Connections and Covariant Derivatives supplies covariant wave operators and curvature commutators.

Helpful background. Events, Intervals, and Minkowski Geometry reviews null covectors; Tensor Fields and Natural Operations reviews one-forms and tensor amplitudes; Timelike Congruences and Spatial Projection prepares the expansion scalar used in transport.

The short-wavelength ansatz

Work in a source-free region and impose Lorenz gauge. The potential obeys

$$\Box A_\mu-R_\mu{}^\nu A_\nu=0, \qquad \nabla_\mu A^\mu=0.$$

Choose a macroscopic scale $L$ no larger than the curvature radius, the amplitude-variation scale, or the scale on which boundary data vary. After measuring coordinates in units of $L$, introduce

$$\varepsilon=\frac{\lambda_{\rm red}}{L}\ll1,$$

and write the complexified potential as

$$A_\mu^{\mathbb C} =e^{i\theta/\varepsilon} \left(a_\mu+\varepsilon b_\mu+O(\varepsilon^2)\right), \qquad k_\mu\equiv\nabla_\mu\theta.$$

The physical field is the real part. The phase and coefficients have derivatives of order unity in the scaled variables. Locally a positive-frequency stationary phase has $\theta/\varepsilon=-\omega t+\cdots$, consistent with the inherited $e^{-i\omega t}$ convention. This ansatz excludes an envelope that changes on the wavelength scale.

The bookkeeping covector $k_\mu$ is the gradient of the slowly scaled phase $\theta$. The physical leading wave covector is $K_\mu=k_\mu/\varepsilon$, so an observer measures $\omega=-u\cdot k/\varepsilon$ in these scaled coordinates. Restoring physical lengths supplies the corresponding factor $L^{-1}$. Keeping this distinction prevents a missing high-frequency factor when the ray equations are compared with local frequency measurements.

For any slow one-form $v_\mu$,

$$\Box\left(v_\mu e^{i\theta/\varepsilon}\right) =e^{i\theta/\varepsilon} \left[ -\varepsilon^{-2}k^2v_\mu +i\varepsilon^{-1} \left(2k^\nu\nabla_\nu v_\mu+(\nabla_\nu k^\nu)v_\mu\right) +\Box v_\mu \right].$$

This identity makes the hierarchy explicit rather than hiding it in the phrase “high frequency.”

Leading order: phase and transversality

At order $\varepsilon^{-2}$, a non-gauge leading amplitude requires

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

Thus constant-phase hypersurfaces are null. The leading $\varepsilon^{-1}$ term in the Lorenz condition is

$$k_\mu a^\mu=0,$$

so the potential amplitude is transverse to the wave covector. Because a multiple of $k_\mu$ is itself transverse when $k^2=0$, this equation has not yet removed gauge freedom. The physical polarization is an equivalence class on the two-dimensional screen transverse to the ray, not a unique four-vector.

The same leading null equation follows from an ansatz for the gauge-invariant Faraday tensor. That route avoids potential-gauge bookkeeping and is worked out in Dolan 2018, §3.2, pp. 7–8, PDF. Agreement of the two routes is an independent check that the characteristic equation is physical.

Next order: transport along the phase gradient

Define $\Theta\equiv\nabla_\mu k^\mu$. At order $\varepsilon^{-1}$, the wave equation gives

$$2k^\nu\nabla_\nu a_\mu+\Theta a_\mu=0.$$

The order $\varepsilon^0$ Lorenz equation is $\nabla_\mu a^\mu+i k_\mu b^\mu=0$, or $k\cdot b=i\nabla\cdot a$. It determines the longitudinal part of the next coefficient without altering the leading transverse class. Splitting $a_\mu=\mathcal A f_\mu$ into a real magnitude and normalized screen polarization yields

$$k^\mu\nabla_\mu\mathcal A=-\frac12\Theta\mathcal A, \qquad k^\mu\nabla_\mu f_\nu=0$$

after a convenient residual-gauge choice. The first equation is flux dilution; the second is polarization transport. Their detailed interpretation belongs to the amplitude-and-polarization page.

Curvature first appears explicitly in the unscaled potential equation at order $\varepsilon^0$ through $\Box a_\mu-R_\mu{}^\nu a_\nu$ and terms involving $b_\mu$. If the series is stopped after leading transport, the differential-equation residual is $O(\varepsilon^0)$, two powers below the $O(\varepsilon^{-2})$ principal term. Solving for $b_\mu$ can move the residual to the next order, but it does not make the asymptotic series convergent or uniformly valid.

Validity, accumulation, and failure

The operational condition is not merely “large frequency.” Along the relevant ray segment one needs $\varepsilon\ll1$, slowly varying coefficients, no wavelength-scale boundary structure, and a beam width large compared with the wavelength. A correction that is locally $O(\varepsilon)$ can accumulate over a propagation distance of order $L/\varepsilon$, so the useful interval must also be controlled.

At a caustic, transverse ray scales collapse and the leading amplitude diverges; the expansion is then nonuniform even if curvature remains gentle. Diffraction, interference between multiple phases, backscattering tails, and sharp edges require a wave treatment or a uniform asymptotic approximation. In a dispersive medium, the leading polynomial is a constitutive dispersion relation rather than $k^2=0$, and $\varepsilon$ must also be compared with microscopic response scales. None of those effects is encoded by simply adding more vacuum null rays.

Exercise

Insert $\Phi=\mathcal A e^{i\theta/\varepsilon}$ into the scalar equation $\Box\Phi=0$ and recover the first two eikonal equations. Then show that the discarded residual is $e^{i\theta/\varepsilon}\Box\mathcal A$.

Solution

Direct differentiation gives

$$\Box\Phi=e^{i\theta/\varepsilon} \left[ -\varepsilon^{-2}k^2\mathcal A +i\varepsilon^{-1} \left(2k\cdot\nabla\mathcal A+\Theta\mathcal A\right) +\Box\mathcal A \right].$$

The coefficients of $\varepsilon^{-2}$ and $\varepsilon^{-1}$ vanish when

$$k^2=0, \qquad k\cdot\nabla\mathcal A=-\frac12\Theta\mathcal A.$$

What remains is exactly $e^{i\theta/\varepsilon}\Box\mathcal A$. It is smaller than the principal term in the equation by $O(\varepsilon^2)$ under the slow-variation assumptions, though the field itself generally has $O(\varepsilon)$ corrections.

References

• Dolan, Sam R. “Geometrical Optics for Scalar, Electromagnetic and Gravitational Waves on Curved Spacetime.” International Journal of Modern Physics D 27 (2018): 1843010. DOI. Open PDF.