Light Propagation and Geometric Optics

Light in general relativity is fundamentally an electromagnetic wave on spacetime, not a collection of point particles. When its reduced wavelength is uniformly short compared with curvature, envelope, and boundary scales, Maxwell’s equations organize into a geometric-optics hierarchy: phase surfaces are null, their generators are affine null geodesics, beam expansion transports wave action, observer frequency and detector geometry determine measured intensity, and the screen polarization is parallel transported. This chapter derives that chain and marks where caustics, diffraction, time dependence, or material dispersion break it.

Helpful background. Classical Fields and Their Actions repairs action and gauge-theory gaps; The Wave Equation and Characteristics reviews eikonal reasoning; Affine Connections and Covariant Derivatives reviews covariant transport. For a shorter foundations refresh, use Events, Intervals, and Minkowski Geometry for null directions and Tensor Fields and Natural Operations for forms. Timelike Congruences and Spatial Projection prepares expansion and shear.

From a field equation to an optical description

The durable problem is to explain when a wave can be represented by rays without turning an approximation into a definition. In source-free vacuum Maxwell theory, the small parameter is

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

where $L$ is no larger than any relevant curvature, amplitude, or boundary scale. The $O(\varepsilon^{-2})$ eikonal equation gives a null phase covector; the $O(\varepsilon^{-1})$ equation gives amplitude and polarization transport; terms beginning at $O(\varepsilon^0)$ supply finite-wavelength corrections. This hierarchy, including the null-fluid stress-energy and optical scalars, is developed in Dolan 2018, §§2–4, pp. 3–11, PDF.

No ray statement here replaces the exact Maxwell initial-value problem. A smooth ray congruence can fail at a caustic while the exact field stays finite. Several phases can interfere, curvature can produce wave tails, and boundaries can diffract a field. A plasma or material medium introduces a constitutive dispersion relation, so its rays need not follow the metric null cone.

The six steps

When learning the derivation, read the chapter in the order shown:

1. Maxwell Fields on Curved Spacetime obtains $dF=0$ and $d\star F=\star j$, where $j$ is the current one-form, translates their signs consistently to components, and shows how curvature enters the potential wave equation.
2. The Eikonal Expansion inserts a controlled phase–amplitude ansatz, separates powers of $\varepsilon$, and identifies the residual left by truncation.
3. Light Rays as Null Geodesics uses both $k^2=0$ and the exact-gradient property $k=d\theta$ to derive affine geodesic motion rather than postulate it.
4. Amplitude and Polarization Transport proves $\nabla_\mu(\mathcal A^2k^\mu)=0$, the amplitude–area law, and parallel transport of the physical screen polarization modulo gauge.
5. Light Bundles and Area Distance constructs the screen Jacobi map, defines $D_A^2=|\det\mathcal D|$, and distinguishes a caustic, parity change, and coordinate distance. The underlying spacetime treatment is given in Perlick 2010, §§2.3–2.4, pp. 13–21, PDF.
6. Fermat Principles in Stationary Spacetimes solves the stationary null condition for a Randers travel-time functional and states the timelike-Killing, spacelike-slice, convexity, and endpoint conditions under which its stationary paths represent rays.

The chapter stops before full wave scattering, black-hole perturbation theory, and observational lens reconstruction. It also does not treat dispersive optical media as if they were vacuum geometry.

Reader routes and recurring checks

For the shortest derivation of why light follows null geodesics, read steps 1–3. For brightness and polarization, add step 4. For angular size, focusing, and caustics, follow steps 3–5. For stationary travel-time methods, read steps 3 and 6; step 5 explains why “stationary” must not be replaced globally by “minimum.”

Three checks recur. First, flat spacetime must recover plane waves, straight null rays, constant collimated amplitude, and $D_A=|\lambda|$ for a vertex bundle. Second, a gauge shift may change the potential or its longitudinal polarization representative but not $F$ or the screen class. Third, the geometric-optics amplitude divergence at $\det\mathcal D=0$ must be interpreted as a nonuniform approximation, not infinite exact energy. Reall’s distinction between general geodesics, affine parameters, and null normalization is useful here; see Reall 2013, §4.3, pp. 45–46, PDF.

Check your preparation

A vacuum beam has $\varepsilon\ll1$ and propagates away from sources. Its infinitesimal cross-sectional area grows by a factor of $9$ between two events, with no caustic between them. What happens to its leading potential amplitude and amplitude squared? When may the same factor be assigned to an observer’s measured energy flux? Which conclusion survives if the beam then reaches a caustic, and which survives if it enters a dispersive plasma?

Solution

Wave-action conservation gives $\mathcal A^2\mathscr S=\text{constant}$. The amplitude therefore falls by $9^{-1/2}=1/3$, and its square falls by $1/9$. An observer measures energy density proportional to $\mathcal A^2\omega^2$, where $\omega=-u\cdot k/\varepsilon$; flux through a detector also contains an incidence-angle projection. Hence the measured flux falls by exactly $1/9$ only if the two measurements use the same measured frequency and detector geometry. At a caustic, the ray equations and Jacobi fields can still be continued, but the single-phase amplitude formula becomes nonuniform and cannot predict a finite exact intensity. In a dispersive plasma, neither the vacuum null condition nor its amplitude law may be imported without the medium’s constitutive dispersion relation; one must rederive the Hamiltonian rays and transport equation at the relevant frequency.

Onward connection

The optical equations measure curvature through beam distortion, but they sample only particular screen projections. Continue with Curvature as Gravitation to reconstruct the full local invariant, separate Ricci from Weyl curvature, and connect light-bundle focusing with tidal measurements and holonomy.

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.
• Perlick, Volker. “Gravitational Lensing from a Spacetime Perspective.” Living Reviews in Relativity 7 (2004): 9; revised arXiv version, 2010. DOI. Open PDF.
• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes, 2013. PDF.