Light Rays as Null Geodesics

A light ray is not an additional object postulated beside the electromagnetic field. In vacuum geometric optics it is an integral curve of the leading eikonal covector, derived from Maxwell’s equations when the reduced wavelength is small compared with all background scales. The derivation produces both nullness and affine geodesic motion; outside that asymptotic regime, an exact field need not admit a unique decomposition into rays.

Required background. Classical Fields and Their Actions supplies Maxwell gauge theory; The Wave Equation and Characteristics supplies the relation between phases and characteristics; Affine Connections and Covariant Derivatives supplies gradients and geodesic differentiation.

Helpful background. Events, Intervals, and Minkowski Geometry reviews null directions; Tensor Fields and Natural Operations distinguishes a covector from its raised vector; Timelike Congruences and Spatial Projection prepares the neighboring-ray description.

From the eikonal equation to affine geodesics

Write the rapidly varying wave factor as $e^{i\theta/\varepsilon}$, with $\varepsilon=\lambda_{\rm red}/L\ll1$ and $L$ a background variation scale. Let $\theta$ be its smooth scaled phase and define

$$k_\mu=\nabla_\mu\theta.$$

The leading Maxwell equation in the eikonal expansion is

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

Because the connection is torsion-free and $k_\mu$ is an exact one-form,

$$\nabla_\mu k_\nu=\nabla_\nu k_\mu.$$

This extra gradient property converts the eikonal equation into an equation of motion:

$$\begin{aligned} k^\nu\nabla_\nu k_\mu &=k^\nu\nabla_\mu k_\nu \\ &=\frac12\nabla_\mu(k^\nu k_\nu)=0. \end{aligned}$$

If $x^\mu(\lambda)$ is defined by $dx^\mu/d\lambda=k^\mu$, then

$$\frac{d^2x^\mu}{d\lambda^2} +\Gamma^\mu{}_{\alpha\beta} \frac{dx^\alpha}{d\lambda} \frac{dx^\beta}{d\lambda}=0.$$

Thus $\lambda$ is affine, not merely an arbitrary ray parameter. This gradient argument and its place in the eikonal hierarchy are stated explicitly in Dolan 2018, §3.1, pp. 6–7, PDF. The general affine-parameter freedom $\lambda\mapsto a\lambda+b$ and the absence of a preferred null proper time are reviewed in Reall 2013, §4.3, pp. 45–46, PDF.

Rays are generators of null wavefronts

The hypersurfaces $\theta=\text{constant}$ are null because their normal $k_\mu$ is null. A peculiarity of null hypersurfaces is that the raised normal is also tangent. Along a ray,

$$\frac{d\theta}{d\lambda} =k^\mu\nabla_\mu\theta=k^2=0,$$

so a ray runs within one constant-phase hypersurface as one of its null generators. Neighboring wavefronts label successive oscillations, whereas neighboring generators within a wavefront form a beam.

The same motion follows from the Hamiltonian

$$H(x,p)=\frac12g^{\mu\nu}(x)p_\mu p_\nu, \qquad H=0.$$

Hamilton’s equations give $\dot x^\mu=p^\mu$ and the affine geodesic equation. This is an independent check because it uses the characteristic Hamiltonian rather than differentiating $k^2=0$. Multiplying $H$ by a nonzero function preserves the unparametrized null curves but generally changes their parameterization.

Frequency, direction, and redshift

For an observer with future-directed unit velocity $u^\mu$, the leading measured angular frequency is

$$\omega_{(u)}=-\frac1\varepsilon u^\mu k_\mu+O(1).$$

It is positive for a future-directed ray under the $e^{-i\omega t}$ convention. The observer decomposes

$$\frac{k^\mu}{-u\cdot k}=u^\mu+n^\mu, \qquad u\cdot n=0, \qquad n\cdot n=1,$$

so $n^\mu$ is the observed spatial propagation direction. For the same eikonal crest emitted and received by observers $u_{\rm e}$ and $u_{\rm r}$,

$$\frac{\omega_{\rm r}}{\omega_{\rm e}} =\frac{(-u\cdot k)_{\rm r}}{(-u\cdot k)_{\rm e}}+O(\varepsilon).$$

This ratio includes both endpoint motion and the way $k_\mu$ is transported through the geometry. It does not assign an observer-independent frequency to a photon.

Why the ray description is approximate

The null geodesic is an exact curve of the background metric, but identifying it with energy transport of a given electromagnetic solution is only the leading asymptotic statement. A truncated field has finite-$\varepsilon$ corrections to its phase, stress-energy, and polarization. At caustics, neighboring generators cross, the leading amplitude diverges, and one phase no longer gives a uniform representation. Diffraction around wavelength-scale structure, interference among several phases, and curvature backscattering require the wave equation.

Vacuum Maxwell characteristics remain null, but that does not turn every exact wave packet into a point following one geodesic. In a plasma or dispersive material, the principal relation may instead be $H(x,k,\omega)=0$ with frequency dependence; the associated Hamiltonian rays can be timelike, spacelike relative to $g_{\mu\nu}$, or polarization-dependent. Such rays are not the null geodesics derived here.

Exercise

Let $K^\mu=fk^\mu$ with $f>0$, where $k^\mu$ is affinely parametrized. Determine when $K^\mu$ is also affine and identify its nonaffinity otherwise.

Solution

Using $k^\nu\nabla_\nu k^\mu=0$,

$$K^\nu\nabla_\nu K^\mu =f k^\nu\nabla_\nu(fk^\mu) =(k^\nu\nabla_\nu f)K^\mu.$$

Thus $K$ traces the same unparametrized geodesics and has nonaffinity $\kappa=k\cdot\nabla f$. It is affine exactly when $f$ is constant along each ray. A constant may differ between rays in a congruence; along any one ray it produces the usual affine rescaling.

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.
• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes, 2013. PDF.