Waves, Optics, and Radiation

Wave physics connects a local field equation to a signal measured elsewhere. That connection is not one approximation but a sequence of questions: Where can influence propagate? How does a packet move? Which inverse represents a prepared source? When can wavefronts be replaced by rays? Which source moments survive far away? How do poles appear in a driven experiment? This chapter keeps those questions separate and then shows how they fit together.

Helpful background. Calculus and Linear Algebra for Relativity supplies multivariable derivatives and Fourier manipulations; Differential Equations for Relativity supplies initial-value and forced-oscillator methods; Classical Fields and Their Actions supplies field equations, currents, and energy flux; Partial Differential Equations and Green Functions supplies characteristics, distributions, and fundamental solutions.

The hierarchy of wave questions

Each row below has a different hypothesis. Passing one row does not automatically pass another.

Question	Control or choice	What it determines
Causal propagation	Principal symbol	Characteristic cone and domain of dependence
Packet motion	Narrow spectral width on one smooth branch	Group transport and dispersive spreading
Source response	Retarded, advanced, or other support condition	Which Green-function inverse is used
Geometric optics	$\varepsilon=\lambda_{\rm phys}/L\ll1$	Rays and transport through a slowly varying background
Source multipoles	$kd\ll1$	Hierarchy of monopole, dipole, quadrupole, and higher moments
Radiation zone	$r\gg d$ and $kr\gg1$	Leading outgoing $1/r$ field and finite far-zone flux
Linear response	Small forcing plus a retarded prescription	Susceptibility, poles, damping, and real-axis peaks

Here $d$ is source size, $r$ is observer distance, $k=\omega$ for a massless monochromatic wave with $c=1$, and $\lambda_{\rm phys}=2\pi/k$. The source condition $kd\ll1$ and observer condition $kr\gg1$ are independent: moving a detector outward does not make a large source multipolarly small.

Propagation conventions

The global metric convention gives

$$\eta_{\mu\nu}=\operatorname{diag}(-1,1,1,1), \qquad \Box=-\partial_t^2+\nabla^2.$$

For source calculations this chapter writes

$$L\psi\equiv(\partial_t^2-\nabla^2)\psi=s, \qquad L=-\Box.$$

Fourier modes use $e^{i(\mathbf k\cdot\mathbf x-\omega t)}$. For a stable system with decaying retarded response, damped-mode poles lie below the real $\omega$ axis and the response is analytic above it; undamped modes require a boundary-value prescription on the real axis. An unstable retarded system can instead have growing-mode poles in the upper half-plane: causality alone is not stability. In three spatial dimensions, $r=|\mathbf x|$ and retarded time is $u=t-r$.

From characteristics to response

The Wave Equation and Characteristics derives the null characteristic condition, a shrinking-cone energy identity, and finite domain of dependence. Fourier Modes, Dispersion, and Group Velocity then builds a narrow packet and separates phase, group, energy, and front velocities. The basic representation and finite-speed results are developed in Oh 2014, §§2–4.2, pp. 2–10 (official PDF).

Retarded and Advanced Green Functions separates the local equation from the support condition imposed on its inverse. A sourced solution still has a homogeneous part fixed by independent initial or incoming data.

The Eikonal and Geometric-Optics Limits derives the eikonal and amplitude-transport equations from a short-wavelength expansion and shows why caustics require a uniform wave approximation. Multipole Expansion and Radiation Zones instead expands a compact source and then takes an exterior far-field limit. The stationary-phase, modulation, and geometric-optics tools are developed in Rosales 2023, §§3.1–4.1, pp. 15–28, §§7.1.2–7.1.3, pp. 40–43, and §7.1.6, pp. 46–47 (official PDF).

Scattering, Resonance, and Linear Response closes the chapter by deriving flux-normalized reflection and transmission, solving a one-dimensional barrier, and connecting lower-half-plane poles to ringdown and forced response. A real-frequency maximum depends on the measured observable and need not equal the real part of a pole.

Check your understanding

A compact monochromatic source obeys $kd\ll1$. A detector lies outside the source with $d\ll r$ but still has $kr\ll1$. Which approximation is controlled, and is the detector in the radiation zone?

Solution

The multipole expansion is controlled because $kd\ll1$: successive source moments are parametrically ordered. The detector is nevertheless in the exterior near zone because $kr\ll1$, not the radiation zone. Its field may vary in time and may contain quasistatic and induction terms, but the far-zone $1/r$ term does not yet dominate. Moving to $kr\gg1$ while retaining $r\gg d$, together with retarded or outgoing conditions, reaches the radiation regime. Source compactness and observer distance answer different questions.

For a fixed wave operator $L$, both retarded and advanced Green functions solve $LG=\delta$. Why does the differential equation alone not choose the laboratory response?

Solution

The two inverses have different support. The retarded Green function vanishes before the source event and matches a source switched on after initial preparation; the advanced Green function has the opposite support. Their difference solves the homogeneous equation, so the inverse equation alone is not unique. Support, boundary, or initial conditions—and any independently specified homogeneous wave—complete the physical problem.

Outcomes and scope

After the chapter, you should be able to read a characteristic cone from a principal symbol, distinguish four wave velocities, normalize a causal Green function, derive eikonal and transport equations, keep $kd$ separate from $kr$, verify flux conservation in scattering, and interpret response poles without confusing them with real-axis peaks.

The setting remains flat spacetime and linear classical waves. Curved-spacetime tails, gravitational-wave generation, and black-hole resonances are later applications, not assumptions here. The next chapter is Thermodynamics, Statistical Mechanics, and Information, where response and dissipation acquire statistical interpretations.

References

• Sung-Jin Oh, Lectures on Wave Equation, University of California, Berkeley, 2014, §§2–4.2, pp. 2–10 (official PDF).
• R. R. Rosales, Lecture Topics for MIT 18.376, Massachusetts Institute of Technology, 2023, §§3.1–4.1, pp. 15–28, §§7.1.2–7.1.3, pp. 40–43, and §7.1.6, pp. 46–47 (official PDF).