Retarded Fields and the Radiation Zone

Retarded boundary conditions turn the linearized Einstein equation into a causal field sourced on the past light cone. For a localized source, expanding that retarded solution at large distance separates near, induction, and radiation terms. Only the leading time-dependent $1/r$ field transports a finite flux through spheres of increasing radius; a static $1/r$ potential can look similar in falloff but has different derivatives, angular structure, and energy transport.

Required background. Weak-Field Gravity is used for the Lorenz-gauge linearized Einstein equation and conserved source. Waves, Green Functions, and Geometric Optics supplies retarded Green functions, causal support, and the wavelength-based zone hierarchy.

Helpful background. Einstein Equations explains the Bianchi compatibility condition $\partial_\mu T^{\mu\nu}=0$. Equivalence Principles and Gravitational Measurement clarifies why a far-zone metric representative must ultimately be translated into curvature or another radiative observable.

The causal solution

In Lorenz gauge,

$$\Box\bar h_{\mu\nu}=-16\pi G T_{\mu\nu}, \qquad \partial^\mu\bar h_{\mu\nu}=0,$$

where $\Box=-\partial_t^2+\nabla^2$. The retarded solution in four-dimensional Minkowski spacetime is

$$\bar h_{\mu\nu}(t,\boldsymbol x) =4G\int d^3x'\, \frac{T_{\mu\nu}\!\left(t-\lvert\boldsymbol x-\boldsymbol x'\rvert, \boldsymbol x'\right)} {\lvert\boldsymbol x-\boldsymbol x'\rvert}.$$

It depends on the source at retarded time

$$t_{\mathrm{ret}}=t-\lvert\boldsymbol x-\boldsymbol x'\rvert,$$

so disturbances propagate on the past null cone. Choosing the advanced Green function would replace this with future source data and describe incoming rather than retarded response. The field equation alone permits both; the physical isolation condition selects the retarded solution.

The formula assumes flat-background propagation and a sufficiently localized, conserved source. In curved spacetime a retarded Green function can have support inside the light cone, producing tails. Cosmological expansion changes distance and redshift relations. Those effects are developed in Propagation, Lensing, and Wave Optics, not hidden inside the flat-space kernel.

Flanagan and Hughes derive the retarded solution, its source conservation identities, and its wave-zone expansion in Flanagan and Hughes 2005, §4.1, eqs. (4.1)–(4.23), pp. 17–20, PDF.

Large-distance expansion

Let the source occupy a region of size $d$, place the origin within it, and observe at $\boldsymbol x=r\boldsymbol n$ with $r\gg d$. Then

$$\lvert\boldsymbol x-\boldsymbol x'\rvert =r-\boldsymbol n\cdot\boldsymbol x' +O(d^2/r),$$

and

$$\frac{1}{\lvert\boldsymbol x-\boldsymbol x'\rvert} =\frac1r+O(d/r^2).$$

The leading field is therefore

$$\bar h_{\mu\nu}(t,r\boldsymbol n) =\frac{4G}{r} \int d^3x'\, T_{\mu\nu}\!\left(u+\boldsymbol n\cdot\boldsymbol x', \boldsymbol x'\right) +O(r^{-2}), \qquad u=t-r.$$

This expression has not yet made a slow-motion approximation. If the source size is also small compared with its characteristic wavelength, $d/\lambda\ll1$, the retardation across the source can be expanded in multipoles. Source conservation then removes independent monopole and dipole radiation in general relativity: total mass-energy and momentum control constrained or stationary pieces, while the leading tensor radiation begins with a time-varying quadrupole.

For slow internal motion, define the mass quadrupole

$$I_{ij}(u)=\int d^3x'\,T^{00}(u,\boldsymbol x')x'_i x'_j.$$

Twice using $\partial_\mu T^{\mu\nu}=0$ and discarding boundary terms for a localized source gives

$$\int d^3x'\,T^{ij} =\frac12\ddot I_{ij}.$$

The intermediate step makes the factor of two clear: one integration by parts gives $\dot I_{ij}=\int(T^{0i}x'_j+T^{0j}x'_i)d^3x'$. Differentiating again and using momentum conservation gives two identical integrals of the symmetric stress $T^{ij}$. For a self-gravitating source, matter alone is not exactly conserved with flat derivatives; gravitational stresses must enter a consistent higher-order source. The identity here is the conserved linear-source calculation, whose systematic extension is described in Flanagan and Hughes 2005, §4.2, pp. 20–21, PDF.

Consequently the leading spatial wave-zone field is

$$\bar h_{ij}(u,r\boldsymbol n) =\frac{2G}{r}\ddot I_{ij}(u)+\cdots,$$

and its physical tensor part is obtained by trace-free TT projection. The systematic distinction among source, canonical, and radiative moments belongs to Multipolar Wave Generation; this leading identity only exposes how retardation and conservation produce the quadrupole baseline.

Near, induction, and radiation behavior

For a source varying at angular frequency $\omega$, compare $r$ with the reduced wavelength $\omega^{-1}$:

Region	Scale	Characteristic behavior
Near zone	$\omega r\ll1$	Quasi-instantaneous constraint fields and the strongest inverse powers of $r$
Transition or induction zone	$\omega r\sim1$	Retardation matters; several inverse powers are comparable
Radiation zone	$\omega r\gg1$	Outgoing phase depends on $u=t-r$ and the radiative field scales as $1/r$

For a quadrupolar source, the schematic hierarchy is

$$h_{\mathrm{quad}} \sim G\left( \frac{I}{r^3} +\frac{\dot I}{r^2} +\frac{\ddot I}{r} \right),$$

with angular projectors and coefficients suppressed. The first term is Coulombic, the second is inductive, and the third is radiative. A derivative of the outgoing $1/r$ term scales as $\omega/r$, so its quadratic flux density scales as $1/r^2$ and integrates to a finite value over a sphere. Faster falloffs do not carry finite energy to infinity.

Falloff alone is insufficient: a static monopolar metric also contains $1/r$. Radiation is identified by retarded-time dependence in the transverse tensor field, nonzero radiative curvature or Bondi news, and outgoing boundary conditions.

Checks on a retarded calculation

A consistent result should pass all of the following:

• Causality: the field at $(t,\boldsymbol x)$ uses no source data later than its past null cone.
• Conservation: the Lorenz condition follows from $\partial_\mu T^{\mu\nu}=0$ with compatible boundary data.
• Dimensions: $G\ddot I/r$ is dimensionless when $c=1$.
• Falloff: the leading outgoing radiative amplitude is $O(r^{-1})$; its curvature is also $O(r^{-1})$ at fixed retarded time.
• Stationary limit: constant mass and momentum produce no oscillatory TT radiation.
• Direction: $f(t-r)/r$ is outgoing, whereas $f(t+r)/r$ is incoming.

This page does not compute nonlinear compact-binary motion or detector noise. The former supplies $T_{\mu\nu}$ or multipoles through Relativistic Two-Body Dynamics; the latter belongs to Gravitational-Wave Astronomy. Retarded propagation is the interface between them, not a substitute for either.

Common pitfalls

Calling every $1/r$ term radiation. A stationary monopole also falls as $1/r$. Check retarded-time variation, TT curvature, and finite outward flux.

Using $r\gg d$ as the only wave-zone condition. A source can be spatially compact yet observed at $r\lesssim\lambda$. The radiation zone additionally requires $\omega r\gg1$ for the relevant frequencies.

Expanding retardation before stating $d/\lambda\ll1$. The far-distance expansion $d/r\ll1$ and the slow-source multipole expansion $d/\lambda\ll1$ are independent approximations.

Exercises

1. Outgoing versus incoming. For $\phi=f(t-r)/r$, show that the leading large-$r$ derivatives satisfy $\partial_r\phi=-\partial_t\phi+O(r^{-2})$. What sign results for $f(t+r)/r$?

Solution

For the outgoing field,

$$\partial_t\phi=\frac{f'(t-r)}{r}, \qquad \partial_r\phi=-\frac{f'(t-r)}{r}-\frac{f(t-r)}{r^2}.$$

Thus $\partial_r\phi=-\partial_t\phi+O(r^{-2})$. For the incoming field $f(t+r)/r$, the leading derivative instead satisfies $\partial_r\phi=+\partial_t\phi+O(r^{-2})$. This derivative test identifies the null orientation without assuming a monochromatic signal.

2. Why no mass-monopole wave? Let $M(t)=\int d^3x\,T^{00}$. Use source conservation and localization to show $\dot M=0$.

Solution

Conservation gives $\partial_tT^{00}=-\partial_iT^{i0}$. Therefore

$$\dot M=-\int d^3x\,\partial_iT^{i0} =-\oint_{\infty}dS_i\,T^{i0}=0,$$

because a localized source has no stress–energy flux through the boundary at spatial infinity. A constant total mass can supply a Coulombic $1/r$ field but not an oscillatory mass-monopole wave. If matter or energy crosses the boundary, the system was not the isolated source assumed here.

References

• Flanagan, Éanna É., and Scott A. Hughes. “The Basics of Gravitational Wave Theory.” New Journal of Physics 7 (2005): 204. DOI. Open PDF.