Gravitational Radiation in Linearized Theory

Linearized gravity is the cleanest place to see what a gravitational wave is and what it is not. Expanding the Einstein equation about Minkowski spacetime reveals two freely propagating tensor degrees of freedom. Gauge fixing makes those modes easy to represent, curvature makes them measurable, a retarded Green function connects them to a localized source, and short-wavelength averaging assigns them an effective energy flux. Each step has a distinct domain; keeping those domains visible prevents the TT metric, tidal field, far-zone strain, and Isaacson tensor from being mistaken for one exact object.

Helpful background. Weak-Field Gravity supplies perturbative metrics and gauge transformations. Waves, Green Functions, and Geometric Optics supplies characteristics, retarded solutions, and wavelength expansions. Einstein Equations supplies constraints and Bianchi consistency. Equivalence Principles and Gravitational Measurement supplies local inertial frames and tidal observables.

Enter the weak-field wave zone

The chapter begins from

$$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}, \qquad \lvert h_{\mu\nu}\rvert\ll1,$$

with signature $(-,+,+,+)$. Define the trace-reversed field

$$\bar h_{\mu\nu} =h_{\mu\nu}-\frac12\eta_{\mu\nu}h.$$

In Lorenz gauge, a conserved source obeys

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

Three restrictions must not be collapsed into “the weak-field approximation”:

• small amplitude: terms beyond the retained order in $h_{\mu\nu}$ are negligible;
• vacuum wave region: $T_{\mu\nu}=0$ locally, although a distant source may set retarded boundary data;
• short wavelength: $\lambda$ is small compared with a background scale, needed for geometric optics and Isaacson averaging but not for every linearized solution.

A linear calculation can be valid without being in the short-wavelength regime. A radiation-zone calculation additionally needs distance large compared with source size and relevant wavelength. A TT representative additionally needs a vacuum region and compatible boundary data.

Flanagan and Hughes provide the common derivational spine used in this chapter, from linearized gauge symmetry through effective wave energy Flanagan and Hughes 2005, §§2–5, pp. 4–29, PDF.

Six questions, in order

Page	Question answered	Independent check
Radiative Degrees of Freedom	Why do ten metric components contain only two free wave polarizations?	Phase-space constraint count and vanishing curvature for pure gauge
Transverse–Traceless Gauge	When can the two polarizations be isolated in a TT representative?	TT-projector transversality, trace, and idempotence
Plane Gravitational Waves	How do null propagation, plus/cross tensors, and spin weight fit together?	Insert $e^{-i\omega(t-z)}$ into the wave equation
Tidal Response to a Passing Wave	What does a freely falling apparatus actually measure?	Gauge-invariant geodesic deviation and trace-free ring deformation
Retarded Fields and the Radiation Zone	How does a localized source produce an outgoing $1/r$ field?	Causality, conservation, falloff, and stationary limit
Isaacson Effective Stress–Energy	Under what averaging does a wave carry energy–momentum?	Positive plane-wave flux and $\lambda\ll\ell\ll L$ scale separation

Read them in this order on a first pass. The logic is cumulative: the degree count precedes gauge specialization; the plane-wave representative precedes its tidal interpretation; the retarded solution precedes the zone hierarchy; and only then does averaged energy become controlled.

The invariant thread

With the passive convention $x'^\mu=x^\mu+\xi^\mu$ used throughout this chapter, the gauge transformation

$$h_{\mu\nu}\mapsto h_{\mu\nu}-2\partial_{(\mu}\xi_{\nu)}$$

changes metric components but leaves the linearized Riemann tensor unchanged. For a vacuum TT wave,

$$R^{(1)}_{0i0j} =-\frac12\ddot h^{\mathrm{TT}}_{ij}.$$

This one equation connects representation to measurement. It shows why a coordinate perturbation can be nonzero with no tidal field, why a freely falling detector responds to curvature, and why two integrations and their constants are required when strain is reconstructed from curvature.

It does not make every use of $h^{\mathrm{TT}}_{ij}$ exact. The local plane-wave field, the leading far-zone strain, and the Bondi strain at null infinity coincide only in their controlled overlap after retarded time, tetrad, radius, gauge, and normalization are aligned. The next chapter, Null Infinity and the Bondi–Sachs Framework, replaces this overlap argument by asymptotic radiative data.

The causal thread

Retarded boundary conditions give

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

At $r\gg d$, with $\boldsymbol x=r\boldsymbol n$, the leading field is proportional to $1/r$ at fixed $u=t-r$. If the source is also slow, $d/\lambda\ll1$, conservation reduces the leading tensor radiation to a time-varying quadrupole. These are two separate expansions:

$$\frac{d}{r}\ll1 \quad\text{and}\quad \frac{d}{\lambda}\ll1.$$

The first is a large-distance expansion. The second is a multipole/slow-source expansion. Their distinction becomes essential in Multipolar Wave Generation, where source, canonical, and radiative moments and their hereditary terms are kept separate.

The energy thread

There is no exact pointwise matter-like stress tensor for the gravitational field. In the high-frequency regime, however, let $\lambda$ be the wave scale, $L$ the background scale, and $\ell$ an averaging scale with

$$\lambda\ll\ell\ll L.$$

Then the leading effective tensor in short-wavelength TT gauge is

$$t^{\mathrm{GW}}_{\mu\nu} =\frac{1}{32\pi G} \left\langle \nabla_\mu h^{\mathrm{TT}}_{\alpha\beta} \nabla_\nu h_{\mathrm{TT}}^{\alpha\beta} \right\rangle.$$

For an outgoing plane wave,

$$\rho_{\mathrm{GW}}=t^{0z}_{\mathrm{GW}} =\frac{1}{16\pi G} \left\langle\dot h_+^2+\dot h_\times^2\right\rangle>0.$$

This positive null flux checks the polarization normalization and outward orientation. The result is averaged and asymptotic; Radiative Energy, Momentum, and Angular Momentum later derives global fluxes at null infinity and horizons without pretending that every construction is the same local tensor.

The nonlinear high-frequency expansion and the effective null stress tensor were established in Isaacson 1968, Part II, pp. 1272–1279; Part I supplies the associated curved-background geometric-optics hierarchy.

Conventions used throughout the chapter

• Coordinates and perturbations inherit $(-,+,+,+)$, and $\Box=-\partial_t^2+\nabla^2$.
• For a wave moving toward $+z$, $u=t-z$ and $e^{ik\cdot x}=e^{-i\omega u}$.
• The plus/cross tensors have norm $e^A_{ij}e^{B\,ij}=2\delta^{AB}$.
• Complex strain is $h_+-i h_\times$ and has spin weight $-2$ under the declared dyad phase rotation.
• Retarded fields are outgoing; $f(t-r)/r$ and $f(t+r)/r$ are not interchangeable.
• Outward energy flux is positive. Source energy loss carries the opposite sign.
• $G$ remains explicit; $c=1$.

The chapter’s curvature sign follows directly from the site’s Riemann convention. A source using the opposite signature or Riemann sign must be translated as a complete set before its $\Psi_4$, strain, or flux formula is imported.

Scope and related topics

The chapter develops source-independent linearized radiation: degrees of freedom, wave-zone gauge, plane waves, tidal curvature, flat-background retarded fields, and short-wavelength effective energy. It does not develop detector noise models or nonlinear compact-binary source dynamics.

Relativistic Two-Body Dynamics determines the orbital or source evolution that supplies moments and stress–energy. Production nonlinear evolution and numerical extraction algorithms are covered in Computational Relativity; finite detector response, calibration, noise, and inference in Gravitational-Wave Astronomy. This chapter supplies a convention-explicit field that those subjects can generate, extract, or measure.

Review the chapter

1. Four statements about one wave. An outgoing weak wave is written in TT gauge as $h^{\mathrm{TT}}_{ij}=A_{ij}\cos\omega(t-z)$. State one check each for gauge content, propagation, measurement, and energy.

Solution

For gauge content, verify $A_{iz}=0$ and $A^i{}_i=0$, leaving two independent screen components; adding pure gauge must not change the linearized Riemann tensor. For propagation, $k^\mu=(\omega,0,0,\omega)$ is null and insertion into $\Box h_{ij}=0$ vanishes. For measurement, compute $R_{0i0j}=-\ddot h^{\mathrm{TT}}_{ij}/2$ and use geodesic deviation. For energy, average over several cycles to obtain a positive $t^{0z}=\omega^2(A_+^2+A_\times^2)/(32\pi G)$.

2. Which approximation failed? A source has size $d\ll r$, but the observer lies at $\omega r=0.2$. May the leading $1/r$ expression be called a radiation-zone waveform solely because $r\gg d$?

Solution

No. The condition $d/r\ll1$ justifies a large-distance expansion relative to source size, but $\omega r=0.2$ places the observer inside the wavelength-scale transition rather than the radiation zone. Near and induction terms can remain comparable to the nominal radiative term. One must keep the full retarded hierarchy or move to $\omega r\gg1$ before interpreting the leading transverse $1/r$ piece as the asymptotic waveform.

3. Averaging test. A wave has wavelength $\lambda=1\,\mathrm{km}$ on a background varying over $L=10\,\mathrm{km}$. Is choosing $\ell=3\,\mathrm{km}$ enough to establish an Isaacson regime?

Solution

No strong scale separation exists: $\lambda/\ell=1/3$ and $\ell/L=0.3$ are not parametrically small. Smoothing may produce a number, but the asymptotic rules that suppress total derivatives and separate fast from slow fields are poorly controlled. The calculation should retain finite-wavelength corrections or state a numerical averaging dependence rather than claim the leading Isaacson tensor as a precise result.

References

• Flanagan, Éanna É., and Scott A. Hughes. “The Basics of Gravitational Wave Theory.” New Journal of Physics 7 (2005): 204. DOI. Open PDF.
• Isaacson, Richard A. “Gravitational Radiation in the Limit of High Frequency. II. Nonlinear Terms and the Effective Stress Tensor.” Physical Review 166 (1968): 1272–1279. DOI.