Gravitational Waves Are Not Pure Coordinate Artifacts

A coordinate transformation can change every visible component of a metric perturbation, but it cannot remove the tidal curvature of a gravitational wave. The safe correction is therefore not “the metric perturbation is observable.” It is that radiative curvature, relative detector motion, and null-infinity news survive the appropriate gauge test, while strain components, tetrads, waveform modes, and finite-radius extractions still require declared frames and conventions.

Gauge freedom and radiative curvature

On a Minkowski background write

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

With the site’s passive convention $x'^\mu=x^\mu+\xi^\mu$, an infinitesimal coordinate change gives

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

Thus an individual component such as $h_{xx}$ is not, by itself, a physical wave. The linearized Riemann tensor

$$R^{(1)}_{\alpha\beta\gamma\delta} =\frac12\left( \partial_\gamma\partial_\beta h_{\alpha\delta} +\partial_\delta\partial_\alpha h_{\beta\gamma} -\partial_\delta\partial_\beta h_{\alpha\gamma} -\partial_\gamma\partial_\alpha h_{\beta\delta} \right)$$

is unchanged: substituting the gauge term makes the third derivatives cancel pairwise. A time-dependent metric written in unusual coordinates can therefore have $R^{(1)}{}_{\alpha\beta\gamma\delta}=0$, whereas a genuine weak wave has radiative curvature that no smooth infinitesimal coordinate change can erase. This is the central distinction in Flanagan and Hughes 2005, §§2–3, pp. 4–17.

Curvature alone is not a complete radiation criterion: a static tidal field also has curvature. Radiation additionally requires the appropriate propagation and boundary conditions—for example, an outgoing $1/r$ field in a wave zone or nonzero news at future null infinity.

What a freely falling detector measures

Let $U^a$ be a reference geodesic and $\zeta^a$ the separation to a neighboring geodesic. In an orthonormal frame carried by the observer,

$$\frac{d^2\zeta^{\hat i}}{d\tau^2} =-R^{\hat i}{}_{\hat0\hat j\hat0}\, \zeta^{\hat j}.$$

For a vacuum wave in a transverse–traceless representative,

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

The coordinates of the test masses can remain fixed in TT coordinates and move in a local proper frame; both descriptions predict the same relative acceleration and light-travel-time response. It is the frame-projected curvature and the complete emission–propagation–reception experiment—not the coordinate motion of one mass—that carries operational meaning. The full calculation is developed in Radiative Degrees of Freedom and Tidal Response to a Passing Wave.

In the long-wavelength, freely suspended limit an arm with unit direction $a^i$, $a^ia_i=1$, has

$$\frac{\delta L_a}{L} =\frac12a^ia^j h^{\mathrm{TT}}_{ij}.$$

A real detector channel also includes photon propagation, antenna geometry, instrument transfer, calibration, sampling, and noise. Gauge invariance of the tidal prediction does not make those response layers optional.

News and flux at future null infinity

For an isolated asymptotically flat system, Bondi coordinates $(u,r,x^A)$ provide a nonlinear radiation test. With $r$ an areal radius,

$$g_{AB}=r^2q_{AB}+rC_{AB}+O(1), \qquad q^{AB}C_{AB}=0, \qquad N_{AB}:=\partial_uC_{AB}.$$

The site’s normalization is twice the tensor news used by Mädler and Winicour, so their news and mass-loss coefficients must be translated together. With the outward orientation used here,

$$\frac{dE_{\rm out}}{du\,d\Omega} =\frac{1}{32\pi G}N_{AB}N^{AB}\ge0.$$

Nonzero $N_{AB}$ is radiative data; $N_{AB}=0$ gives zero Bondi luminosity even when a time-independent shear remains. The asymptotic construction, including the conformal frame and BMS action, is given in Mädler and Winicour 2018, §§2–4, pp. 3–20. The site’s complete normalization is collected in Gravitational-Wave Strain, News, Flux, and Ψ₄.

This positive flux is an asymptotic balance quantity, not a generally covariant pointwise stress tensor for gravity. In the additional short-wavelength regime, an averaged Isaacson tensor reproduces the leading wave-zone flux; outside that scale separation it must not be promoted to an exact local gravitational energy density Flanagan and Hughes 2005, §5.3, pp. 27–29.

What remains frame dependent

Gauge invariance does not mean that every waveform representation is unique.

Object	What must still be declared
$h_+$ and $h_\times$	Observer, oriented screen, polarization basis, wave-zone approximation, and strain convention
$R_{0i0j}$	Observer tetrad and the distinction between radiative and stationary curvature
Bondi shear and news	Conformal Bondi frame, retarded time, areal radius, cuts, sphere dyad, and normalization
$\Psi_4$	Curvature sign, null-tetrad normalization and orientation, and the two integrations needed to recover strain
Finite-radius waveform	Coordinate sphere, radius, tetrad, near-zone error, extrapolation or characteristic extraction, and convergence
Harmonic modes	Spin-weight convention, angular basis, phases, and BMS frame

Rotations, boosts, translations, and supertranslations can mix modes or shift the shear while describing physically equivalent asymptotic data. Comparisons must transform the fields and cuts together, rather than compare equal coordinate labels. The required BMS transformations are derived in Boyle 2016, §§II–VI, pp. 2–17. Finite-radius extraction methods and their gauge, tetrad, and asymptotic errors are reviewed in Bishop and Rezzolla 2016, §§3–7.

Three discriminating checks

Pure gauge. Insert $h_{\mu\nu}=-2\partial_{(\mu}\xi_{\nu)}$ into the linearized Riemann tensor. It vanishes without imposing a wave equation, so a time-dependent coordinate pattern alone proves nothing.

Plane wave. For $h^{\mathrm{TT}}_{ij}=A_{ij}\cos\omega(t-z)$, the wave vector is null, $R^{(1)}_{0i0j}\ne0$, neighboring geodesics have a transverse tidal response, and the cycle-averaged outward flux is positive. Adding a pure-gauge perturbation changes none of those statements.

Static field. A stationary Newtonian tidal field can have $R_{0i0j}\ne0$ but has no outgoing Bondi news. This prevents the opposite mistake of calling every nonzero curvature component a gravitational wave.

The canonical derivation and degree count are in Gravitational Radiation in Linearized Theory.

Common pitfalls

“Gauge dependent” does not mean “unphysical.” A representative may depend on gauge while relations among curvature, boundary data, and detector observables remain invariant or covariant.

A local inertial frame does not remove a wave. It can set connection coefficients to zero on one worldline, but not the Riemann tensor over an extended apparatus.

Detection does not make strain a scalar. A measured channel is invariant under a consistent change of description, but its decomposition into polarizations and modes still depends on declared frames.

References

• Bishop, Nigel T., and Luciano Rezzolla. “Extraction of Gravitational Waves in Numerical Relativity.” Living Reviews in Relativity 19 (2016): 2. DOI. Open PDF.
• Boyle, Michael. “Transformations of Asymptotic Gravitational-Wave Data.” Physical Review D 93 (2016): 084031. DOI. Open PDF.
• Flanagan, Éanna É., and Scott A. Hughes. “The Basics of Gravitational Wave Theory.” New Journal of Physics 7 (2005): 204. DOI. Open PDF.
• Mädler, Thomas, and Jeffrey Winicour. “Bondi–Sachs Formalism.” arXiv:1609.01731v3, 2018. Open PDF.