Gauge-Invariant Radiative Quantities

No single variable is the canonical description of gravitational radiation in every domain. Linearized curvature is natural for local tidal physics, Bondi news for flux at null infinity, master variables for perturbations of a specified black-hole background, and a response channel for an instrument. Each is invariant only under a stated class of gauge or frame changes. This page compares those domains and fixes the derivative chain among strain, news, and $\Psi_4$ without treating detector inference or empirical bounds on non-GR polarizations.

Helpful background. Einstein Equations supplies the distinction between coordinate gauge and curvature. Waves, Green Functions, and Geometric Optics supplies radiation-zone and short-wavelength regimes. Gravitational Radiation and Waveforms supplies the volume’s strain, news, and curvature dictionary. Tests of Gravity supplies later theory-dependent observables.

Gauge invariance is a qualified statement

Split a field into background and perturbation. Under an infinitesimal identification change generated by $\xi^a$, a first-order tensor perturbation obeys

$$\delta T\mapsto\delta T-\mathcal L_\xi T^{(0)}.$$

It is automatically invariant only when the relevant background tensor vanishes, is constant in the needed sense, or when a combination cancels the Lie-derivative term. A quantity can also be coordinate gauge invariant but depend on a tetrad, polarization dyad, Bondi frame, or detector worldline. “Observable” does not erase those physical choices; it makes their transformation law part of the definition.

The linearized-gravity discussion and its curvature checks are given in Flanagan and Hughes 2005, §2, pp. 4–13, PDF.

Local curvature and radiation-zone strain

On a Minkowski background, $R_{abcd}^{(0)}=0$, so the linearized Riemann tensor is unchanged by

$$h_{ab}\mapsto h_{ab}-2\partial_{(a}\xi_{b)}.$$

Its observer projection $\mathcal E_{ij}=R_{0i0j}^{(1)}$ controls geodesic deviation. In a TT radiation-zone representative,

$$\mathcal E_{ij}=-\frac12\ddot h_{ij}^{\rm TT}.$$

The curvature is invariant under linear coordinate gauge changes, while the displayed strain is a convenient potential reconstructed after fixing a radiation frame and two integration constants. On a curved background, $\mathcal L_\xi R_{abcd}^{(0)}$ generally does not vanish, so raw perturbed curvature components need not be gauge invariant.

Bondi shear, news, and radiative Weyl curvature

At future null infinity the volume uses

$$H=C_{AB}\bar m^A\bar m^B =\lim_{r\to\infty}r(h_+-ih_\times), \qquad \mathcal N=\partial_uH, \qquad \Psi_4^0=\partial_u\mathcal N=\partial_u^2H.$$

$H$ is the Bondi shear component in a chosen conformal Bondi frame and dyad. A supertranslation can add a retarded-time-independent, angle-dependent shear term; a dyad rotation gives the expected spin phase. News removes the time-independent shear offset and directly controls flux, but it remains a field on a specified cut and transforms covariantly when the BMS frame changes. $\Psi_4^0$ adds a derivative but requires a normalized asymptotic tetrad.

The finite-radius-to-asymptotic distinctions are reviewed in Bishop and Rezzolla 2016, §3.3, pp. 22–27, and §§6.4–6.5, pp. 58–62, PDF, while explicit BMS and tetrad transformations are given in Boyle 2016, §III, pp. 9–11, eqs. (17)–(23), PDF.

Background-adapted master variables

On Schwarzschild spacetime, Regge–Wheeler and Zerilli–Moncrief master functions combine metric-perturbation components so that infinitesimal coordinate changes cancel for radiative multipoles. Their meaning depends on the chosen background, parity split, harmonic normalization, and master-function normalization. They are not universal scalar fields on an arbitrary dynamical spacetime.

On a type-D rotating background, Teukolsky variables built from perturbed Weyl scalars exploit background algebraic structure. Their coordinate-gauge robustness does not remove tetrad freedom: the background tetrad, spin-boost, extraction radius, and conversion to asymptotic strain must still be controlled. The equations and spectra are developed in the black-hole perturbation volume; here they serve as examples of domain-specific invariant construction.

Fluxes and detector channels

An integrated Bondi energy flux,

$$\frac{dE_{\rm out}}{du} =\frac{1}{32\pi G}\int_{S^2}N_{AB}N^{AB}\,d\Omega,$$

is independent of a polarization-screen rotation and nonnegative in the declared outward orientation. Angular-momentum fluxes require more frame care, and finite-radius pseudotensors are not pointwise gauge-invariant local energy densities.

A detector channel is invariant under a coordinated relabeling of the field basis and antenna response. Yet it depends physically on the detector trajectory, arm geometry, light travel time, and calibration. Thus “the detector measures $h$” is shorthand for a controlled response limit, not an identity between a spacetime field and stored data.

Comparison axes and choosing the right quantity

Compare the candidates along four axes: spacetime domain, allowed gauge or frame transformations, boundary data needed for reconstruction, and the physical operation being predicted.

• Use $\mathcal E_{ij}$ for local relative acceleration and polarization patterns.
• Use $H$ when waveform phase and memory offsets are needed in a fixed asymptotic frame.
• Use $\mathcal N$ for radiation intervals and Bondi flux.
• Use $\Psi_4^0$ for outgoing radiative curvature, provided the tetrad and two integrations are controlled.
• Use a Regge–Wheeler, Zerilli, or Teukolsky variable for perturbations of its declared black-hole background.
• Use the full response channel when making an instrument-level statement.

The correct choice is the least frame-dependent object that still answers the physical question. Replacing every variable by curvature can discard memory integration constants; replacing every curvature statement by strain can conceal gauge and boundary choices.

Exercises

1. Pure-gauge check. Add $h^{\rm g}_{ab}=-2\partial_{(a}\xi_{b)}$ to a flat-background perturbation. Why is the linearized curvature unchanged?

Solution

The linearized Riemann tensor contains antisymmetrized second derivatives of $h_{ab}$. Substituting $h^{\rm g}_{ab}$ produces third partial derivatives of $\xi_a$ in pairs with opposite signs. Because flat-background partial derivatives commute, every term cancels. In covariant language the change is $-\mathcal L_\xi R_{abcd}^{(0)}=0$. The same argument would fail for a generic curved background with nonzero $R_{abcd}^{(0)}$.

2. What curvature cannot reconstruct. Suppose $\Psi_4^0(u)$ is known on an interval. Integrate the derivative chain and identify the missing data.

Solution

Twice integrating gives

$$H(u)=A+B(u-u_0) +\int_{u_0}^{u}(u-u')\Psi_4^0(u')\,du'.$$

The complex constant $A$ fixes the initial shear or strain offset, while $B$ fixes the initial news. Curvature data on the interval cannot determine either. Physical early-time conditions, matching information, or a specified Bondi frame must supply them; arbitrarily filtering the low-frequency sector can erase genuine memory.

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.