Polarizations and Gauge-Invariant Wave Observables

Gravitational radiation is physically read through curvature, but it is often represented by strain components, asymptotic fields, master variables, or detector channels. This chapter separates those levels. It first fixes the two tensor polarizations of general relativity, then constructs the local tidal observable, enlarges the kinematic response space to six possible metric-wave patterns, develops polarization coherency and Stokes parameters, and closes by comparing gauge-invariant quantities across their proper domains. Empirical bounds on non-GR polarizations remain outside the chapter.

Helpful background. Einstein Equations supplies the vacuum restriction to two tensor modes. Waves, Green Functions, and Geometric Optics supplies null propagation, Fourier components, and the transverse screen. Gravitational Radiation and Waveforms supplies the inherited strain-curvature conventions. Tests of Gravity supplies theory-specific dynamics and the later comparison with observations.

Enter through the observable

The safest organizing question is: what operation could an observer perform? For nearby freely falling bodies, the answer is relative acceleration,

$$\ddot\xi^i=-\mathcal E^i{}_j\xi^j, \qquad \mathcal E_{ij}=R_{0i0j}.$$

The symmetric $3\times3$ tidal matrix has six real entries. Plane-null-wave kinematics sorts them into two tensor, two vector, one breathing, and one longitudinal pattern. Field equations then restrict that kinematic space. In weak vacuum general relativity,

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

and only the transverse, trace-free plus and cross patterns remain. This logical order avoids both common errors: treating a gauge-dependent metric component as a measurement, and treating the six-pattern catalog as six degrees of freedom in every theory.

Shared screen and tetrad conventions

A wave propagating along the declared unit vector $\boldsymbol z$ uses an oriented transverse screen $(\boldsymbol x,\boldsymbol y)$. The unit complex dyad is

$$m^a=\frac{x^a+iy^a}{\sqrt2}, \qquad m\mathbin{\cdot}\bar m=1.$$

The volume’s complex strain is $h=h_+-ih_\times$. Under $m\mapsto e^{i\psi}m$ it transforms as $h\mapsto e^{-2i\psi}h$. With observer $u^a$, define

$$\ell^a=\frac{u^a+z^a}{\sqrt2}, \qquad n^a=\frac{u^a-z^a}{\sqrt2}, \qquad \ell\mathbin{\cdot}n=-1,$$

and fix

$$\Psi_4=-C_{abcd}n^a\bar m^b n^c\bar m^d.$$

For an outgoing plane wave with site time dependence $e^{-i\omega t}$,

$$\Psi_4=\ddot h_+-i\ddot h_\times.$$

These choices align the screen rotation, Newman–Penrose classification, asymptotic derivative chain, and detector response. A source using the opposite Riemann sign, propagation direction, or complex strain must be translated before formulas are combined.

Six pages, one chain of reasoning

1. Plus and Cross Polarizations builds the oriented tensor basis, proves the doubled-angle rotation, and distinguishes field amplitudes from antenna factors.
2. Curvature and Detector Observables derives geodesic deviation, checks flat-background linear gauge invariance, and marks where the long-wavelength response approximation ends.
3. Newman–Penrose Polarization Classification projects curvature onto the normalized null tetrad, maps six real tidal entries into E(2) classes, and recovers the two-mode GR limit. The foundational classification is Eardley et al. 1973, pp. 884–886.
4. Additional Polarizations Beyond GR constructs six linearly independent response matrices and explains why a detector network, rather than one pattern drawing, is needed to separate them.
5. Gravitational-Wave Stokes Parameters packages second-order polarization statistics into a positive coherency matrix and derives their spin-four screen transformation. The random-field formulation is treated in Romano and Cornish 2017, Appendix A.3, pp. 204–205, eqs. (A.12)–(A.18), PDF.
6. Gauge-Invariant Radiative Quantities compares local curvature, asymptotic shear and news, Weyl scalars, background master variables, fluxes, and response channels.

The sequence moves from a GR field basis to the measured curvature, then broadens the kinematic possibilities before returning to invariant comparison. Readers interested only in standard GR should still read pages 1, 2, and 6; the intermediate pages prevent polarization claims from outrunning their assumptions.

Comparison axes

The chapter compares every quantity along four axes.

Domain. TT strain belongs to a controlled radiation-zone description; Bondi fields belong to null infinity; Regge–Wheeler, Zerilli, and Teukolsky variables belong to specified black-hole backgrounds; detector channels belong to specified worldlines and light paths.

Allowed transformations. Coordinate gauge, polarization-screen rotations, tetrad spin-boosts, BMS frame changes, and detector-frame relabelings are different groups. Invariance under one does not imply invariance under all.

Boundary data. Curvature determines strain only after integrations. In the asymptotic convention,

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

The two integrations from $\Psi_4^0$ leave a constant shear and an initial-news term. Those are physical boundary/frame questions, not numerical clutter to remove automatically.

Measurement map. A field polarization becomes a detector channel only after contraction with geometry and, at finite frequency, a transfer function. Calibration, noise, network likelihoods, and empirical verdicts belong to Gravitational-Wave Astronomy and Tests of Gravity.

Coherency as a basis-covariant summary

For one frequency-direction element, define $J_{AB}=\langle h_Ah_B^*\rangle$. Its Stokes decomposition is

$$J=\frac12 \begin{pmatrix} I+Q&U+iV\\ U-iV&I-Q \end{pmatrix}.$$

Positivity gives $I^2\ge Q^2+U^2+V^2$. Under a screen rotation, $I$ and $V$ are invariant while $Q+iU\mapsto e^{4i\psi}(Q+iU)$. The density matrix therefore retains all second-order polarization information without pretending that plus and cross labels are absolute. A deterministic wave saturates the inequality; an unpolarized ensemble has $Q=U=V=0$.

Chapter diagnostic

An approximately plane wave traveling along $+z$ has measured tidal matrix

$$\mathcal E= A\begin{pmatrix}1&0&0\\0&1&0\\0&0&0\end{pmatrix} +B\begin{pmatrix}0&0&1\\0&0&0\\1&0&0\end{pmatrix}.$$

Identify the polarization patterns. Can vacuum GR support this matrix at leading plane-wave order? Would an equal-arm Michelson lying in the $xy$ plane and directly facing the wave respond to the $A$ term in the long-wavelength limit?

Solution

The first matrix is the scalar breathing pattern $P^b$; the second is the vector-$x$ pattern $P^x$. They are linearly independent because one is diagonal and the other occupies the $xz$ pair. Vacuum GR permits neither as a leading plane null wave: its Ricci and vector NP entries vanish, leaving only plus and cross.

For orthogonal arms $x$ and $y$, the detector tensor is

$$D=\frac12\operatorname{diag}(1,-1,0).$$

Its contraction with the breathing matrix is $D\mathbin{:}P^b=(1-1)/2=0$. Thus this specially aligned single channel is blind to $A$ even though the spacetime curvature is nonzero. That degeneracy is why mode identification requires varying antenna responses, independent channels, or a network; absence in one channel is not absence in the field.

Connections to other topics

This chapter develops the mathematical polarization basis, curvature response, and domain-qualified invariants. Alternative-theory field equations, empirical bounds on non-GR polarizations, calibration, searches, and likelihoods are treated elsewhere. The chapter also does not replace the detailed asymptotic geometry of Global Structure, the master equations of Black-Hole Perturbation Theory, or the production algorithms of Computational Relativity. The goal is a reliable translation layer: once a neighboring subject supplies a field, frame, or detector model, the quantities here show exactly what may be compared.

References

• Eardley, Douglas M., David L. Lee, Alan P. Lightman, Robert V. Wagoner, and Clifford M. Will. “Gravitational-Wave Observations as a Tool for Testing Relativistic Gravity.” Physical Review Letters 30 (1973): 884–886. DOI.
• Flanagan, Éanna É., and Scott A. Hughes. “The Basics of Gravitational Wave Theory.” New Journal of Physics 7 (2005): 204. DOI. Open PDF.
• Romano, Joseph D., and Neil J. Cornish. “Detection Methods for Stochastic Gravitational-Wave Backgrounds: A Unified Treatment.” Living Reviews in Relativity 20 (2017): 2. DOI. Open PDF.