Polarizations, Speeds, and Propagation

Radiation content is determined twice: the field equations decide which physical eigenmodes propagate, while matter coupling decides which metric tidal patterns a detector can see. The same theory can also change phase speed, group delay, damping, or mixing during travel. These are logically distinct statements, and none follows merely from counting fields in the action.

Helpful background. Einstein Equations supplies the two tensor-mode baseline; Gravitational Actions and Boundary Terms supplies kinetic normalization; Formulations of General Relativity supplies gauge and constraint reduction; Relativistic Two-Body Dynamics supplies source-frame frequencies; Gravitational Radiation and Waveforms supplies detector response and cosmological redshift; and Gravity Beyond General Relativity supplies the theory-specific mode and matter-coupling record.

Six metric tidal patterns

For freely falling nearby test masses, the observable is geodesic deviation,

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

For a plane wave traveling along $+z$, a symmetric $3\times3$ tidal matrix has six independent entries. A useful basis is

Pattern	Nonzero tidal components	Motion
tensor plus	$\mathcal E_{xx}=-\mathcal E_{yy}$	transverse stretch and squeeze along $x,y$
tensor cross	$\mathcal E_{xy}=\mathcal E_{yx}$	transverse stretch and squeeze on axes rotated by $45^\circ$
vector $x$	$\mathcal E_{xz}=\mathcal E_{zx}$	mixed transverse–longitudinal shear
vector $y$	$\mathcal E_{yz}=\mathcal E_{zy}$	orthogonal mixed shear
scalar breathing	$\mathcal E_{xx}=\mathcal E_{yy}$	isotropic transverse expansion and contraction
scalar longitudinal	$\mathcal E_{zz}$	motion along the propagation direction

Eardley and collaborators established the polarization classification for null plane waves in metric theories in Eardley et al. 1973, pp. 884–886. Its original null-wave classification must not be imported unchanged for massive dispersion. The six symmetric-matrix patterns above still provide a local tidal basis; they do not assert that every theory propagates six independent modes. GR in vacuum has only plus and cross.

One scalar gives a useful example. On a locally flat background, let the matter metric have the small conformal perturbation $\delta\widetilde g_{\mu\nu}=2\alpha_0\delta\varphi\,\eta_{\mu\nu}$ and let $\delta\varphi\propto e^{-i\omega t+ikz}$. Linearized curvature gives

$$\mathcal E_{ij}=\alpha_0 \left(\partial_i\partial_j\delta\varphi -\delta_{ij}\partial_t^2\delta\varphi\right).$$

For $\omega^2=k^2+m_s^2$ this yields

$$\mathcal E_{xx}=\mathcal E_{yy}=\alpha_0\omega^2\delta\varphi, \qquad \mathcal E_{zz}=\alpha_0 m_s^2\delta\varphi.$$

The longitudinal-to-breathing ratio is $m_s^2/\omega^2$ when the coupling is nonzero. Both patterns are fixed by the same scalar amplitude. The massless limit is purely breathing, and a vanishing matter coupling makes this scalar invisible to the metric detector even if it propagates.

An interferometer measures a contraction with its detector tensor, schematically $h_D=D^{ij}h_{ij}$, or equivalently the integrated tidal response. A network is needed to separate patterns because antenna functions can be degenerate. More importantly, a polarization seen in the metric need not correspond one-to-one with a fundamental field: kinetic and matter-coupling mixing can rotate the propagation eigenbasis before detection.

The principal symbol decides what propagates

Linearize the complete equations about a declared background, fix gauge, solve nondynamical constraints, and retain physical amplitudes $a^A$. Their high-frequency equation has principal symbol

$$P_{AB}^{\mu\nu}k_\mu k_\nu a^B=0, \qquad \det P_{\rm phys}(k)=0.$$

Gauge zeros must be removed before interpreting the determinant. On a homogeneous isotropic background, an eigenmode may have

$$\omega^2=c_A^2k^2+\mu_A^2+O(k^4/\Lambda^2).$$

The phase speed is $\omega/k$. At leading two-derivative order, with locally constant coefficients, the group speed is

$$v_{g,A}=\frac{\partial\omega}{\partial k}=\frac{c_A^2k}{\omega},$$

and the front speed is a high-frequency property of the complete theory. They coincide only for a nondispersive massless mode. Real $\omega$ is not enough: the kinetic matrix must have the correct sign, the reduced system must be hyperbolic, and $k/\Lambda$ must remain small. Concrete tensor, vector, and scalar mode speeds in Einstein–æther and khronometric gravity illustrate the background- and coupling-dependent spectrum in Berti, Yagi, and Yunes 2018, § 2.4 and Table 1, pp. 10–11.

Arrival time and dispersive phase

With the site’s Fourier convention $e^{-i\omega t}$, propagation over a static distance $D$ contributes $e^{ik(\omega)D}$. Relative to a luminal massless wave,

$$\Delta\Psi_{\rm prop}(\omega)=D\,[k(\omega)-\omega], \qquad \Delta t_g=D\left(\frac1{v_g}-1\right).$$

For $c_A=1$ and $\omega\gg \mu_A$,

$$k=\sqrt{\omega^2-\mu_A^2} =\omega-\frac{\mu_A^2}{2\omega}+O(\mu_A^4/\omega^3),$$

so

$$\Delta\Psi_{\rm prop}=-\frac{D\mu_A^2}{2\omega}+\cdots, \qquad \Delta t_g=\frac{D\mu_A^2}{2\omega^2}+\cdots.$$

The phase derivative gives the positive delay, providing an independent sign check. Berti, Yagi, and Yunes give the generic modified dispersion relation, group velocity, and cosmological distance mapping in Berti, Yagi, and Yunes 2018, § 3.2, eqs. (20)–(22), pp. 17–18.

On an expanding background, use the leading luminal redshift $\omega(z)\simeq2\pi f_{\rm obs}(1+z)$ when evaluating a first-order small dispersion correction. For a substantially massive mode the exact frequency instead follows its dispersion law at redshifting comoving momentum. To first order in a small delay,

$$\Delta t_{g,\rm obs} =\int_0^{z_s}\frac{dz}{H(z)} \left[\frac1{v_g(\omega(z),z)}-1\right].$$

Using detector-frame frequency in a source-frame dispersion law without the factor $1+z$ is a dimensional-looking but physically wrong substitution.

Damping and mode mixing on cosmological backgrounds

A useful propagation equation in conformal time is

$$h_A''+(2+\nu_A)\mathcal H h_A' +\left(c_A^2k^2+a^2\mu_A^2\right)h_A +\sum_{B\ne A}\mathcal M_{AB}h_B=0.$$

$\nu_A$ changes amplitude damping, while $\mathcal M_{AB}$ can convert or birefringently rotate modes. For the tensor mode in the same matter frame and field normalization used in Cosmological Gravitational-Wave Propagation, $\nu_T=\alpha_M=d\ln M_*^2/d\ln a$. The present chapter supplies the emitted amplitudes and generic mode-resolved propagation data; that cosmology page derives the FLRW transfer from its quadratic tensor action. In the subhorizon, unmixed, luminal, massless case,

$$\frac{d_L^{\rm GW}(z)}{d_L^{\rm EM}(z)} =\exp\left[\frac12\int_0^z\frac{\nu(z')}{1+z'}\,dz'\right].$$

Belgacem and collaborators use $2\mathcal H[1-\delta]$ instead, so $\nu=-2\delta$ and their exponent is $-\int\delta\,dz/(1+z)$. The derivation and sign appear in Belgacem et al. 2018, § II, eqs. (9)–(13), pp. 1–2. This luminosity-distance formula does not apply unchanged when modes mix, leave the WKB regime, or have different characteristic cones.

Checks and interpretation limits

The GR propagation limit is two tensor modes with $c_A=1$, $\mu_A=0$, $\nu_A=0$, and $\mathcal M_{AB}=0$. The dimensions are $[Dk]=1$, $[\mu_A]=[\omega]=L^{-1}$, and the delay has dimension $L$. A subluminal massive wave has $k\lt\omega$, hence negative relative phase and positive group delay in the chosen convention.

Polarization counting alone does not establish kinetic stability, causality, source excitation, or detector distinguishability. An arrival-time relation also requires an emission-time model; an electromagnetic counterpart or two frequency components do not automatically share the same intrinsic launch time. Cosmological damping must be matched to the same background and Planck-mass convention used in waveform generation.

Exercises

Massive-mode sign diagnostic. A mode obeys $\omega^2=k^2+m^2$ and travels a distance $D$. Derive its leading propagation phase and group delay, then verify their relation.

Solution

At fixed observed angular frequency, $k=\omega-m^2/(2\omega)+O(m^4/\omega^3)$. Thus $\Delta\Psi=D(k-\omega)=-Dm^2/(2\omega)$. The relative group delay is the frequency derivative of the propagation phase relative to the luminal phase: $d\Delta\Psi/d\omega=Dm^2/(2\omega^2)$. Directly, $v_g=k/\omega=1-m^2/(2\omega^2)+\cdots$, so $D(1/v_g-1)=Dm^2/(2\omega^2)$, in agreement.

References

• Belgacem, Enis, Yves Dirian, Stefano Foffa, and Michele Maggiore. “The Gravitational-Wave Luminosity Distance in Modified Gravity Theories.” Physical Review D 97 (2018): 104066. DOI; Open PDF.
• Berti, Emanuele, Kent Yagi, and Nicolás Yunes. “Extreme Gravity Tests with Gravitational Waves from Compact Binary Coalescences: (I) Inspiral–Merger.” General Relativity and Gravitation 50 (2018): 46. DOI; Open PDF.
• 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.