Polarization Transport

A gravitational wave carries two tensor polarizations, but the labels “plus” and “cross” have meaning only after a transverse screen basis is chosen. Along a short-wavelength ray, the physical polarization tensor and a chosen screen dyad can be parallel transported. At detection, the transported dyad generally differs from the detector’s dyad by a rotation, so the recorded plus and cross components mix with twice that angle. This kinematic mixing must be distinguished from helicity conversion, diffraction-induced apparent components, and genuinely new propagating polarizations.

Required background. Waves, Green Functions, and Geometric Optics supplies eikonal transport and transverse wave spaces; Curvature supplies parallel transport and path-dependent holonomy.

Helpful background. Einstein Equations supplies the linearized tensor constraints; Relativistic Cosmology supplies source and detector rest frames on an expanding background.

The two-dimensional screen

Let

$$k_\mu=\nabla_\mu\theta, \qquad k^\mu k_\mu=0, \qquad k^\nu\nabla_\nu k^\mu=0$$

be the wave covector and the tangent to a future-directed affinely parametrized ray. The local phase convention is $e^{i\theta}=e^{-i\omega t+i\boldsymbol{k}\cdot\boldsymbol{x}}$. For an observer with four-velocity $u^\mu$, define

$$\omega=-k_\mu u^\mu>0, \qquad n^\mu=\frac{k^\mu}{\omega}-u^\mu.$$

Then $u_\mu n^\mu=0$ and $n_\mu n^\mu=1$. The observer’s screen projector is

$$S_{\mu\nu}=g_{\mu\nu}+u_\mu u_\nu-n_\mu n_\nu.$$

Choose real screen vectors $s_1^\mu,s_2^\mu$ satisfying

$$s_A^\mu s_{B\mu}=\delta_{AB}, \qquad s_A^\mu u_\mu=s_A^\mu n_\mu=0.$$

For transport along the whole ray, it is convenient to replace the observer-dependent pair $(u,n)$ by an auxiliary null vector $\ell^\mu$ with $k\cdot\ell=-1$. Parallel transport $\ell^\mu$ and the two $s_A^\mu$ from the source. Their span is the quotient of vectors orthogonal to $k^\mu$ by the gauge direction proportional to $k^\mu$. At the detector, project into its observer screen if necessary; this projection changes representatives but not the physical transverse tensor.

Parallel transport preserves the polarization space

Impose

$$k^\nu\nabla_\nu s_A^\mu=0.$$

Metric compatibility immediately gives

$$k^\rho\nabla_\rho(s_A\cdot s_B)=0.$$

Geodesic transport of $k^\mu$ also gives

$$k^\rho\nabla_\rho(k\cdot s_A) =(k^\rho\nabla_\rho k_\mu)s_A^\mu +k_\mu k^\rho\nabla_\rho s_A^\mu=0.$$

Thus orthonormality and transversality, once imposed, persist along the affine ray. Form the normalized tensor basis

$$e^+_{\mu\nu}=s_{1\mu}s_{1\nu}-s_{2\mu}s_{2\nu}, \qquad e^\times_{\mu\nu}=s_{1\mu}s_{2\nu}+s_{2\mu}s_{1\nu}.$$

It obeys $e^+\!\cdot e^+=e^\times\!\cdot e^\times=2$, $e^+\!\cdot e^\times=0$, and $k^\mu e^{+,\times}_{\mu\nu}=0$. Write the leading eikonal amplitude as

$$A^{(0)}_{\mu\nu} =\mathcal A\left(c_+e^+_{\mu\nu}+c_\times e^\times_{\mu\nu}\right).$$

The geometric-optics transport equation separates into focusing of the common magnitude $\mathcal A$ and parallel transport of the normalized tensor. In a parallel screen, $dc_+/d\lambda=dc_\times/d\lambda=0$. Flanagan and Hughes derive this short-wavelength separation for curved backgrounds in Flanagan and Hughes 2005, §5, pp. 23–29, PDF.

Source and detector bases

The emitter defines a source-frame dyad using its four-velocity $u_s^\mu$ and a source orientation. Parallel transport that dyad along the ray. The detector defines another dyad from $u_o^\mu$, the arriving direction, and its own orientation. After projection into the same detector screen, the two orthonormal dyads differ by some angle $\chi$:

$$\begin{aligned} s'_1&=\cos\chi\,s_1-\sin\chi\,s_2,\\ s'_2&=\sin\chi\,s_1+\cos\chi\,s_2. \end{aligned}$$

For the site’s complex strain convention

$$h=h_+-ih_\times,$$

the components in the rotated basis are

$$h'=e^{-2i\chi}h,$$

or equivalently

$$\begin{pmatrix}h'_+\\h'_\times\end{pmatrix} = \begin{pmatrix} \cos2\chi&-\sin2\chi\\ \sin2\chi&\cos2\chi \end{pmatrix} \begin{pmatrix}h_+\\h_\times\end{pmatrix}.$$

The factor of two is the spin-two signature. The norm $h_+^2+h_\times^2$ is unchanged. A curved path can give a nonzero relative angle between separately prescribed source and detector frames; different lensed paths can give different angles. Calling that angle “polarization rotation” is meaningful only after both endpoint frames and the path have been stated.

This endpoint comparison is also why a coordinate-component change is insufficient evidence for rotation. One must first construct physical orthonormal tetrads, project the arriving tensor into the detector screen, and compare it with the dyad obtained by transporting the declared source screen. Coordinate shear, aberration from changing observer velocity, and an actual path-dependent screen angle are logically separate effects.

Basis mixing versus new physics

A screen rotation mixes plus and cross coordinates but does not mix the two circular helicities dynamically: each helicity merely acquires the corresponding spin phase. In leading general-relativistic geometric optics, parallel transport supplies no scalar breathing mode, longitudinal mode, or vector mode.

Finite-wavelength propagation requires more care. Diffraction smears the single propagation direction used to define a two-dimensional screen. Projecting the resulting field relative to one observer can then yield apparent vector or scalar pieces at subleading eikonal order. Cusin and Lagos calculate these pieces and show that their classification depends on the observer in Cusin and Lagos 2020, §IV, pp. 8–9, and Appendix A, pp. 11–12, PDF. They do not represent additional graviton degrees of freedom. Genuine extra polarizations require extra dynamical fields or altered constraints in a specified theory; theory-specific propagation tests and modified dispersion lie outside this chapter.

In flat spacetime, choose a constant ray $k^\mu=(\omega,0,0,\omega)$ and constant Cartesian screen vectors along $x$ and $y$. Their connection derivatives vanish, so the transported dyad, $c_+$, and $c_\times$ are constant. Any evolution found in this setup is a basis artifact or an algebraic error.

Exercise

A wave is purely plus polarized in one screen: $h_+=H$ and $h_\times=0$. A detector dyad is rotated by $\chi=\pi/4$ according to the convention above. Find the detector components and verify that the polarization norm is preserved.

Solution

Here $2\chi=\pi/2$, so

$$h'_+=H\cos\frac\pi2=0, \qquad h'_\times=H\sin\frac\pi2=H.$$

Equivalently, $h=H$ and $h'=e^{-i\pi/2}H=-iH$, which indeed means $h'_+=0$ and $h'_\times=H$ because $h'=h'_+-ih'_\times$. Finally,

$$(h'_+)^2+(h'_\times)^2=H^2=h_+^2+h_\times^2.$$

The detector calls the same tensor “cross” rather than “plus”; no physical helicity conversion or new polarization has occurred.

References

• Éanna É. Flanagan and Scott A. Hughes, “The Basics of Gravitational Wave Theory,” New Journal of Physics 7 (2005) 204, arXiv:gr-qc/0501041v3, DOI, Open PDF.
• Giulia Cusin and Macarena Lagos, “Gravitational Wave Propagation Beyond Geometric Optics,” Physical Review D 101 (2020) 044041, arXiv:1910.13326v2, DOI, Open PDF.