Gravitational-Wave Stokes Parameters

Gravitational-wave Stokes parameters describe the intensity and polarization state of a narrow frequency-direction element without assigning physical meaning to an arbitrary plus-cross screen. They are the four real coefficients of a positive $2\times2$ coherency matrix. The same algebra applies to a deterministic analytic signal and to an ensemble of stochastic radiation; only the averaging and spectral normalization differ. Detector estimation of these fields and empirical bounds on non-GR polarizations are not treated here.

Helpful background. Einstein Equations supplies the GR tensor field being decomposed. Waves, Green Functions, and Geometric Optics supplies Fourier modes and transverse screens. Gravitational Radiation and Waveforms supplies $h_+-ih_\times$ and its spin weight. Tests of Gravity supplies the later interpretation of non-tensor polarization content.

The coherency matrix

For one positive-frequency component in a declared propagation direction, assemble the complex amplitudes into

$$\boldsymbol h= \begin{pmatrix}h_+\\h_\times\end{pmatrix}.$$

For a deterministic signal, the coherency matrix is the outer product $J=\boldsymbol h\boldsymbol h^\dagger$. For a random field, replace that product by an ensemble or suitable time average:

$$J_{AB}=\left\langle h_A h_B^*\right\rangle, \qquad A,B\in\{+,\times\}.$$

At fixed frequency and sky direction, delta functions and one- versus two-sided spectral factors can be restored according to the declared Fourier convention. They multiply every entry and do not change the polarization algebra. Hermiticity follows immediately, $J_{AB}=J_{BA}^*$, and positivity follows from $v^\dagger Jv=\langle\lvert v^\dagger\boldsymbol h\rvert^2\rangle\ge0$.

The coherency-matrix construction is developed in Romano and Cornish 2017, Appendix A.3, pp. 204–205, eqs. (A.12)–(A.18), PDF. The sign translation to this page is stated below.

Four real Stokes parameters

Define

$$\begin{aligned} I&=J_{++}+J_{\times\times}, &Q&=J_{++}-J_{\times\times},\\ U&=J_{+\times}+J_{\times+}, &V&=-i\left(J_{+\times}-J_{\times+}\right). \end{aligned}$$

Thus $U=2\operatorname{Re}J_{+\times}$ and $V=2\operatorname{Im}J_{+\times}$, and

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

$I$ is total intensity in the selected spectral element; $Q$ and $U$ describe linear polarization relative to the screen; $V$ describes circular polarization with a sign fixed by propagation direction, Fourier sign, and dyad orientation. The convention here uses site time dependence $e^{-i\omega t}$ and the ordered screen $(p,q)$ associated with $h=h_+-ih_\times$.

For the same complex amplitude components, Romano and Cornish define $V_{\mathrm{RC}}=i(J_{+\times}-J_{\times+})$, so $V=-V_{\mathrm{RC}}$. Their screen-rotation angle is opposite to the dyad phase used here: $\psi_{\mathrm{RC}}=-\psi$. These two explicit translations reconcile the matrix off-diagonal signs and the spin-four rotation law. A change of Fourier sign would also require conjugating the positive-frequency amplitudes of each real polarization.

Positivity and degree of polarization

The eigenvalues of $J$ are

$$\lambda_\pm=\frac12\left(I\pm \sqrt{Q^2+U^2+V^2}\right).$$

Nonnegative eigenvalues imply

$$I\ge0, \qquad I^2\ge Q^2+U^2+V^2.$$

For $I\gt0$, define the polarization fraction

$$\mathcal P=\frac{\sqrt{Q^2+U^2+V^2}}{I}, \qquad 0\le\mathcal P\le1.$$

A single deterministic Jones vector gives a rank-one matrix, so $\det J=0$ and $\mathcal P=1$. An unpolarized ensemble has $J=(I/2)\mathbf1$ and $Q=U=V=0$. A partially polarized ensemble lies between those limits. “Unpolarized” is therefore a covariance statement, not the claim that every realization has zero strain.

Rotation of the polarization screen

Rotate the transverse screen by $\psi$. The real plus-cross vector transforms through $2\psi$:

$$\boldsymbol h'= \begin{pmatrix} \cos2\psi&-\sin2\psi\\ \sin2\psi&\cos2\psi \end{pmatrix}\boldsymbol h, \qquad J'=RJR^{\mathsf T}.$$

Direct substitution gives

$$I'=I, \qquad V'=V, \qquad Q'+iU'=e^{4i\psi}(Q+iU).$$

Intensity and circular polarization are screen-rotation scalars, while the linear-polarization pair has spin weight $+4$ in this passive component convention. This is exactly what is needed for the same physical coherency tensor to be expressed in different plus-cross bases.

Linear, circular, and mixed limits

A pure plus amplitude $\boldsymbol h=(A,0)$ has $(I,Q,U,V)=(\lvert A\rvert^2,\lvert A\rvert^2,0,0)$. Equal real amplitudes have $U\ne0$ and represent a linearly polarized state rotated on the screen.

For $\boldsymbol h=(A,iA)$,

$$(I,Q,U,V)= (2\lvert A\rvert^2,0,0,-2\lvert A\rvert^2).$$

Changing the relative phase from $+\pi/2$ to $-\pi/2$ reverses $V$. Calling either sign right- or left-handed additionally requires stating whether the observer looks along propagation or back toward the source. A parity-symmetric ensemble can have zero mean $V$ even when individual realizations are circularly polarized.

Spectral and sky conventions

For a stationary random field, $J_{AB}$ becomes a spectral density $J_{AB}(f,\boldsymbol n)$, possibly with angular correlations beyond a direction-local model. Its physical units depend on whether a one-sided or two-sided spectrum is used and on where factors such as $1/2$ and $4\pi$ are placed. A frequency integral of $I$ must recover the declared mean-square strain normalization.

Stokes fields belong to the radiation itself. Antenna patterns map them into cross-spectra between detector channels; noise weighting and likelihoods then belong to Gravitational-Wave Astronomy. Keeping this order prevents an instrument convention from being mistaken for a polarization convention.

The construction is also deliberately agnostic about generation. A source model predicts a coherency matrix; it does not alter Hermiticity or the positivity cone. Conversely, a measured point inside that cone is a compact description of second-order polarization statistics, not by itself an identification of the source population or gravitational action.

Exercises

1. Derive the positivity bound. Use the determinant of $J$ to prove the Stokes inequality.

Solution

From the displayed matrix,

$$\det J=\frac14 \left(I^2-Q^2-U^2-V^2\right).$$

A positive-semidefinite Hermitian $2\times2$ matrix has nonnegative trace and determinant. Therefore $I=\operatorname{tr}J\ge0$ and $I^2\ge Q^2+U^2+V^2$. For $I\gt0$, equality means one eigenvalue vanishes, which is the fully polarized rank-one limit. At $I=0$, the whole matrix vanishes and a polarization fraction is undefined.

2. Rotate a pure plus state. Begin with $(I,Q,U,V)=(I,I,0,0)$ and rotate the screen by $\psi=\pi/8$.

Solution

Since $4\psi=\pi/2$,

$$Q'+iU'=iI.$$

Thus $(Q',U')=(0,I)$, while $I'=I$ and $V'=0$. The apparent plus-cross mixture changes, but $Q'^2+U'^2=I^2$ and the state remains fully linearly polarized.

The calculation also shows why $Q$ alone is not a basis-independent measure of linear polarization. The invariant linear magnitude is $\sqrt{Q^2+U^2}$; rotation merely redistributes it between the two coordinates.

References

• 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.