Additional Polarizations Beyond GR

Beyond the two tensor patterns of general relativity, a metric wave can kinematically produce two vector and two scalar tidal patterns. The clean comparison is made with the measured electric-curvature matrix, not with gauge-dependent metric components. This page constructs all six matrices and explains what an ideal detector projects from them. It does not assert that every gravity theory propagates every pattern, and it leaves observational bounds to Tests of Gravity.

Helpful background. Einstein Equations supplies the GR field-equation restriction. Waves, Green Functions, and Geometric Optics supplies the plane-wave and null-propagation assumptions. Gravitational Radiation and Waveforms supplies plus-cross strain and tidal response. Tests of Gravity supplies theory-specific dynamics and empirical constraints.

Six measurable tidal patterns

Take a nearly plane wave propagating along $+z$ and an oriented orthonormal triad $(x,y,z)$. Expand the electric curvature as

$$\mathcal E_{ij}=\sum_M A_M P^M_{ij}, \qquad M\in\{+,\times,x,y,b,l\}.$$

A convenient real basis is

$$\begin{aligned} P^+&=\begin{pmatrix}1&0&0\\0&-1&0\\0&0&0\end{pmatrix}, &P^\times&=\begin{pmatrix}0&1&0\\1&0&0\\0&0&0\end{pmatrix}, &P^x&=\begin{pmatrix}0&0&1\\0&0&0\\1&0&0\end{pmatrix},\\[4pt] P^y&=\begin{pmatrix}0&0&0\\0&0&1\\0&1&0\end{pmatrix}, &P^b&=\begin{pmatrix}1&0&0\\0&1&0\\0&0&0\end{pmatrix}, &P^l&=\begin{pmatrix}0&0&0\\0&0&0\\0&0&1\end{pmatrix}. \end{aligned}$$

The letters $b$ and $l$ denote breathing and longitudinal. Overall normalization and the sign of each amplitude are conventional; the invariant content is the deformation pattern obtained from $\ddot\xi^i=-\mathcal E^i{}_j\xi^j$. The sixfold kinematic classification originates with Eardley et al. 1973, pp. 884–886.

Tensor and vector responses

The plus and cross matrices have no $z$ row or column and zero trace. They distort a transverse ring into an ellipse whose area is unchanged to first order in strain, with the two patterns related by a $45^\circ$ screen rotation.

The vector matrices mix one transverse direction with the propagation direction. For $P^x$, a separation initially along $x$ acquires longitudinal acceleration and a separation along $z$ acquires transverse acceleration; $P^y$ does the analogous job in the other screen direction. They are therefore neither purely transverse nor a rotated version of plus or cross. Under a screen rotation their two amplitudes mix through the ordinary angle $\psi$, while tensor amplitudes mix through $2\psi$.

Names such as “vector-$x$” refer to the screen convention. The two-dimensional vector subspace is physical, but its individual component labels rotate with $(x,y)$.

Breathing and longitudinal responses

The breathing matrix is isotropic on the screen. A transverse ring expands or contracts without changing its circular shape. It has a nonzero transverse trace and is therefore linearly distinct from both tensor matrices.

The longitudinal matrix accelerates separations only along the propagation direction. In a null plane-wave classification these two scalar patterns are independent. In a specified massive scalar theory, however, field equations and dispersion can tie their relative amplitudes together as a frequency-dependent combination. A measured mixture is not by itself evidence for two independent scalar fields.

These statements describe local curvature. A visual drawing of test masses at one phase does not encode propagation speed, action, stability, or coupling to matter.

Linear independence and completeness

Use the Frobenius product $A\mathbin{:}B=A_{ij}B^{ij}$. The six displayed matrices are mutually orthogonal: the three diagonal matrices $P^+$, $P^b$, and $P^l$ have orthogonal diagonal vectors, while the three off-diagonal matrices occupy distinct symmetric index pairs. Since every matrix has nonzero norm, they are linearly independent.

A symmetric $3\times3$ tidal matrix has six independent entries, so this basis is also complete. The counting does not mean a general wave can choose all six amplitudes freely. Bianchi identities, the propagation regime, and a theory’s field equations impose relations. Vacuum GR reduces the basis to $P^+$ and $P^\times$.

From field patterns to detector patterns

In the long-wavelength approximation an ideal detector channel is

$$s=\sum_M F_M A_M^{(h)}, \qquad F_M=D^{ij}e^M_{ij},$$

where $e^M_{ij}$ is the appropriate metric-response tensor and $D^{ij}$ describes the instrument. Equivalently, one may work directly with curvature and integrate using stated boundary data. Antenna patterns vary with sky direction, polarization basis, arm geometry, and frequency. A single channel generally measures one linear combination, not six separately identifiable amplitudes.

For example, take an equal-arm Michelson in the $xy$ plane and a wave directly along $z$. Its trace-free detector tensor has $D^{zz}=0$. Consequently its direct overhead contractions with both $P^b$ and $P^l$ vanish, even though those matrices are linearly independent fields.

There is also a stronger scalar degeneracy for any sky direction in this long-wavelength limit. With the normalizations used here,

$$P^b_{ij}=\delta_{ij}-n_in_j, \qquad P^l_{ij}=n_in_j, \qquad D^{ij}\delta_{ij}=0,$$

so $D:P^b=-D:P^l$. Changing the orientation of an equal-arm differential Michelson changes the common scalar response but preserves this proportionality. Even a network of such channels cannot kinematically separate two otherwise arbitrary scalar amplitudes with identical time dependence. Additional information must break the degeneracy—for example a different response geometry, resolved finite-arm transfer functions, or a theory relating the scalar waveforms. Linear independence of spacetime patterns is therefore necessary but insufficient for independent reconstruction by a specified detector network.

What this comparison does not decide

The six matrices are phenomenological response templates. They do not specify an alternative theory’s kinetic terms, constraints, strong-field generation, propagation speed, or screening. They also do not establish an empirical detection or bound. Those questions require a theory-to-waveform map and a calibrated statistical analysis, treated in Tests of Gravity and Gravitational-Wave Astronomy.

Exercises

1. Prove independence. Suppose $\sum_M c_M P^M=0$. Show that every coefficient vanishes.

Solution

The $xy$, $xz$, and $yz$ entries immediately give $c_\times=c_x=c_y=0$. The $zz$ entry gives $c_l=0$. The remaining $xx$ and $yy$ equations are

$$c_++c_b=0, \qquad -c_++c_b=0,$$

so $c_+=c_b=0$. Hence the six matrices are linearly independent and, by dimension counting, span all symmetric tidal matrices.

2. Identify transverse trace. Which basis elements contribute to $\mathcal E_{xx}+\mathcal E_{yy}$?

Solution

Plus contributes $1-1=0$, cross and both vectors have zero diagonal, and longitudinal contributes only to $zz$. Breathing alone contributes

$$\mathcal E_{xx}+\mathcal E_{yy}=2A_b.$$

Thus the transverse trace isolates the breathing coefficient in this plane-wave basis. This is a curvature statement; whether a particular detector can reconstruct it depends on its response matrix and available independent channels.

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.