Newman–Penrose Polarization Classification

The Newman–Penrose description classifies the tidal content of an approximately plane, null metric wave by projecting curvature onto a normalized null tetrad. In that regime, six real response amplitudes organize into two tensor, two vector, and two scalar patterns. The classification is kinematic: field equations decide which entries actually propagate, and general relativity in vacuum retains only the two tensor entries. It neither supplies alternative-theory dynamics nor reports empirical bounds.

Helpful background. Einstein Equations supplies the vacuum restriction that selects the GR class. Waves, Green Functions, and Geometric Optics supplies the plane-null-wave approximation. Gravitational Radiation and Waveforms supplies the volume’s strain, curvature, and Newman–Penrose signs. Tests of Gravity supplies the theory-specific interpretation beyond this kinematic classification.

A fully normalized wave tetrad

Let $u^a$ be a unit future timelike observer and $z^a$ a unit spatial vector in the wave-propagation direction. With transverse unit vectors $x^a,y^a$, define

$$\ell^a=\frac{u^a+z^a}{\sqrt2}, \qquad n^a=\frac{u^a-z^a}{\sqrt2}, \qquad m^a=\frac{x^a+iy^a}{\sqrt2}.$$

For signature $(-,+,+,+)$,

$$\ell\mathbin{\cdot}n=-1, \qquad m\mathbin{\cdot}\bar m=1,$$

and all other inner products vanish. The outgoing wave has phase $u_{\rm ret}=t-z$, tangent direction $\ell^a$, and the volume defines

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

For the other entries below, fix the complete projections

$$\begin{aligned} \Psi_2&=-C_{abcd}\ell^a m^b\bar m^c n^d,\\ \Psi_3&=-C_{abcd}\ell^a n^b\bar m^c n^d,\\ \Phi_{22}&=-\frac12R_{ab}n^an^b. \end{aligned}$$

The Weyl-scalar conventions and symmetric null-tetrad normalization are those of Bishop and Rezzolla 2016, pp. 22–23, eqs. (95)–(103), PDF. The Ricci projection is stated separately because its sign also varies between references.

In the radiation zone this gives $\Psi_4=\ddot h_+-i\ddot h_\times$ for $h=h_+-ih_\times$. Reversing the propagation vector, interchanging $m$ and $\bar m$, or rescaling $\ell$ and $n$ changes component signs or boost weights, so a list of NP scalars without its tetrad is incomplete.

Six entries of the tidal matrix

The observable object is the electric curvature $\mathcal E_{ij}=R_{0i0j}$ in the real triad $(x,y,z)$. In a common Eardley–Newman–Penrose normalization adapted to the tetrad above, the nearly plane null-wave matrix is

$$\mathcal E_{ij}= \begin{pmatrix} -\tfrac12(\operatorname{Re}\Psi_4+\Phi_{22}) &\tfrac12\operatorname{Im}\Psi_4&-2\operatorname{Re}\Psi_3\\ \tfrac12\operatorname{Im}\Psi_4 &\tfrac12(\operatorname{Re}\Psi_4-\Phi_{22})&2\operatorname{Im}\Psi_3\\ -2\operatorname{Re}\Psi_3&2\operatorname{Im}\Psi_3&-6\Psi_2 \end{pmatrix}.$$

Here $\Phi_{22}$ is real, $\Psi_3$ is complex, and the plane-wave restrictions make the displayed $\Psi_2$ real. For example, the projections give $\Psi_3=(-\mathcal E_{xz}+i\mathcal E_{yz})/2$ and $\Psi_2=-\mathcal E_{zz}/6$, directly checking the vector and longitudinal factors. Other NP references sometimes absorb factors or reverse the Riemann sign. The six real coefficients are therefore

curvature data	tidal patterns	character
$\operatorname{Re}\Psi_4,\operatorname{Im}\Psi_4$	plus, cross	transverse tensor
$\operatorname{Re}\Psi_3,\operatorname{Im}\Psi_3$	vector-$x$, vector-$y$	mixed transverse–longitudinal
$\Phi_{22}$	breathing	transverse scalar
$\Psi_2$	longitudinal	scalar along propagation

The classification through measurable curvature, rather than coordinate components, is the central construction of Eardley et al. 1973, pp. 884–886.

The E(2) classes

For a plane null wave, the little group preserving its propagation direction is the two-dimensional Euclidean group E(2). Null rotations can mix some NP components, so the invariant question is not simply whether one component vanishes in one arbitrary tetrad. The standard classes can be summarized as follows:

class	invariant leading condition	possible patterns
$\mathrm{II}_6$	$\Psi_2\ne0$	all six
$\mathrm{III}_5$	$\Psi_2=0$, $\Psi_3\ne0$	two vector, two tensor, breathing
$\mathrm N_3$	$\Psi_2=\Psi_3=0$, $\Psi_4\ne0$, $\Phi_{22}\ne0$	two tensor plus breathing
$\mathrm N_2$	only $\Psi_4\ne0$	two tensor
$\mathrm O_1$	only $\Phi_{22}\ne0$	breathing
$\mathrm O_0$	all radiative entries vanish	no wave

The subscript counts the maximum number of observable polarization patterns in the class. A particular solution can occupy fewer patterns than its class permits.

Recovering general relativity

For a weak vacuum wave in general relativity, $R_{ab}=0$ removes $\Phi_{22}$, while the radiative peeling hierarchy and field equations remove the leading $\Psi_2$ and $\Psi_3$ entries from the plane-wave tidal field. What remains is

$$\mathcal E_{ij}^{\rm GR}= \begin{pmatrix} -\tfrac12\operatorname{Re}\Psi_4& \tfrac12\operatorname{Im}\Psi_4&0\\ \tfrac12\operatorname{Im}\Psi_4& \tfrac12\operatorname{Re}\Psi_4&0\\ 0&0&0 \end{pmatrix},$$

which is transverse and trace free. Using $\Psi_4=\ddot h_+-i\ddot h_\times$ reproduces $\mathcal E_{ij}=-\tfrac12\ddot h^{\rm TT}_{ij}$ exactly. Thus the GR result is class $\mathrm N_2$, not an assumption smuggled into the six-mode list.

Domain of validity

E(2) classification assumes an approximately plane wave with a null propagation vector and wavelengths short compared with background-curvature scales. A massive or dispersive mode has a timelike wave vector and a different little group; its longitudinal and breathing response can also be frequency dependent. Curvature corrections beyond geometric optics can generate observer-dependent projections without introducing a new fundamental degree of freedom. Those cases require returning to the field equations and the full detector response rather than relabeling the six null patterns.

Common pitfalls

Treating six patterns as six universal degrees of freedom. Six is the maximum kinematic dimension of the symmetric tidal response under the stated metric and null-wave assumptions. A specified theory and background determine the propagating subset.

Reading an NP component without its tetrad. Boosts, spins, and null rotations alter components. State normalization, propagation direction, and dyad orientation before comparing signs.

Exercises

1. Count and recover the GR entries. Starting from the displayed matrix, impose vacuum GR plane-wave conditions and verify the remaining independent components.

Solution

Vacuum removes the Ricci scalar $\Phi_{22}$. The GR plane-wave equations leave $\Psi_2=\Psi_3=0$, so only the real and imaginary parts of $\Psi_4$ remain. The resulting matrix is symmetric, has $\mathcal E_{zz}=0$, and obeys

$$\mathcal E_{xx}+\mathcal E_{yy}=0.$$

It therefore has two independent transverse, trace-free components. Substitution of $\Psi_4=\ddot h_+-i\ddot h_\times$ gives $\mathcal E_{xx}=-\ddot h_+/2$ and $\mathcal E_{xy}=-\ddot h_\times/2$, matching the volume’s plus-cross convention.

2. Identify a breathing-only response. Set every NP entry except $\Phi_{22}$ to zero. What does a transverse ring do?

Solution

Then $\mathcal E_{xx}=\mathcal E_{yy}=-\Phi_{22}/2$, with all off-diagonal and longitudinal entries zero. Geodesic deviation accelerates every point of a transverse ring radially with the same sign at a given phase, so the ring expands and contracts isotropically rather than becoming an ellipse. This is the single pattern of class $\mathrm O_1$.

References

• Bishop, Nigel T., and Luciano Rezzolla. “Extraction of Gravitational Waves in Numerical Relativity.” Living Reviews in Relativity 19 (2016): 2. 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.