Plane Waves & Polarizations

1. Introduction: Vacuum Solutions to the Linearized Equations

In the Lorenz gauge $\partial^\mu \bar{h}_{\mu\nu} = 0$, the linearized vacuum Einstein equations reduce to the wave equation

where $\bar{h}_{\mu\nu} = h_{\mu\nu} - \frac{1}{2}\eta_{\mu\nu}h$ is the trace-reversed perturbation and $\Box = -\frac{1}{c^2}\partial_t^2 + \nabla^2$ is the flat-space d'Alembertian. This is a massless wave equation identical in form to the electromagnetic wave equation in Lorenz gauge. The most general solution is a superposition of monochromatic plane waves, so we begin by studying a single Fourier mode.

The key questions we address in this chapter are: How many independent polarization states does a gravitational wave possess? What is the physical significance of each? How much energy and momentum does a gravitational wave carry? The answers — two polarizations, helicity $\pm 2$, and a well-defined effective stress-energy tensor — are foundational to all of gravitational wave physics.

2. Monochromatic Plane Wave Solution

We seek a monochromatic plane-wave solution to the vacuum wave equation by adopting the ansatz

where $A_{\mu\nu}$ is a constant symmetric polarization tensor (10 independent components for a symmetric $4 \times 4$ matrix) and $k^\mu = (\omega/c, \vec{k})$ is the wave 4-vector.

Dispersion Relation

Substituting the ansatz into $\Box \bar{h}_{\mu\nu} = 0$:

For a non-trivial solution ($A_{\mu\nu} \neq 0$), we require

Lorenz Gauge Constraint: Transversality

The Lorenz gauge condition $\partial^\mu \bar{h}_{\mu\nu} = 0$ applied to our plane-wave ansatz gives

This imposes 4 conditions (one for each value of $\nu$), reducing the independent components of $A_{\mu\nu}$ from 10 to 6. The condition$k^\mu A_{\mu\nu} = 0$ means the polarization tensor is transverse to the propagation direction — the wave oscillations are perpendicular to the direction of travel, just like electromagnetic waves.

Counting Degrees of Freedom So Far

We will see next that further gauge fixing reduces these 6 to just 2 physical polarizations.

3. The Transverse-Traceless (TT) Gauge

The Lorenz gauge condition $\partial^\mu \bar{h}_{\mu\nu} = 0$ does not completely fix the gauge. We still have residual gauge freedom: we can perform an additional infinitesimal coordinate transformation $x^\mu \to x^\mu + \xi^\mu$ provided$\Box \xi^\mu = 0$, which preserves the Lorenz condition.

Residual Gauge Freedom

Under $x^\mu \to x^\mu + \xi^\mu$, the metric perturbation transforms as

For a plane-wave gauge parameter $\xi^\mu = C^\mu e^{ik_\rho x^\rho}$ with$\Box \xi^\mu = 0$ (automatically satisfied since $k_\mu k^\mu = 0$), the polarization tensor shifts as

The 4 free parameters $C^\mu$ allow us to impose 4 additional conditions. We choose:

Imposing the TT Conditions

Condition 1: Temporal components vanish. We use $C^0$ and the spatial$C^i$ to set

This eliminates $A_{00}$ and $A_{0i}$ (4 components), but we already had the Lorenz conditions. The net effect of the combined gauge fixing is:

Condition 2: Tracelessness. We further require

Note that when $A_{0\mu} = 0$ and $A^i{}_i = 0$, the trace-reversed perturbation equals the perturbation itself: $\bar{h}_{\mu\nu} = h_{\mu\nu}$, since the trace $h = \eta^{\mu\nu}h_{\mu\nu} = -h_{00} + h_{ii} = 0$. This simplifies all subsequent calculations enormously.

Summary of TT Gauge Conditions

Explicit Form for Propagation Along z

Consider a wave propagating in the $+z$ direction: $k^\mu = (\omega/c)(1, 0, 0, 1)$. The transversality condition $k^i A_{ij} = 0$ requires $A_{3j} = 0$ for all $j$. Combined with $A_{0\mu} = 0$, the polarization tensor has nonzero components only in the$xy$-plane:

where tracelessness imposes $A_{yy} = -A_{xx}$. We define the two polarization amplitudes:

The complete TT metric perturbation for a monochromatic plane wave traveling along $z$ is therefore:

The TT Projection Operator

For a general gravitational wave (not necessarily along $z$), the TT projection can be performed using the Lambda tensor (TT projector):

where $\hat{n}$ is the propagation direction. Then$h^{TT}_{ij} = \Lambda_{ij,kl}\,h_{kl}$. The projector $P_{ij}$ projects onto the plane transverse to $\hat{n}$, and the $-\frac{1}{2}P_{ij}P_{kl}$term removes the trace.

4. Helicity and the Spin-2 Nature of Gravitons

The polarization structure of gravitational waves reveals that the graviton — the quantum of the gravitational field — is a massless spin-2 particle. We demonstrate this by examining how the polarization modes transform under rotations about the propagation axis.

Rotation Transformation

Consider a rotation by angle $\psi$ about the $z$-axis (the propagation direction). In the transverse plane, the coordinates transform as

As a rank-2 tensor in the transverse plane, $h^{TT}_{ij}$ transforms via the rotation matrix $R(\psi)$ as $h'_{ij} = R_{ik}R_{jl}\,h_{kl}$. Carrying out the matrix multiplication explicitly:

Complex Polarization and Helicity States

Define the complex polarization combinations (circular polarization basis):

Under the rotation by $\psi$:

A field that acquires a phase $e^{is\psi}$ under rotation by $\psi$ has helicity $s$. Therefore $h_R$ has helicity$+2$ and $h_L$ has helicity $-2$. A massless particle with helicity $\pm s$ has spin $s$. The graviton is a massless spin-2 particle.

Comparison with Electromagnetism (Spin-1)

Newman-Penrose Scalar $\Psi_4$

In the Newman-Penrose formalism, gravitational radiation is encoded in the Weyl scalar $\Psi_4$, defined as

where $C_{\mu\nu\rho\sigma}$ is the Weyl tensor and $(l^\mu, n^\mu, m^\mu, \bar{m}^\mu)$is a null tetrad. For linearized waves, $\Psi_4$ relates to the polarizations as

$\Psi_4$ is a complex scalar of spin weight $-2$ and is invariant under the residual gauge transformations that preserve the TT gauge. It is the primary observable extracted in numerical relativity simulations, from which waveforms are computed. The five complex Weyl scalars$\Psi_0, \ldots, \Psi_4$ classify the algebraic type of the gravitational field (Petrov classification); for radiation zones, only $\Psi_4$ is non-vanishing at leading order in $1/r$ (peeling theorem).

5. Energy and Momentum of Gravitational Waves: The Isaacson Tensor

A deep question in general relativity is whether gravitational waves carry energy and momentum. The difficulty is that there is no local, gauge-invariant energy density for the gravitational field. Isaacson (1968) resolved this by showing that an effective stress-energy tensor emerges when one averages over several wavelengths.

Derivation of the Isaacson Tensor

We expand the Einstein equations to second order in $h_{\mu\nu}$. The second-order terms act as an effective source for the first-order equations. After averaging over several wavelengths (denoted $\langle \cdots \rangle$), the effective stress-energy tensor of gravitational waves in TT gauge is

The averaging is essential: it makes the expression gauge-invariant at leading order and gives a physically meaningful, non-negative energy density.

Step-by-Step Derivation

Step 1: Write the full metric as $g_{\mu\nu} = \bar{g}_{\mu\nu} + h_{\mu\nu}$, where $\bar{g}_{\mu\nu}$ is a slowly varying background and $h_{\mu\nu}$is a rapidly oscillating perturbation.

Step 2: Expand the Einstein tensor to second order:

Step 3: The first-order piece satisfies the linearized vacuum equations$G_{\mu\nu}^{(1)} = 0$. The averaged second-order piece defines the effective stress-energy tensor:

Step 4: Explicit computation in TT gauge gives the Isaacson formula above.

Energy Density and Flux

The energy density (the $T^{GW}_{00}$ component) for a wave propagating along $z$ is:

where the factor of 2 arises because $h^{TT}_{\alpha\beta} h^{TT\,\alpha\beta} = 2(h_+^2 + h_\times^2)$for the TT metric. The energy flux (power per unit area) along the propagation direction is:

For a monochromatic wave $h_+ = A_+\cos(\omega t - kz)$, the time average gives$\langle \dot{h}_+^2 \rangle = \frac{1}{2}\omega^2 A_+^2$, so:

Properties of the Isaacson Tensor

• ●Conservation: $\nabla^\mu T^{GW}_{\mu\nu} = 0$ — GW energy-momentum is conserved, just like matter
• ●Positivity: $\rho_{GW} \geq 0$ — gravitational waves always carry positive energy
• ●Backreaction: The Isaacson tensor sources the background curvature, allowing GWs to curve spacetime
• ●Gauge invariance: The averaging makes it invariant under gauge transformations that preserve the short-wavelength/long-wavelength split

6. Applications: Polarization Analysis

Stokes Parameters for Gravitational Waves

By analogy with optics, we define Stokes parameters for gravitational waves to characterize the polarization state:

where $\delta$ is the phase difference between the two polarizations. These satisfy$I^2 \geq Q^2 + U^2 + V^2$, with equality for fully polarized waves.

Circular vs. Linear Polarization

Linear polarization ($V = 0$): A purely plus-polarized wave stretches particles along $x$ and compresses along $y$ (and vice versa). A purely cross-polarized wave does the same but rotated by $45°$. Most astrophysical sources produce a mixture depending on inclination angle.

Circular polarization ($Q = U = 0$): When $h_+ = h_\times$and $\delta = \pm\pi/2$, the wave is circularly polarized. A ring of test particles deforms into a rotating ellipse. For a face-on binary ($\iota = 0$), the emitted GWs are circularly polarized; for edge-on ($\iota = \pi/2$), they are linearly polarized.

Polarization Tests of GR vs. Alternative Theories

General relativity predicts exactly 2 tensor polarizations ($h_+$ and $h_\times$). Alternative theories of gravity can predict up to 6 polarization modes:

With a network of at least 5 non-coplanar detectors, all 6 polarization modes can in principle be separated. Current LIGO-Virgo-KAGRA analyses find results consistent with pure tensor polarization, constraining many alternative theories. Future detectors like LISA and the Einstein Telescope will enable even more stringent polarization tests.

7. Historical Context

The physical reality of gravitational waves was debated for decades after Einstein's 1916 prediction. The central question was whether they were genuine physical phenomena or mere coordinate artifacts.

The Sticky Bead Argument (1957)

At the 1957 Chapel Hill conference, Hermann Bondi presented the decisive "sticky bead" thought experiment (often attributed also to Richard Feynman). The argument is elegant in its simplicity:

Pirani's Contribution

Felix Pirani (1956) provided the rigorous mathematical framework by showing that gravitational waves produce measurable tidal forces through the geodesic deviation equation. He demonstrated that the Riemann tensor — a coordinate-invariant quantity — oscillates in the presence of a gravitational wave, proving their physical reality beyond any coordinate ambiguity. Pirani's work directly connected the abstract mathematics of linearized gravity to observable physical effects.

Resolution of the Energy Controversy

Bondi, along with van der Burg and Metzner (1962), established the Bondi mass and the Bondi news function — a fully nonlinear, coordinate-invariant measure of gravitational radiation at null infinity. The key result: the Bondi mass decreases monotonically when gravitational radiation is emitted, providing an exact proof that GWs carry positive energy away from isolated systems. Isaacson's (1968) tensor provided the complementary short-wavelength description that we derived in Section 5.