Linearized Gravity and Graviton Polarizations

A massless graviton is not a generic symmetric-tensor quantum. Covariantly, gauge choices and constraint equations reduce ten on-shell tensor components to two helicities in four dimensions; canonically, four first-class constraints reduce the metric phase space to the same two polarizations. This page derives the result about Minkowski spacetime, fixes the plane-wave conventions, and marks where the weak-field description fails.

Required background. Flat-Spacetime Quantum Field Theory supplies free-field quantization. Weak-Field Gravity supplies metric perturbations. Gravitational Actions and Boundary Terms supplies the variational starting point.

Helpful background. Einstein Equations supplies the field equations, while Quantum Information supplies the physical-state interpretation.

Quadratic Einstein dynamics

Write

$$g_{\mu\nu}=\eta_{\mu\nu}+\kappa h_{\mu\nu}, \qquad \kappa^2=32\pi G,$$

on four-dimensional Minkowski spacetime. The canonically normalized $h_{\mu\nu}$ has mass dimension one. Expanding $2\sqrt{-g}R/\kappa^2$ through quadratic order and discarding a total derivative gives the Fierz–Pauli density,

$$\mathcal L_2= -\frac12\partial_\lambda h_{\mu\nu}\partial^\lambda h^{\mu\nu} +\partial_\mu h^{\mu\nu}\partial^\lambda h_{\lambda\nu} -\partial_\mu h^{\mu\nu}\partial_\nu h +\frac12\partial_\lambda h\partial^\lambda h.$$

It is invariant, up to a boundary term, under

$$\delta_\xi h_{\mu\nu}=\partial_\mu\xi_\nu+\partial_\nu\xi_\mu.$$

The interaction terms omitted here begin at $O(\kappa h^3)$ and contain two derivatives. The covariant quadratic and interacting expansion is developed in DeWitt 1967, §§2–5.

Harmonic gauge and wave equation

Define the trace reverse

$$\bar h_{\mu\nu}=h_{\mu\nu}-\frac12\eta_{\mu\nu}h.$$

The harmonic condition is $\partial^\mu\bar h_{\mu\nu}=0$. Under the gauge transformation,

$$\delta_\xi(\partial^\mu\bar h_{\mu\nu})=\Box\xi_\nu,$$

so a suitable solution of $\Box\xi_\nu=-\partial^\mu\bar h_{\mu\nu}$ reaches harmonic gauge. In vacuum, the linearized Einstein equation then reduces to

$$\Box\bar h_{\mu\nu}=0.$$

Use the site time convention and a plane wave

$$\bar h_{\mu\nu}(x)=\varepsilon_{\mu\nu}e^{-i\omega t+i\mathbf k\cdot\mathbf x}, \qquad k^\mu=(\omega,\mathbf k).$$

The equations imply $k^2=0$ and $k^\mu\varepsilon_{\mu\nu}=0$. Residual gauge transformations satisfy $\Box\xi_\mu=0$ and can remove four further components of an on-shell polarization.

Transverse-traceless polarizations

Take propagation in the $+z$ direction. Residual gauge freedom and the constraints allow transverse-traceless gauge:

$$h_{0\mu}=0,\qquad h_{z\mu}=0,\qquad h^i{}_i=0.$$

The nonzero spatial polarization matrix is

$$\varepsilon_{ij}= \begin{pmatrix} \varepsilon_+ & \varepsilon_\times & 0\\ \varepsilon_\times & -\varepsilon_+ & 0\\ 0&0&0 \end{pmatrix}.$$

Thus $\varepsilon_+$ and $\varepsilon_\times$ span the two real polarizations. The circular combinations

$$\varepsilon_{\pm2}=\frac{1}{\sqrt2} \left(\varepsilon_+\mp i\varepsilon_\times\right)$$

transform with helicity $\pm2$ under rotations about the momentum axis.

The count can be checked in two ways. Covariantly, ten on-shell symmetric components minus four gauge choices and four constraint equations leave two. Canonically, $h_{ij}$ has twelve phase-space dimensions; four first-class constraints remove eight phase-space dimensions, leaving two canonical pairs per point. Both routes require a massless gauge symmetry; a massive spin-2 field instead has five physical polarizations.

Quantization and consistency checks

Quantize first in a periodic box of volume $V$ and promote the transverse-traceless modes to operators,

$$h_{ij}^{\mathrm{TT}}(x)= \sum_{\lambda=\pm2}\sum_{\mathbf k}\frac{1}{\sqrt{2\omega V}} \left[a_\lambda(\mathbf k)\varepsilon_{ij}^{(\lambda)}e^{ik\cdot x} +a_\lambda^\dagger(\mathbf k)\varepsilon_{ij}^{(\lambda)*}e^{-ik\cdot x}\right],$$

with $\omega=\lvert\mathbf k\rvert\gt0$, orthonormal tensors $\varepsilon_{ij}^{(\lambda)*}\varepsilon_{ij}^{(\lambda')}=\delta_{\lambda\lambda'}$, and $[a_\lambda(\mathbf k),a_{\lambda'}^\dagger(\mathbf k')]=\delta_{\lambda\lambda'}\delta_{\mathbf k,\mathbf k'}$. Here $k\cdot x=-\omega t+\mathbf k\cdot\mathbf x$, so the annihilation term has the required positive-frequency dependence $e^{-i\omega t}$. The zero-momentum mode is not a radiative oscillator in this expansion. The infinite-volume limit replaces the momentum sum by the usual integral. A pure-gauge polarization $\varepsilon_{\mu\nu}=k_\mu\zeta_\nu+k_\nu\zeta_\mu$ must decouple from a conserved source because $k_\mu T^{\mu\nu}=0$.

The soft limit $\omega\to0$ preserves the two helicities but can invalidate a finite-particle Fock description because infinitely many unresolved soft quanta contribute. The strong-field limit $\kappa h\sim1$ invalidates the truncation. Curved backgrounds can still have two local tensor polarizations, but global positive frequency, particle number, and asymptotic helicity require additional structure.

The source coupling provides an independent physical-state check. At leading order it is

$$S_{\mathrm{source}}=+\frac{\kappa}{2}(h,T),$$

where $(h,T)$ denotes the spacetime pairing of $h_{\mu\nu}$ with $T^{\mu\nu}$.

Under a linearized gauge transformation, integration by parts changes this action by a term proportional to $\xi_\nu\partial_\mu T^{\mu\nu}$ plus a boundary term. A conserved stress tensor therefore makes the coupling gauge invariant. In momentum space, the same statement says that replacing an external polarization with $k_\mu\zeta_\nu+k_\nu\zeta_\mu$ gives zero. This Ward check should be applied to any proposed graviton emission amplitude.

The two tensor polarizations also have a direct tidal meaning. For slowly moving freely falling test bodies, geodesic deviation gives a relative acceleration proportional to the electric part of the linearized Riemann tensor. A plus-polarized wave alternately stretches the $x$ direction while squeezing the $y$ direction, then reverses; a cross-polarized wave produces the same pattern rotated by forty-five degrees. The Riemann tensor is gauge invariant at linear order about flat spacetime, so this observable description agrees with the gauge-fixed count without assigning physical meaning to every component of $h_{\mu\nu}$.

Normalization conventions deserve care. Rescaling $h_{\mu\nu}$ can move factors of $\kappa$ between the kinetic term and the interaction, but it cannot change the number of polarizations or a normalized amplitude. Here the canonical normalization places no Newton constant in the free kinetic operator and one power of $\kappa$ in the matter–graviton vertex. Polarization tensors are chosen orthonormal, so summing over helicities projects a conserved source onto its massless spin-two part.

Two common miscounts are worth avoiding. First, imposing ten component equations and then subtracting four gauge parameters without separating constraints from evolution can count dependent equations twice. Second, reading $h_{00}$ or $h_{0i}$ as additional quantum polarizations confuses nondynamical fields that enforce constraints with radiative oscillators. In covariant gauges these components propagate inside intermediate expressions, but ghosts and Ward identities prevent them from appearing as negative-norm external gravitons.

Exercises

Exercise. For a wave moving along $+z$, impose $\varepsilon_{0\mu}=0$, $k^i\varepsilon_{ij}=0$, symmetry, and tracelessness. Starting from a general $3\times3$ symmetric spatial matrix, derive the displayed plus-and-cross form.

Solution

Transversality with $k^i=(0,0,\omega)$ sets $\varepsilon_{zx}=\varepsilon_{zy}=\varepsilon_{zz}=0$. Symmetry also sets $\varepsilon_{xz}=\varepsilon_{yz}=0$. The remaining $2\times2$ block has entries $\varepsilon_{xx}$, $\varepsilon_{xy}=\varepsilon_{yx}$, and $\varepsilon_{yy}$. Tracelessness gives $\varepsilon_{yy}=-\varepsilon_{xx}$. Naming $\varepsilon_{xx}=\varepsilon_+$ and $\varepsilon_{xy}=\varepsilon_\times$ leaves exactly two independent components.

This explicit component count agrees with the canonical constraint count and is independent of the normalization chosen for the two basis tensors. Rotating the transverse plane mixes plus and cross, while their circular combinations acquire phases appropriate to helicity plus or minus two.

References

• DeWitt, Bryce S. “Quantum Theory of Gravity. II. The Manifestly Covariant Theory.” Physical Review 162 (1967): 1195–1239. DOI.