Radiative Degrees of Freedom

A metric perturbation has ten components, but a gravitational wave in four-dimensional general relativity has only two independent polarizations. The difference is not a convenient gauge trick: it reflects the first-class constraints of the Einstein equation. This page performs the count, identifies the freely propagating data, and shows why linearized curvature—not an arbitrary component of the metric perturbation—is the gauge-invariant test for radiation.

Required background. Weak-Field Gravity is used to linearize the metric and distinguish infinitesimal gauge transformations from physical changes. Waves, Green Functions, and Geometric Optics is used to interpret the characteristic and plane-wave solutions of the resulting hyperbolic equation.

Helpful background. Einstein Equations clarifies how the Bianchi identity organizes constraints and evolution. Equivalence Principles and Gravitational Measurement supplies the distinction between removable connection coefficients and measurable tidal curvature.

Linearized fields and gauge transformations

Work to first order about Minkowski spacetime:

$$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}, \qquad \lvert h_{\mu\nu}\rvert\ll1.$$

Indices on first-order quantities are moved with $\eta_{\mu\nu}$. Define the trace $h=\eta^{\mu\nu}h_{\mu\nu}$ and trace reversal

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

Under an infinitesimal passive coordinate change $x'^\mu=x^\mu+\xi^\mu$, the perturbation at the same coordinate label transforms as

$$h'_{\mu\nu} =h_{\mu\nu}-\partial_\mu\xi_\nu-\partial_\nu\xi_\mu,$$

and hence

$$\bar h'_{\mu\nu} =\bar h_{\mu\nu}-\partial_\mu\xi_\nu-\partial_\nu\xi_\mu +\eta_{\mu\nu}\partial_\rho\xi^\rho.$$

The four functions $\xi^\mu$ change the representative $h_{\mu\nu}$ without changing the perturbed spacetime geometry. Imposing Lorenz gauge,

$$\partial^\mu\bar h_{\mu\nu}=0,$$

reduces the linearized Einstein equation to

$$\Box\bar h_{\mu\nu}=-16\pi G T_{\mu\nu}, \qquad \Box=\eta^{\rho\sigma}\partial_\rho\partial_\sigma.$$

Stress–energy conservation is the compatibility condition: applying $\partial^\mu$ to the wave equation gives $\partial^\mu T_{\mu\nu}=0$. Lorenz gauge is not a complete gauge fixing. A transformation generated by a vector satisfying $\Box\xi^\mu=0$ preserves it, leaving residual freedom available in vacuum.

The derivation and its relation to gauge symmetry are developed in Flanagan and Hughes 2005, §2, eqs. (2.3)–(2.16), pp. 5–6, PDF. The source uses the same metric and curvature signs; the formulas here restore $G$.

Why the count gives two

A safe count uses phase space. On a spacelike slice, the spatial metric perturbation $h_{ij}$ has six components and six conjugate momenta, for twelve phase-space functions. The linearized Hamiltonian and momentum constraints give four first-class constraints. Each first-class constraint removes one constrained phase-space direction and one gauge direction. Therefore

$$12-2\times4=4$$

physical phase-space functions remain: two configuration variables and their two momenta. Thus there are two local propagating degrees of freedom.

The familiar covariant mnemonic—ten symmetric components, four coordinate functions, and four constraint relations—also ends at two, but phase-space language explains why gauge and constraints are paired rather than eight unrelated algebraic conditions. The lapse and shift perturbations $h_{00}$ and $h_{0i}$ act as nondynamical multipliers in this description; they can encode constrained fields near matter without becoming additional radiation.

The same result is visible in Fourier space. In vacuum let

$$\bar h_{\mu\nu}=\Re\!\left[A_{\mu\nu}e^{ik_\rho x^\rho}\right].$$

The wave equation and Lorenz condition imply

$$k^\rho k_\rho=0, \qquad k^\mu A_{\mu\nu}=0.$$

For a wave moving toward increasing $z$, residual gauge freedom removes the time and longitudinal amplitudes and the trace. What remains can be written

$$A^{\mathrm{TT}}_{ij} =A_+ e^+_{ij}+A_\times e^\times_{ij},$$

where, in the $(x,y,z)$ basis,

$$e^+_{ij}= \begin{pmatrix} 1&0&0\\ 0&-1&0\\ 0&0&0 \end{pmatrix}, \qquad e^\times_{ij}= \begin{pmatrix} 0&1&0\\ 1&0&0\\ 0&0&0 \end{pmatrix}.$$

The coefficients $A_+$ and $A_\times$ are the two tensor polarizations. The TT-gauge page constructs this representative and states where it is available; the count itself does not assume that TT gauge exists globally.

Curvature separates radiation from representation

The linearized Riemann tensor is

$$R^{(1)}_{\alpha\beta\gamma\delta} =\frac12\left( \partial_\gamma\partial_\beta h_{\alpha\delta} +\partial_\delta\partial_\alpha h_{\beta\gamma} -\partial_\delta\partial_\beta h_{\alpha\gamma} -\partial_\gamma\partial_\alpha h_{\beta\delta} \right).$$

Substituting a pure-gauge perturbation

$$h_{\mu\nu}=\partial_\mu\xi_\nu+\partial_\nu\xi_\mu$$

makes every term cancel by commutation of partial derivatives. Therefore $R^{(1)}_{\alpha\beta\gamma\delta}$ is invariant under linearized gauge transformations about flat spacetime. A nonzero oscillatory metric component need not be radiation; a nonzero radiative curvature cannot be removed by a smooth infinitesimal coordinate change. Nonzero curvature alone is insufficient to identify radiation: a static Newtonian tidal field also has curvature. One must additionally identify the freely propagating transverse field, or the outgoing asymptotic news.

For a TT wave, the locally measurable component becomes

$$R^{(1)}_{0i0j}=-\frac12\ddot h^{\mathrm{TT}}_{ij}.$$

This relation supplies the bridge to tidal response. It also exposes a zero-frequency subtlety: curvature depends on two time derivatives, whereas a lasting change of strain can encode memory. One must use the appropriate asymptotic observable rather than infer that zero late-time curvature means that no radiation passed.

Constraints are fields, but not free waves

Near a source, constrained scalar and vector components can be physically important. They produce Newtonian gravity, frame dragging, and other near-zone fields. The distinction is dynamical:

• constrained variables on a slice are determined by matter and the freely specified data through elliptic or constraint equations;
• radiative variables admit independent initial data and obey hyperbolic propagation equations;
• gauge variables change under a coordinate transformation and have no independent curvature content.

The split can depend on background and boundary conditions. “Two polarizations” is the local vacuum degree count for four-dimensional general relativity, not a claim that every metric theory, dimension, or background has the same spectrum. Nor does it say that a general source field is everywhere a plane TT wave. It says that the freely propagating vacuum content of the linearized Einstein equation has two independent tensor amplitudes.

Common pitfalls

Subtracting four gauge functions and stopping at six. The Einstein equation also contains four first-class constraints. The physical count is most transparent in phase space, where each first-class constraint removes both a constrained and a gauge direction.

Calling every time-dependent metric component radiation. Coordinates can make a flat metric time dependent. Test the linearized curvature, the far-zone $1/r$ field with outgoing boundary conditions, or the corresponding asymptotic news.

Treating TT gauge as the definition of a gravitational wave. TT gauge is a particularly transparent representative in a vacuum wave zone. Curvature and asymptotic radiative data are the invariant content, including situations where a global TT representative is unavailable.

Exercises

1. Pure-gauge plane perturbation. Let $\xi_\mu=b_\mu e^{ik_\rho x^\rho}$ and

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

Show directly that its linearized Riemann tensor vanishes, without imposing $k^2=0$.

Solution

The amplitude is $h_{\mu\nu}=i(k_\mu b_\nu+k_\nu b_\mu)e^{ik\cdot x}$. Each second derivative in $R^{(1)}_{\alpha\beta\gamma\delta}$ contributes a product of three $k$ factors with one $b$. Terms cancel pairwise because multiplication of the $k$ components is commutative. Equivalently, substituting $h_{\mu\nu}=2\partial_{(\mu}\xi_{\nu)}$ into the curvature gives sums of commuting third derivatives with opposite signs. No wave equation is needed: the vanishing follows from gauge form alone.

2. Phase-space count. Suppose a hypothetical four-dimensional metric theory had the same six spatial metric components but only three independent first-class constraints. If there were no second-class constraints, how many configuration-space degrees of freedom would it have?

Solution

The spatial metric and its momentum give twelve phase-space functions. Three first-class constraints remove $2\times3=6$, leaving six physical phase-space functions. Dividing by two to pair coordinates with momenta gives three configuration-space degrees of freedom. This arithmetic alone does not determine their spins or stability; the field equations and constraint algebra would be needed.

References

• Flanagan, Éanna É., and Scott A. Hughes. “The Basics of Gravitational Wave Theory.” New Journal of Physics 7 (2005): 204. DOI. Open PDF.