Linearized Gravity

Core Idea

Linearized gravity is the weak-field approximation of general relativity, where the metric is written as g_μν = η_μν + h_μν with |h_μν| << 1. Expanding the Einstein equations to first order in h_μν yields linear equations resembling Maxwell's equations for electromagnetism — a profound structural parallel known as gravitoelectromagnetism. In the Lorenz (harmonic) gauge, the linearized vacuum equations reduce to a wave equation □h̄_μν = 0 for the trace-reversed perturbation h̄_μν, with solutions that propagate at the speed of light. The residual gauge freedom can be used to impose the transverse-traceless (TT) gauge, isolating the two physical polarizations of gravitational waves. Linearized gravity provides the framework for gravitational wave physics, the post-Newtonian expansion, and the connection between GR and Newtonian gravity.

Explainer

The full Einstein field equations are 10 coupled, nonlinear partial differential equations — too complex to solve analytically except in cases with high symmetry. Linearized gravity tames this complexity by restricting attention to weak gravitational fields, where the spacetime metric is close to the flat Minkowski metric: g_μν = η_μν + h_μν with the perturbation h_μν much smaller than 1 in magnitude. Expanding the Einstein equations to first order in h_μν discards all products of perturbations (h × h terms), producing a set of linear equations for h_μν. This approximation is excellent for the gravitational field of the Sun at the Earth's orbit (h ~ 10⁻⁸), for gravitational waves far from their source (h ~ 10⁻²¹), and for post-Newtonian corrections to planetary orbits.

The linearized equations have a remarkable structural similarity to Maxwell's equations of electromagnetism. Define the trace-reversed perturbation h̄_μν = h_μν - (1/2)η_μν h (where h = η^{μν}h_μν is the trace). The linearized Einstein equations become □h̄_μν - ∂_μ∂^α h̄_αν - ∂_ν∂^α h̄_αμ + η_μν ∂^α∂^β h̄_αβ = -16πG T_μν / c⁴. Imposing the Lorenz gauge condition ∂^μ h̄_μν = 0 (analogous to the Lorenz gauge ∂^μ A_μ = 0 in electromagnetism) simplifies this to □h̄_μν = -16πG T_μν / c⁴ — a wave equation with source, identical in structure to □A_μ = -μ₀ J_μ in electrodynamics. This is gravitoelectromagnetism: the time-time component h̄₀₀ plays the role of the gravitational analog of the electric potential, and the time-space components h̄₀ᵢ play the role of a gravitomagnetic vector potential.

The Lorenz gauge does not completely fix the coordinates — there remains a residual gauge freedom of coordinate transformations x^μ → x^μ + ξ^μ(x) where □ξ^μ = 0. For plane-wave solutions in vacuum, this residual freedom can be used to impose the transverse-traceless (TT) gauge, in which h₀₀ = h₀ᵢ = 0 (no temporal components), hⁱ_i = 0 (traceless), and k^j h_{ij} = 0 (transverse to the propagation direction k). In this gauge, the 10 original components reduce to just 2 independent physical degrees of freedom: the plus and cross polarizations h₊ and h×. For a wave propagating in the z-direction, h₊ produces differential stretching in the x and y directions, while h× does the same along axes rotated by 45 degrees. These two polarizations are the measurable content of a gravitational wave.

The TT gauge also reveals why gravitational waves are transverse and traceless — these are not arbitrary gauge choices but physical properties. Transversality (no longitudinal component) follows from the gauge constraints and means gravitational waves do not compress space along their direction of travel. Tracelessness means they produce no overall volume change — only shape distortion. Both properties are consequences of the masslessness of the graviton (or equivalently, the fact that gravity propagates at c). In the linearized theory with sources, the retarded-potential solution and its far-field multipole expansion lead to the quadrupole formula for gravitational wave emission, which is the quantitative basis for predicting signals from astrophysical sources and for interpreting LIGO/Virgo observations.