Linearized Gravity and Gravitational Waves

Far from any strong source, the metric is nearly flat and the Einstein equations can be linearized, turning the intractable nonlinear theory into a manageable wave equation. The result is one of general relativity's boldest predictions (Einstein, 1916): gravitational waves — ripples in the geometry of spacetime that propagate at the speed of light, carry energy, and stretch and squeeze matter as they pass. A century later they became observable: LIGO's 2015 detection of two merging black holes opened gravitational-wave astronomy.

We use $c=1$, keep $G$ explicit, mostly-plus signature.

Linearizing the field equations

Write the metric as flat plus a small perturbation,

$$g_{\mu \nu }={\eta }_{\mu \nu }+h_{\mu \nu },|h_{\mu \nu }|\ll 1,$$

and keep only first order in $h$. Indices are now raised and lowered with $\eta$, as in special relativity. Introducing the trace-reversed perturbation ${\bar{h}}_{\mu \nu }=h_{\mu \nu }-\frac{1}{2}{\eta }_{\mu \nu }h$ (with $h={\eta }^{\mu \nu }{h}_{\mu \nu }$), the linearized Einstein tensor simplifies dramatically. Imposing the Lorenz (harmonic) gauge ${\partial }^{\mu }{\bar{h}}_{\mu \nu }=0$ — the gravitational analogue of the electromagnetic Lorenz gauge — the field equations become a decoupled wave equation:

$$\boxed{\square {\bar{h}}_{\mu \nu }=-16\pi G{T}_{\mu \nu },}$$

where $\square ={\eta }^{\mu \nu }{\partial }_{\mu }{\partial }_{\nu }$ is the flat d'Alembertian. This is exactly the structure of electromagnetism ($\square A_{\mu }=-{\mu }_0{J}_{\mu }$): retarded-potential solutions, radiation at speed $c$, the works. Linearized gravity is a spin-2 field theory on flat spacetime, and its quantization would give the massless spin-2 graviton — the bridge to field theory foreshadowed in SR.

Gauge freedom and the physical polarizations

Just as electromagnetism has gauge freedom $A_{\mu }\to A_{\mu }+{\partial }_{\mu }\lambda$, linearized gravity inherits it from coordinate (diffeomorphism) invariance: an infinitesimal coordinate change $x^{\mu }\to x^{\mu }+{\xi }^{\mu }$ shifts $h_{\mu \nu }\to h_{\mu \nu }-{\partial }_{\mu }{\xi }_{\nu }-{\partial }_{\nu }{\xi }_{\mu }$. The Lorenz gauge leaves residual freedom that can be used, in vacuum, to reach the transverse–traceless (TT) gauge:

$$h_{0\mu }^{\text{TT}}=0,{\partial }^i{h}_{ij}^{\text{TT}}=0,h^{\text{TT}i}{}_i=0.$$

Of the ten components of $h_{\mu \nu }$, gauge fixing removes eight, leaving exactly two physical polarizations, conventionally called plus ($+$) and cross ($\times$). For a wave traveling in the $z$-direction they act in the transverse $x$–$y$ plane. The two-polarization count is the classical signature of a massless spin-2 field (helicity $\pm 2$), just as light's two polarizations signal massless spin-1.

What a wave does: the ring of test masses

A gravitational wave is detectable through the tidal (geodesic-deviation) forces it exerts. A ring of freely floating test masses in the path of a wave is alternately stretched and squeezed transverse to the propagation direction, $90°$ out of phase along the two axes:

• the plus polarization stretches horizontally while squeezing vertically, then reverses;
• the cross polarization does the same along axes rotated $45°$.

graph LR
  A["circle<br/>(no wave)"] --> B["+ pol:<br/>stretch x, squeeze y"]
  B --> C["+ pol:<br/>squeeze x, stretch y"]
  A --> D["× pol:<br/>stretch/squeeze<br/>at 45°"]

The fractional length change is the strain $h\sim \Delta L/L$. It is minuscule: astrophysical waves at Earth have $h\sim 10^{-21}$, a length change smaller than a proton's width over LIGO's kilometre arms — which is why detection took a century.

Generation: the quadrupole formula

Solving the wave equation with a source gives the radiated field. Unlike electromagnetism, gravity has no monopole radiation (mass is conserved — Birkhoff's theorem) and no dipole radiation (momentum conservation kills the mass dipole). The leading term is the mass quadrupole. Einstein's quadrupole formula gives the luminosity

$$\boxed{L=\frac{G}{5}⟨\frac{d^3{Q}_{ij}}{d{t}^3}\frac{d^3{Q}^{ij}}{d{t}^3}⟩,}$$

with $Q_{ij}$ the source's traceless mass-quadrupole moment. Consequences:

• Only non-spherical, accelerating mass distributions radiate — a spinning perfectly axisymmetric star does not; a binary system does.
• The strongest sources are compact binaries (neutron stars, black holes) in tight, fast orbits, where the third derivative of $Q_{ij}$ is enormous.
• Gravitational radiation carries energy away, so a binary's orbit decays, the stars spiral inward, and the frequency and amplitude chirp upward to merger.

The evidence

• Indirect (Hulse–Taylor, 1974). The binary pulsar PSR B1913+16's orbital period shrinks at exactly the rate the quadrupole formula predicts from gravitational-wave energy loss — agreement to better than $0.2\%$, and the 1993 Nobel Prize. The first proof that gravitational waves are real and carry energy.
• Direct (LIGO, 14 Sept 2015 — event GW150914). The two LIGO interferometers recorded the chirp of two $\sim 30M_{\odot }$ black holes merging $\sim 1.3$ billion light-years away, converting $\sim 3M_{\odot }$ into gravitational-wave energy in a fraction of a second. Confirmed general relativity in the strong-field, dynamical regime and won the 2017 Nobel Prize.
• Multi-messenger (GW170817, 2017). A binary neutron-star merger seen in gravitational waves and across the electromagnetic spectrum. The near-simultaneous arrival of gravitational waves and gamma rays confirmed that gravity propagates at the speed of light to $\sim 10^{-15}$, and tied gravitational-wave astronomy to conventional astrophysics (kilonovae, heavy-element nucleosynthesis).

Gravitational-wave detectors are now a standard astronomical tool: ground-based interferometers (LIGO/Virgo/KAGRA), pulsar-timing arrays sensing nanohertz waves from supermassive black-hole binaries (NANOGrav's 2023 stochastic-background evidence), and the planned space-based LISA.

Energy of the waves

That gravitational waves carry energy is subtle: gravitational energy has no local tensorial density (it can be transformed away pointwise by the EEP). The energy flux is instead captured by the Isaacson stress–energy tensor, a gauge-invariant average of $⟨\partial h^{\text{TT}}\partial h^{\text{TT}}⟩$ over several wavelengths. Averaged over regions large compared to a wavelength, it yields a well-defined, positive energy flux — reconciling the local non-localizability of gravitational energy with the unmistakable orbital decay of binaries.

Summary

• Linearizing $g_{\mu \nu }={\eta }_{\mu \nu }+h_{\mu \nu }$ in Lorenz gauge gives the wave equation $\square {\bar{h}}_{\mu \nu }=-16\pi G{T}_{\mu \nu }$ — gravity as a massless spin-2 field on flat spacetime.
• Gauge (diffeomorphism) freedom reduces ten components to two polarizations, plus ($+$) and cross ($\times$), the hallmark of helicity-$\pm 2$.
• A wave produces transverse tidal stretching/squeezing (strain $h\sim \Delta L/L\sim 10^{-21}$) on a ring of free masses.
• Radiation is quadrupolar (no monopole/dipole); the quadrupole formula gives the luminosity; compact binaries chirp as they inspiral.
• Confirmed by Hulse–Taylor (indirect), GW150914 (first direct, black holes), and GW170817 (multi-messenger, $v_{\text{GW}}=c$); energy is captured by the averaged Isaacson tensor.

Next

The variational and canonical underpinnings — the Einstein–Hilbert action and the 3+1 / ADM formulation that makes gravitational-wave simulation (numerical relativity) possible — are developed in foundations/. The quantization of the spin-2 field, and why it fails at high energies, opens Frontiers.