---
title: Linearized Gravity and Wave Solutions
module: Gravitational Waves
moduleNumber: 9
lessonNumber: 1
order: 901
summary: >
  Weak gravity is a small perturbation of flat spacetime, and the linearized
  Einstein equation in the Lorenz gauge is an ordinary wave equation propagating
  at the speed of light. The trace-reversed perturbation carries the dynamics,
  residual gauge freedom fixes the transverse-traceless form, and the two
  physical polarizations deform a ring of freely falling masses into oscillating
  ellipses whose fractional size change is the strain.
topics: [Gravitational Waves]
draft: false
sources:
  - book: Schutz
    ref: "A First Course in General Relativity, Ch. 9 — Gravitational Radiation"
  - book: Hartle
    ref: "Gravity, Ch. 16 — Gravitational Waves"
---

The full Einstein equation is nonlinear, but far from any strong source the
metric differs only slightly from Minkowski. Writing that small difference as a
field on a fixed flat background reduces the field equation to a linear wave
equation, exactly the structure that governs electromagnetic radiation. This
lesson develops linearized gravity from $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$,
identifies the gauge freedom that removes the coordinate artefacts, fixes the
transverse-traceless gauge in which the two physical degrees of freedom stand
alone, and computes what a passing wave does to a set of test masses.

## The weak-field metric

Take coordinates in which the metric is Minkowski plus a small perturbation,

$$
g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}, \qquad |h_{\mu\nu}| \ll 1,
$$

with signature $(-,+,+,+)$. Every quantity is expanded to first order in
$h_{\mu\nu}$ and higher orders are dropped. Indices are raised and lowered with
$\eta_{\mu\nu}$ rather than the full metric, since the correction from using
$g_{\mu\nu}$ is second order. In particular the inverse metric is
$g^{\mu\nu}=\eta^{\mu\nu}-h^{\mu\nu}$ to first order, where
$h^{\mu\nu}=\eta^{\mu\alpha}\eta^{\nu\beta}h_{\alpha\beta}$.

The Christoffel symbols are first order in $h$,

$$
\Gamma^\lambda{}_{\mu\nu}
= \tfrac{1}{2}\eta^{\lambda\rho}
\left(\partial_\mu h_{\rho\nu} + \partial_\nu h_{\rho\mu} - \partial_\rho h_{\mu\nu}\right),
$$

so the Riemann tensor, which is $\partial\Gamma+\Gamma\Gamma$, keeps only the
$\partial\Gamma$ piece at linear order:

$$
R^{(1)}{}_{\mu\nu\rho\sigma}
= \tfrac{1}{2}\left(
\partial_\nu\partial_\rho h_{\mu\sigma}
+ \partial_\mu\partial_\sigma h_{\nu\rho}
- \partial_\nu\partial_\sigma h_{\mu\rho}
- \partial_\mu\partial_\rho h_{\nu\sigma}\right).
$$

Contracting gives the linearized Ricci tensor, and contracting once more the
Ricci scalar. With the shorthand $\Box=\eta^{\mu\nu}\partial_\mu\partial_\nu$ for
the flat d'Alembertian and $h=\eta^{\mu\nu}h_{\mu\nu}$ for the trace, the
linearized Einstein tensor is

$$
G^{(1)}_{\mu\nu}
= \tfrac{1}{2}\Big(
\partial^\alpha\partial_\mu h_{\nu\alpha}
+ \partial^\alpha\partial_\nu h_{\mu\alpha}
- \Box h_{\mu\nu}
- \partial_\mu\partial_\nu h
- \eta_{\mu\nu}\,\partial^\alpha\partial^\beta h_{\alpha\beta}
+ \eta_{\mu\nu}\,\Box h\Big).
$$

Six terms are a nuisance; a single change of variable collapses them.

## The trace-reversed perturbation

Define the **trace-reversed perturbation**

$$
\bar h_{\mu\nu} = h_{\mu\nu} - \tfrac{1}{2}\eta_{\mu\nu}\,h.
$$

The name records that its trace is the negative of the original,
$\bar h = \eta^{\mu\nu}\bar h_{\mu\nu} = h - 2h = -h$, so the map is its own
inverse: $h_{\mu\nu}=\bar h_{\mu\nu}-\tfrac{1}{2}\eta_{\mu\nu}\bar h$. Substituting
into $G^{(1)}_{\mu\nu}$ and grouping terms,

$$
G^{(1)}_{\mu\nu}
= -\tfrac{1}{2}\Big(
\Box \bar h_{\mu\nu}
+ \eta_{\mu\nu}\,\partial^\alpha\partial^\beta \bar h_{\alpha\beta}
- \partial^\alpha\partial_\mu \bar h_{\nu\alpha}
- \partial^\alpha\partial_\nu \bar h_{\mu\alpha}\Big).
$$

The last three terms all contain a divergence $\partial^\alpha\bar h_{\alpha\nu}$.
A gauge choice will annihilate them.

## Gauge freedom

The split $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ is not unique: an infinitesimal
change of coordinates $x^\mu \to x'^\mu = x^\mu + \xi^\mu(x)$, with $\xi^\mu$ of
the same small order as $h$, leaves the background flat but shifts the
perturbation,

$$
h_{\mu\nu} \to h_{\mu\nu} - \partial_\mu\xi_\nu - \partial_\nu\xi_\mu.
$$

This is the exact analogue of the electromagnetic gauge transformation
$A_\mu \to A_\mu - \partial_\mu\chi$, which changes the potential without
changing the fields. Here the physical, coordinate-independent content is the
linearized Riemann tensor, and one checks directly that $R^{(1)}{}_{\mu\nu\rho\sigma}$
is invariant under the shift. Four functions $\xi^\mu$ are free, so four of the
ten components of $h_{\mu\nu}$ carry no physics.

$$
% caption: An infinitesimal coordinate shift x -> x + xi changes the metric
% perturbation without altering the curvature, exactly as an electromagnetic
% gauge transformation changes the potential but not the field. The invariant
% content is the linearized Riemann tensor.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black, thick] (0,1.6) rectangle (2.4,2.6);
  \node[black, anchor=center] at (1.2,2.1) {gravity h};
  \draw[black, thick] (5.0,1.6) rectangle (7.8,2.6);
  \node[black, anchor=center] at (6.4,2.1) {shifted h};
  \draw[->, black, thick] (2.45,2.1) -- (4.95,2.1);
  \node[black, anchor=south] at (3.7,2.15) {add grad of xi};
  \draw[black, thick] (0,-0.2) rectangle (2.4,0.8);
  \node[black, anchor=center] at (1.2,0.3) {Riemann};
  \draw[black, thick] (5.0,-0.2) rectangle (7.8,0.8);
  \node[black, anchor=center] at (6.4,0.3) {same Riemann};
  \draw[->, black] (1.2,1.55) -- (1.2,0.85);
  \draw[->, black] (6.4,1.55) -- (6.4,0.85);
  \draw[<->, black, densely dotted] (2.45,0.3) -- (4.95,0.3);
  \node[black, anchor=south] at (3.7,0.35) {unchanged};
\end{tikzpicture}
$$

## The Lorenz gauge and the wave equation

Impose the **Lorenz gauge** (also called the harmonic or de Donder gauge),

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

That this is always reachable follows from the transformation law: under
$x^\mu\to x^\mu+\xi^\mu$ the divergence shifts by $\Box\xi_\nu$, so any starting
$\bar h_{\mu\nu}$ can be brought to zero divergence by solving $\Box\xi_\nu =
\partial^\mu\bar h_{\mu\nu}$, an inhomogeneous wave equation that always has a
solution. In this gauge the three divergence terms in $G^{(1)}_{\mu\nu}$ vanish,
leaving

$$
G^{(1)}_{\mu\nu} = -\tfrac{1}{2}\Box\bar h_{\mu\nu}.
$$

The Einstein equation $G_{\mu\nu}=\dfrac{8\pi G}{c^4}T_{\mu\nu}$ becomes a wave
equation sourced by the stress–energy tensor,

$$
\Box \bar h_{\mu\nu} = -\frac{16\pi G}{c^4}\,T_{\mu\nu}.
$$

In vacuum the right side is zero and each component of $\bar h_{\mu\nu}$ satisfies
the homogeneous wave equation $\Box\bar h_{\mu\nu}=0$, whose disturbances travel
at speed $c$. Weak gravitational fields propagate as waves at the speed of light,
a prediction with no Newtonian counterpart, where the potential responds
instantaneously.

$$
% caption: In the Lorenz gauge the ten-term linearized Einstein tensor collapses
% to a single d'Alembertian acting on the trace-reversed perturbation, turning
% the field equation into a sourced wave equation identical in form to
% electromagnetism.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black, thick] (0,1.4) rectangle (3.2,2.6);
  \node[black, anchor=center] at (1.6,2.0) {linearized Einstein};
  \draw[->, black, thick] (3.25,2.0) -- (4.35,2.0);
  \node[black, anchor=south] at (3.8,2.05) {gauge};
  \draw[black, thick] (4.4,1.4) rectangle (7.6,2.6);
  \node[black, anchor=center] at (6.0,2.0) {box of h};
  \draw[->, black, thick] (6.0,1.35) -- (6.0,0.55);
  \node[black, anchor=west] at (6.1,0.95) {source};
  \draw[acc, thick, fill=acc!10] (4.4,-0.7) rectangle (7.6,0.5);
  \node[acc, anchor=center] at (6.0,-0.1) {wave equation};
  \node[black, anchor=east] at (3.9,-0.1) {travels at c};
  \draw[->, black] (3.95,-0.1) -- (4.35,-0.1);
\end{tikzpicture}
$$

## The transverse-traceless gauge

The Lorenz condition is four equations, but it does not exhaust the freedom: any
further shift with $\Box\xi_\nu=0$ preserves it. These residual functions remove
four more components. For a plane wave travelling in the $z$ direction, write

$$
\bar h_{\mu\nu}(x) = A_{\mu\nu}\,\exp\!\big(i\,k_\alpha x^\alpha\big),
\qquad k^\alpha = \frac{\omega}{c}\,(1,0,0,1).
$$

The vacuum equation $\Box\bar h_{\mu\nu}=0$ forces $k_\alpha k^\alpha=0$: the wave
vector is null, confirming propagation at $c$. The Lorenz condition $k^\mu
A_{\mu\nu}=0$ makes the amplitude transverse to $k$. The residual gauge freedom is
then spent to set

$$
A_{0\mu}=0 \quad(\text{purely spatial}),
\qquad
\eta^{\mu\nu}A_{\mu\nu}=0 \quad(\text{traceless}),
\qquad
A_{\mu\nu}k^\nu=0 \quad(\text{transverse}).
$$

This is the **transverse-traceless (TT) gauge**. Since the perturbation is now
traceless, $\bar h^{\text{TT}}_{\mu\nu}=h^{\text{TT}}_{\mu\nu}$, and the two names
coincide. For the wave along $z$ the surviving amplitude has only $x$ and $y$
components, symmetric and traceless in that $2\times2$ block:

$$
h^{\text{TT}}_{\mu\nu} =
\begin{pmatrix}
0 & 0 & 0 & 0\\
0 & h_+ & h_\times & 0\\
0 & h_\times & -h_+ & 0\\
0 & 0 & 0 & 0
\end{pmatrix}
\cos\!\big(\omega(t - z/c)\big).
$$

Two independent amplitudes remain, $h_+$ and $h_\times$, the two physical
polarizations. The counting is worth stating plainly: ten components of a
symmetric $h_{\mu\nu}$, minus four Lorenz conditions, minus four residual gauge
functions, leaves two.

$$
% caption: A gravitational plane wave in the transverse-traceless gauge
% propagates along z with its physical amplitude confined to the transverse x-y
% plane; each wavefront carries the same symmetric traceless deformation pattern.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black, thick] (0,0) -- (8.2,0) node[anchor=west] {z};
  \node[black, anchor=north] at (6.3,-0.1) {propagation};
  % wavefront planes as vertical ellipses
  \foreach \x in {1.4,3.4,5.4} {
    \draw[acc, thick] (\x,0) ellipse (0.35 and 1.15);
  }
  \node[acc, anchor=south] at (1.4,1.2) {wavefront};
  % transverse deformation arrows on middle front
  \draw[<->, black] (3.4,0.75) -- (3.4,-0.75);
  \draw[<->, black] (3.05,0) -- (3.75,0);
  \node[black, anchor=west] at (5.7,1.0) {transverse x-y plane};
  % sinusoid overhead indicating oscillation
  \draw[densely dotted] (0.6,1.7) sin (1.4,2.0) cos (2.2,1.7) sin (3.0,1.4) cos (3.8,1.7) sin (4.6,2.0) cos (5.4,1.7);
  \node[anchor=west] at (5.7,1.75) {amplitude};
\end{tikzpicture}
$$

## What a wave does to test masses

The TT gauge has a subtle feature: the coordinates are tied to the freely falling
test masses themselves, so a mass initially at rest stays at fixed coordinate
$x^i$ even as the wave passes. The coordinate positions do not move. What changes
is the proper distance between masses, because the metric that measures distance
oscillates. This is not a coordinate artefact; it is measured directly by the
invariant relative acceleration of geodesics.

The physical effect is the equation of geodesic deviation. For two nearby masses
with separation vector $\xi^i$, both freely falling and slow,

$$
\frac{\d^2 \xi^i}{\d t^2}
= -R^i{}_{0j0}\,\xi^j
= \tfrac{1}{2}\,\frac{\d^2 h^{\text{TT}}_{ij}}{\d t^2}\,\xi^j,
$$

using the linearized curvature $R^i{}_{0j0}=-\tfrac{1}{2}\ddot h^{\text{TT}}_{ij}$
in the TT gauge. Integrating twice for a small, nearly constant separation
$\xi^j_{(0)}$,

$$
\xi^i(t) = \xi^i_{(0)} + \tfrac{1}{2}\,h^{\text{TT}}_{ij}(t)\,\xi^j_{(0)}.
$$

The proper separation of two masses a coordinate distance $L$ apart along the
$x$-axis, with only the plus polarization present, is

$$
\Delta L(t) = L\left(1 + \tfrac{1}{2}h_+\cos\omega t\right)^{1/2}
\approx L\left(1 + \tfrac{1}{4}h_+\cos\omega t\right).
$$

The **strain** is the fractional length change, $h \sim \Delta L / L$, and its
amplitude is set directly by $h_+$ or $h_\times$. Because $h$ is dimensionless
and of order $10^{-21}$ for astrophysical sources reaching Earth, the length
change over a kilometre baseline is a small fraction of a proton radius, which
is what makes detection difficult.

$$
% caption: Two freely falling masses hold fixed TT coordinates while the passing
% wave stretches and squeezes the proper distance between them; the fractional
% change is the strain, of order ten to the minus twenty-one at Earth.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % rest separation
  \fill[black] (0,1.6) circle (0.09);
  \fill[black] (3.0,1.6) circle (0.09);
  \draw[black, thick] (0.15,1.6) -- (2.85,1.6);
  \node[black, anchor=south] at (1.5,1.65) {rest length L};
  % stretched
  \fill[acc] (0,0.6) circle (0.09);
  \fill[acc] (3.5,0.6) circle (0.09);
  \draw[acc, very thick] (0.15,0.6) -- (3.35,0.6);
  \node[acc, anchor=north] at (1.75,0.55) {stretched};
  \draw[->, acc] (3.05,0.6) -- (3.45,0.6);
  % strain bracket
  \draw[black] (3.0,1.45) -- (3.0,0.75);
  \draw[black] (3.5,0.75) -- (3.5,0.45);
  \node[black, anchor=west] at (3.65,1.05) {change in L over L = strain};
\end{tikzpicture}
$$

## The two polarizations

The polarizations act on a ring of test masses in the plane transverse to the
propagation. Set $\xi=(x,y)$ and apply
$\xi^i \to \xi^i + \tfrac{1}{2}h^{\text{TT}}_{ij}\xi^j$.

- **Plus polarization** ($h_+\ne0$, $h_\times=0$): the map is
  $x\to x(1+\tfrac{1}{2}h_+\cos\omega t)$, $y\to y(1-\tfrac{1}{2}h_+\cos\omega t)$.
  A circular ring becomes an ellipse elongated along $x$, returns to a circle a
  quarter period later, then elongates along $y$. The distortion pattern has the
  shape of a plus sign, naming the mode.
- **Cross polarization** ($h_\times\ne0$, $h_+=0$): the map mixes the axes,
  $x\to x+\tfrac{1}{2}h_\times y\cos\omega t$, $y\to y+\tfrac{1}{2}h_\times
  x\cos\omega t$. The ellipse major axis lies along the $45^\circ$ diagonals,
  the same pattern rotated by $45^\circ$.

Under a rotation by angle $\psi$ about the propagation axis, the pair
$(h_+,h_\times)$ transforms with $\cos 2\psi$ and $\sin 2\psi$. A rotation of
$45^\circ$ interchanges the two modes, and a rotation of $90^\circ$ returns the
pattern to itself. This double-angle behaviour is the signature of a spin-2
field, the tensor character of the graviton, in contrast to the spin-1 photon
whose polarization repeats every $180^\circ$.

$$
% caption: The plus polarization deforms a ring of test masses into an ellipse
% along the coordinate axes, oscillating between horizontal and vertical over one
% period; a quarter period after maximum stretch the ring is momentarily circular.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \foreach \cx/\xs/\ys/\lab in {0/1.0/0.55/phase 0, 3.0/0.78/0.78/quarter, 6.0/0.55/1.0/half} {
    \begin{scope}[shift={(\cx,0)}, xscale=\xs, yscale=\ys]
      \draw[acc, thick] (0,0) circle (1);
      \foreach \a in {0,45,...,315} {
        \fill[black] ({cos(\a)}, {sin(\a)}) circle (0.06);
      }
    \end{scope}
    \node[black, anchor=north] at (\cx,-1.25) {\lab};
  }
  \draw[->, black] (1.15,0) -- (1.85,0);
  \draw[->, black] (4.15,0) -- (4.85,0);
\end{tikzpicture}
$$

$$
% caption: The cross polarization produces the same oscillating ellipse rotated
% by forty-five degrees; a rotation of the pattern by that angle interchanges the
% two modes, the double-angle behaviour that marks a spin-two field.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \foreach \cx/\rot/\lab in {0/45/phase 0, 3.0/0/quarter, 6.0/-45/half} {
    \begin{scope}[shift={(\cx,0)}, rotate=\rot]
      \draw[acc, thick] (0,0) ellipse (1.0 and 0.55);
      \foreach \a in {0,45,...,315} {
        \fill[black] ({1.0*cos(\a)}, {0.55*sin(\a)}) circle (0.05);
      }
    \end{scope}
    \node[black, anchor=north] at (\cx,-1.25) {\lab};
  }
  \draw[->, black] (1.25,0) -- (1.75,0);
  \draw[->, black] (4.25,0) -- (4.75,0);
\end{tikzpicture}
$$

The wave equation, the null propagation vector, and these two transverse modes
are everything linearized theory says about a wave once it has left its source.
What sets the amplitude and the waveform is the source itself, and the leading
term of that expansion is the mass quadrupole, taken up next.

[^schutz-lin]: **Schutz**, _A First Course in General Relativity_, Ch. 9 — the linearized field equations, the trace-reversed perturbation, the Lorenz and transverse-traceless gauges, and the polarization content of a plane wave.

[^hartle-lin]: **Hartle**, _Gravity_, Ch. 16 — weak gravitational waves as perturbations of flat spacetime, the wave equation, and the effect of the two polarizations on a ring of free test masses.