Linearized Gravity and Wave Solutions

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,

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 $h$,

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

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

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

The trace-reversed perturbation

Define the trace-reversed perturbation

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: $h_{\mu \nu }={\bar{h}}_{\mu \nu }-\frac{1}{2}{\eta }_{\mu \nu }\bar{h}$. Substituting into $G_{\mu \nu }^{(1)}$ and grouping terms,

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 $x^{\mu }\to x^{\prime \mu }=x^{\mu }+{\xi }^{\mu }(x)$, with ${\xi }^{\mu }$ of the same small order as $h$, leaves the background flat but shifts the perturbation,

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.

The Lorenz gauge and the wave equation

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

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

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

In vacuum the right side is zero and each component of ${\bar{h}}_{\mu \nu }$ satisfies the homogeneous wave equation $\square {\bar{h}}_{\mu \nu }=0$, 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.

The transverse-traceless gauge

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

The vacuum equation $\square {\bar{h}}_{\mu \nu }=0$ 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

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

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.

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,

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

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

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.

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 ${\xi }^i\to {\xi }^i+\frac{1}{2}{h}_{ij}^{\text{TT}}{\xi }^j$.

• Plus polarization ($h_{+}\ne 0$, $h_{\times }=0$): the map is $x\to x(1+\frac{1}{2}{h}_{+}\cos \omega t)$, $y\to y(1-\frac{1}{2}{h}_{+}\cos \omega t)$. 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 ($h_{\times }\ne 0$, $h_{+}=0$): the map mixes the axes, $x\to x+\frac{1}{2}{h}_{\times }y\cos \omega t$, $y\to y+\frac{1}{2}{h}_{\times }x\cos \omega t$. 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 $180^{\circ }$.

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.