Linearized Einstein equations

The Linearized Einstein equations retain first-order terms in a metric perturbation $g_{\mu \nu }={\eta }_{\mu \nu }+h_{\mu \nu }$ about the Minkowski metric. Use metric signature $(-,+,+,+)$, coordinates $x^0=ct$, andIn Lorenz gauge in linearized gravity, ${\partial }^{\mu }{\bar{h}}_{\mu \nu }=0$, the first-order Einstein tensor is $G_{\mu \nu }^{(1)}=-\square {\bar{h}}_{\mu \nu }/2$. HenceThe source must satisfy background stress-energy conservation, ${\partial }^{\mu }{T}_{\mu \nu }=0$, since taking the divergence of the left-hand side gives zero. In units $G=c=1$ this becomes $\square {\bar{h}}_{\mu \nu }=-16\pi T_{\mu \nu }$.

The equation is a sourced wave equation, not just a static Poisson equation. A retarded solution additionally needs a radiation boundary condition for linearized gravity; homogeneous gravitational waves can otherwise be added. The expansion is controlled only where the neglected higher-order metric perturbations remain small.

The source-free Linearized Einstein equations. For a metric perturbation about the Minkowski metric, imposing the Lorenz gauge in linearized gravity reduces them to $\square h_{\mu \nu }=0$ together with ${\partial }^{\mu }{\bar{h}}_{\mu \nu }=0$, where $\bar{h}$ is the trace-reversed metric perturbation. A linearized coordinate gauge transformation preserves $G^{(1)}[h]=0$, while preserving the imposed gauge additionally requires its generator to satisfy the wave equation.

For signature $(+---)$ and the curvature convention giving $\square {\bar{h}}_{ik}=-16\pi G{ϵ}^{-1}{T}_{ik}$, take $u=t-x$, ${\ell }_i=(1,-1,0,0)$ and $T_{ik}=ϵ\rho (u){\ell }_i{\ell }_k$. A trace-free Lorenz-gauge particular solution has the displayed form. The vector ${\ell }^i=(1,1,0,0)$ is null, and ${\Gamma }^i{}_{jk}{\ell }^j{\ell }^k=0$, so co-propagating test photons are undeflected by this beam field. For a transversely uniform source, $H=4\pi G\rho (u)(y^2+z^2)$ is one particular solution. A function of $u$ alone has vanishing wave operator and cannot source nonzero $\rho$; homogeneous gravitational perturbations and boundary conditions remain separate data.

The retarded source solution of the Linearized Einstein equations is selected by excluding added incoming or freely specified homogeneous gravitational waves. A homogeneous wave can satisfy Lorenz gauge in linearized gravity while having no source mass quadrupole moment. Thus a formula for the retarded source field is not an identity for every solution of the sourced wave equation.

For a compact slowly moving source in the weak-field approximation, with no incoming homogeneous wave and $G=c=1$, the retarded Linearized Einstein equations give this leading far-zone trace-reversed metric perturbation. The key consequence of stress-energy conservation is ${\ddot{I}}_{ij}=2\int T_{ij}{d}^{3}x$. The physical gravitational wave is obtained by the transverse-traceless projector.

Dropping the position-dependent source delay in the leading far-zone retarded field requires source size $d$ to be small compared with its variation time $\tau$, in units $c=1$. The retarded quadrupole field also requires $d/r\ll 1$. For a typical nonrelativistic bound source, $\tau \sim d/v$ implies this long-wavelength condition; an arbitrarily rapidly oscillating small displacement is not controlled merely by its instantaneous speed.