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The Christoffel trace identity converts the covariant derivative of a vector field into a coordinate divergence weighted by the metric volume density. This is valid for the Levi-Civita connection. Applied to the raised gradient of a scalar field, it gives $\square \varphi =|g{|}^{-1/2}{\partial }_{\mu }(\sqrt{|g|}{g}^{\mu \nu }{\partial }_{\nu }\varphi )$ in Lorentzian signature, the curved d'Alembert operator.

For Minkowski spacetime with unit light speed, $L_i=t{\partial }_i+x_i{\partial }_t$ generates a Lorentz boost. It commutes with the d'Alembert operator and controls derivatives transverse to time slices in the vector field method for wave equations.

Raise perturbation indices with the background Minkowski metric. To first order, the inverse metric is $g^{ik}={\eta }^{ik}-ϵh^{ik}+O({ϵ}^2)$ and the Levi-Civita connection isQuadratic connection products in the Riemann curvature tensor are of order ${ϵ}^2$. Substitution into the paper's curvature convention givesThe signs follow the convention on the cover sheet; reversing the curvature definition would reverse this expression.

The gauge freedom is the freedom to identify points of the perturbed spacetime with points of the flat background in slightly different coordinates. Under the passive infinitesimal change $x^{\prime i}=x^i+ϵ{\xi }^i(x)$, the metric transformation law gives the linearized coordinate gauge transformationThus a nonzero perturbation can partly represent a coordinate change rather than physical curvature. Substituting this variation into the displayed curvature formula givesEach third derivative has a partner differing only by the order of its commuting partial derivatives. This explicitly proves gauge invariance of the linearized Riemann tensor. The restriction to first order matters: a coordinate change acts on nonzero background curvature as well, but the background here is flat.

Define the trace-reversed metric perturbation and d'Alembert operator byIn four dimensions $\bar{h}=-h$ and $h_{ik}={\bar{h}}_{ik}-\frac{1}{2}{\eta }_{ik}\bar{h}$. Contract the curvature formula using $R_{ik}=R^m{}_{imk}$:up to $O({ϵ}^2)$. Consequently the Einstein tensor isChoose Lorenz gauge in linearized gravity, ${\partial }^k{\bar{h}}_{ik}=0$. It can be imposed locally by solving $\square {\xi }_i={\partial }^k{\bar{h}}_{ik}$, since the gauge transformation changes this divergence by $-\square {\xi }_i$. The remaining coordinate freedom satisfies $\square {\xi }_i=0$. In this gauge,This is the requested Linearized Einstein equations in the stated sign convention. Their divergence requires ${\partial }^k{T}_{ik}=0$ to this order.

For the photon beam, introduce $u=t-x$ and the constant null covector ${\ell }_i=(1,-1,0,0)$, so ${\ell }^i=(1,1,0,0)$. The stress-energy tensor isIt is trace-free and conserved: ${\ell }^i{\partial }_i\rho (u)=0$. A beam-adapted particular solution, with no independently added homogeneous gravitational field, isThe equality follows because its trace is zero, and the Lorenz condition holds because ${\ell }^k{\partial }_{k}H=0$. The field equation becomes a transverse Poisson equation:For the transversely uniform source in the question, one particular solution is $H=4\pi G\rho (u)(y^2+z^2)$. Thus the sourced components can be chosen asBoundary conditions and homogeneous solutions determine additional freedom; the source does not uniquely specify every component in every gauge. In particular, taking $H$ to depend only on $t-x$ would give $\square H=0$ and cannot solve the equation for nonzero $\rho$. The uniform idealization is not an asymptotically flat finite beam, and its quadratic particular solution is a weak-field description only where $|ϵH|\ll 1$.

Finally consider another photon with tangent proportional to ${\ell }^i$. Its tangent remains null because $h_{ik}{\ell }^i{\ell }^k=0$. Moreover $h_{ik}{\ell }^k=0$, and since $\ell$ is constant, the connection contraction isThe geodesic equation is therefore satisfied by a straight affinely parametrized ray along the beam. Hence co-propagating photons experience no deflection from the beam's gravitational field. This is parallel photon beams in linearized gravity; it does not assert absence of forces on counter-propagating photons or effects from unrelated homogeneous gravitational perturbations.

Keep only first order in the metric perturbation and use the Minkowski metric to raise and lower indices; set $G=c=1$. The Levi-Civita connection hasThe quadratic products of Christoffel symbols in the Riemann curvature tensor can be dropped at this order. Contracting the remaining derivatives giveswhere $\square ={\eta }^{\rho \sigma }{\partial }_{\rho }{\partial }_{\sigma }=-{\partial }_t^2+{\nabla }^2$ is the d'Alembert operator. In four dimensions the trace-reversed metric perturbation satisfies $\bar{h}=-h$ and $h_{\mu \nu }={\bar{h}}_{\mu \nu }-\frac{1}{2}{\eta }_{\mu \nu }\bar{h}$. Writing the Linearized Einstein equations in terms of it givesIn Lorenz gauge in linearized gravity, the first, second and fourth terms vanish. Therefore $G_{\mu \nu }^{(1)}=8\pi T_{\mu \nu }$ becomesThe divergence of this equation also gives ${\partial }^{\mu }{T}_{\mu \nu }=0$, the leading-order stress-energy conservation required by the contracted Bianchi identity.

For the wave equation with d'Alembert operator $\square =-{\partial }_t^2+\Delta$, the wave speed is one. The domain of dependence of $(t,x)$ is the initial ball $\overline{B}(x,t)$. More precisely, two solutions whose Cauchy data agree on a neighbourhood of that ball have the same value at $(t,x)$. Thus, if both Cauchy data have support in $K$, finite propagation speed givesThe Strong Huygens principle is sharper in three spatial dimensions: only the initial data on the sphere $|y-x|=t$ contribute, including the first spatial derivatives of the initial displacement there. The Kirchhoff formula makes this precise:In particular, if the Cauchy data have compact support in $\overline{B}(0,R)$, thenFor $t>R$, the solution therefore vanishes behind the inward edge $|x|=t-R$ as well as outside $|x|=t+R$. The absence of an interior tail is the additional content of the Strong Huygens principle.

One version of the Klainerman-Sobolev inequality, for a sufficiently decaying smooth function $\psi$ on ${\mathbb{R}}^{1+3}$, isHere $I$ denotes a word of length $|I|$ in the following eleven commutation vector fields for the wave equation:The first four are spacetime translation vector fields; the next three are spatial rotation vector fields; the next three are Lorentz boost vector fields; and $S$ is the scaling vector field. The vector field method for wave equations uses these vector fields because their commutators with the d'Alembert operator obeyIn particular the Klainerman-Sobolev inequality impliesThe displayed L2 norms are spatial norms at fixed time; the spacetime vector fields can contain time derivatives. No wave equation assumption is needed for the Klainerman-Sobolev inequality itself.

For the minimally coupled scalar field, keep the metric tensor fixed and vary $\varphi$. A scalar's first covariant derivative equals its ordinary derivative, soApply integration by parts for tensor fields. The boundary contribution is proportional to $\delta \varphi$ and vanishes under the given boundary condition, leavingThe interior variation is arbitrary. Thus the scalar equation of motion isHere the d'Alembert operator contains the Levi-Civita connection in the second covariant derivative, even though the first derivative of the scalar field is ordinary.

For each fixed coordinate function $x^{\alpha }$, the expression $\square x^{\alpha }$ is a scalar: $\alpha$ labels four chosen functions and is not a free tensor index. The equation is therefore coordinate invariant when those functions are held fixed. However, the demand that the currently chosen coordinates themselves be harmonic coordinates is a chart condition, not a four-vector equation invariant under arbitrary changes of coordinate functions.

Indeed, for new functions ${\bar{x}}^{\alpha }=f^{\alpha }(x)$ the d'Alembert operator givesThe second term prevents a vector transformation law. An affine redefinition preserves harmonic coordinates; an arbitrary nonlinear one does not. For example, a Cartesian spatial coordinate $x$ in Minkowski spacetime has $\square x=0$, whereas $\square (x^2)=2$, and $x^2$ is a valid new coordinate on $x>0$. The equations are scalar equations for specified functions, while harmonic gauge is chart dependent.

Use geometrized units $G=c=1$ and signature $(-,+,+,+)$. The original PDF starts with the d'Alembert operator $\square {\bar{h}}_{\mu \nu }=-16\pi T_{\mu \nu }$; the local TeX incorrectly transcribes its derivative indices. The harmonic condition is the Lorenz gauge in linearized gravity.

Choose the retarded solution, excluding an incoming homogeneous wave. For a spatially localized source, the Linearized Einstein equations giveLet the source size be $D$ and the characteristic angular frequency be $\omega$. Assume the weak-field approximation, nonrelativistic source velocities, and $D\ll {\omega }^{-1}\ll r$ for a radiative far-zone measurement. Then $|\mathbf{x}-\mathbf{y}|=r-\mathbf{n}\cdot \mathbf{y}+O(D^2/r)$. At leading order in $D/r$ and $\omega D$, one may replace the denominator by $r$ and the retarded time throughout the source by $t-r$, obtaining

To identify this integral, use stress-energy conservation ${\partial }_{\mu }{T}^{\mu \nu }=0$ for a symmetric source tensor. Compact support or sufficient decay permits integration by parts with no boundary flux. Because $T^{00}=T_{00}$ in this signature, the second mass moment tensor satisfiesSpatial indices are raised with ${\delta }_{ij}$, so $T^{ij}=T_{ij}$. Substitution gives the retarded quadrupole field:The assumptions are an isolated conserved source, retarded boundary conditions, weak gravity, a source small compared with the wavelength, and observation far from it. The physical radiative field follows by the transverse-traceless projector applied to this trace-reversed metric perturbation.

Define the stress-energy tensor by metric variation with the scalar held fixed:Use compactly supported variations to remove boundary terms. The first covariant derivative of a scalar field is an ordinary derivative, so it has no connection variation. Varying the kinetic term and the volume form gives

For the curvature term set $f={\Phi }^2$. Using part (a), its variation isApplying integration by parts twice moves each pair of derivatives from the metric variation onto $f$. The result iswhere $G_{ab}=R_{ab}-R{g}_{ab}/2$ is the Einstein tensor and $\square ={\nabla }^c{\nabla }_c$ is the d'Alembert operator. Comparing with the defining variation gives the stress-energy tensor of a nonminimally coupled scalar field:The factor $2\xi$ follows from the action's coupling $-\xi R{\Phi }^2$; a convention using $-\xi R{\Phi }^2/2$ would instead give $\xi$ in that bracket. At $\xi =0$ this reduces to the massless Klein-Gordon scalar stress-energy tensor.

The wave operator has principal symbolThe conormal to the initial plane is $dt$, and $p(dt)=-1\ne 0$. The plane is therefore non-characteristic.

At $ϵ=0$ the metric is the constant Minkowski metric, whose Christoffel symbols and curvature vanish, soInsert $R=ϵR^{(1)}+O({ϵ}^2)$ into the Penrose wave equation. Every curvature-square term is $O({ϵ}^2)$, while the covariant wave operator reduces at first order to the flat d'Alembert operator. ThusThe linearized Riemann curvature operator is unchanged by $h_{\mu \nu }\mapsto h_{\mu \nu }+{\partial }_{\mu }{\xi }_{\nu }+{\partial }_{\nu }{\xi }_{\mu }$, because the resulting third derivatives cancel pairwise. The equation therefore requires no gauge choice for $h_{\mu \nu }$.

The vector field $S=t{\partial }_t+x\cdot \nabla$ generates simultaneous spacetime dilation. For the flat d'Alembert operator, $[\square ,S]=2\square$. Consequently it commutes with the homogeneous wave equation at the level of its solution set, although its operator commutator is nonzero.

The vector fields ${\partial }_t$ and ${\partial }_i$ generate time and space translations and commute with the flat d'Alembert operator. They form the unweighted part of the commutation vector fields for the wave equation.

The vector field ${\Omega }_{ij}=x_i{\partial }_j-x_j{\partial }_i$ is an infinitesimal spatial rotation. It commutes with the flat d'Alembert operator and supplies angular derivatives in the Klainerman-Sobolev inequality.

If $\square u=F$ with d'Alembert operator $\square =-{\partial }_t^2+\Delta$, then $\Vert \partial u(t){\Vert }_2\le \Vert \partial u(0){\Vert }_2+{\int }_0^t\Vert F(s){\Vert }_{2}ds$, where $\partial u=(u_t,\nabla u)$. Test the wave equation against $u_t$, use integration by parts, and apply the Cauchy-Schwarz inequality. The same energy estimate applies to a finite system of wave equations.