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Put $\mu =GM/c^2$, $A=1-2\mu /r$ and $B=1+2\mu /r$. For an affinely parametrized geodesic, use the unhalved quadratic Lagrangian $\mathcal{L}=-c^{2}A{\dot{t}}^2+B{\dot{\mathbf{r}}}^{,2}$. Its Euler-Lagrange equations are unchanged by this overall normalization. Since $t$ is a cyclic coordinate, $d(-2c^{2}A\dot{t})/d\tau =0$, equivalentlyThe spatial derivatives give $\partial \mathcal{L}/\partial {\dot{x}}^i=2B{\dot{x}}^i$ and $\partial \mathcal{L}/\partial x^i=-c^2{\dot{t}}^2{A}_i+{\dot{\mathbf{r}}}^{,2}{B}_i$, where $A_i={\partial }_{i}A$. Therefore

For the connection coefficients, take $x^0=ct$, so $g_{00}=-A$ and $g_{ij}=B{\delta }_{ij}$. Substituting the diagonal inverse metric into the Christoffel symbol formula gives all nonzero component types, including lower-index symmetry:All other component types vanish. If the time coordinate is $t$ rather than $ct$, replace ${\Gamma }^i{}_{00}$ by ${\Gamma }^i{}_{tt}=c^2{\Gamma }^i{}_{00}$; the displayed time–space mixed coefficients are unchanged. Spatial indices in these formulas are raised with the background Euclidean metric.

For the circular orbit let $u=\mu /R$ and $\Omega =d\varphi /dt=\omega /\dot{t}$. Both $A$ and $B$ are constant along it. The radial spatial equation gives $(1+u){\omega }^2=(GM/R^3){\dot{t}}^2$, so, treating the displayed metric coefficients algebraically, ${\Omega }^2=GM/[R^3(1+u)]$. The weak-field approximation securely determines the leading Keplerian orbit resultThe time normalization is $-c^2=-c^{2}A{\dot{t}}^2+B{R}^2{\omega }^2$, yielding $\dot{t}=1+3u/2+O(u^2)$ and $P_{\tau }/P_t=1-3u/2+O(u^2)$. One must distinguish the proper-time angular frequency in the supplied path from the coordinate-time angular frequency. The formal factor $1+u$ in the frequency equation comes from treating the truncated metric as exact; a complete relative post-Newtonian correction to the orbital period would also require the omitted higher-order temporal metric terms.

Geodesic deviation compares neighboring members of a family of freely falling trajectories. Subtracting their geodesic equations to first order in the connecting vector gives a covariant relative-acceleration equation governed by the Riemann curvature tensor. For a slowly moving satellite, $u^0=c\dot{t}\simeq c$ dominates the spatial components. Choose simultaneous spatial separation in a locally transported frame, take the weak-field approximation, and neglect terms quadratic in the small separation and higher velocity corrections. The leading spatial equation becomesReplacing the transported covariant derivatives by ordinary derivatives is a leading local or Newtonian approximation; a rotating orbital frame would introduce additional inertial terms. In the convention established above, the instantaneous tidal acceleration stretches radial separations and compresses transverse ones.

The growth expected for nearby different-radius circular orbits is especially transparent from Kepler's third law. If the radii are $R$ and $R+\delta R$, their coordinate angular frequencies differ by $\delta \Omega \simeq -(3/2)\Omega \delta R/R$. Initially aligned orbits therefore acquire an along-track separation $R\delta \Omega t\simeq -(3/2)\Omega \delta Rt$. Over the interval in which the nearby-orbit approximation remains valid, this produces a growing separation without needing to solve the deviation equation.

This growth is not universal or indefinitely unbounded. Equal-radius co-orbiting geodesics with a fixed phase offset have constant separation, and slightly inclined equal-radius circular orbits have bounded, periodically varying separation. Even different-radius circular orbits have bounded exact positions; their relative angle eventually wraps around. The secular drift statement assumes distinct radii and describes the local differential-orbit regime.

Use metric signature $(-,+,+,+)$, $x^0=ct$, and retain terms linear in $\Phi /c^2$ or $\Psi /c^2$. The inverse metric is $g^{00}=-(1-2\Phi /c^2)$ and $g^{ij}=(1+2\Psi /c^2){\delta }^{ij}$. In the Christoffel symbol formulathe differentiated metric perturbation is already first order, so the prefactor can be replaced by the flat inverse metric. For example ${\Gamma }^0{}_{ij}=\frac{1}{2}{\partial }_0{g}_{ij}=-{\delta }_{ij}{\partial }_0\Psi /c^2$ and ${\Gamma }^k{}_{00}=-\frac{1}{2}{\partial }_k{g}_{00}={\partial }_k\Phi /c^2$. Evaluating the remaining components givesThe lower-index symmetry follows from the torsion-free connection. Products of these first-order Christoffel symbols are beyond the retained weak-field approximation.

For a massive particle parametrized by proper time, the geodesic equation isThe weak-field approximation alone does not imply Newtonian motion. Also take a nonrelativistic limit, $|v|/c\ll 1$, in the usual quasistatic regime so the potentials' time variation does not compensate for the small velocity factors. Reparametrizing by coordinate time and retaining the leading velocity-independent spatial term givesIn empty space the first field equation gives ${\nabla }^2\Psi =0$. For static potentials, the spatial equations imply ${\nabla }^2(\Phi -\Psi )=0$ and then ${\partial }_i{\partial }_j(\Phi -\Psi )=0$. The difference is at most affine; isolated-system boundary conditions fixing both potentials to zero at infinity remove it. Thus $\Phi =\Psi$ andthe empty-space Newton-Poisson equation. With the same quasistatic assumptions and dust present it becomes ${\nabla }^2\Phi =4\pi G\rho$. Keeping rapid time dependence instead leaves ${\nabla }^2\Phi =-3{\partial }_0^2\Psi$ in vacuum, so the static qualification matters.

The apple's proper time $\tau$ is the time read by a clock carried with it, with $d{\tau }^2=-d{s}^2/c^2$ for metric signature $-+++$. At any event choose a local inertial frame with coordinates ${\xi }^{\alpha }$, in which free motion satisfies $d^2{\xi }^{\alpha }/d{\tau }^2=0$ at that event. The chain rule givesMultiplying by the inverse coordinate Jacobian matrix yields the exact geodesic equationHere $x^{\mu }$ are the observer's coordinates, $u^{\mu }$ is the apple's four-velocity, and the Christoffel symbols describe the Levi-Civita connection. The last expression applies when the $\xi$ coordinates are locally inertial at the event considered; one such chart need not remain inertial throughout a finite region.

Choose $x^0=ct$, so $u^0=cdt/d\tau$. In the weak-field approximation and at small spatial speed compared with $c$, the leading spatial geodesic equation isIt is exact at the release event when all spatial $u^i$ vanish, and is the intended approximation to the displayed expression in the source. Terms $2{\Gamma }_{0j}^{\mu }{u}^0{u}^j$ and ${\Gamma }_{ij}^{\mu }{u}^i{u}^j$ cannot generally be dropped; the time component must also be retained to find changing $dt/d\tau$. For a concrete counterexample to the unrestricted expression, a straight free trajectory $x=v\gamma \tau$, $y=b$ in flat spacetime has polar radius $r=\sqrt{b^2+v^2{\gamma }^2{\tau }^2}$ and $d^{2}r/d{\tau }^2=v^2{\gamma }^2{b}^2/r^3\ne 0$, although ${\Gamma }_{00}^r=0$. The missing angular term ${\Gamma }_{\varphi \varphi }^r(d\varphi /d\tau {)}^2$ supplies that radial coordinate acceleration.

Write $\mu =GM/c^2$ and $r_0=R_E+h$. The Christoffel symbols follow fromThe relevant time and radial entries, using $x^0=ct$, areFor the angular geodesic equations, ${\Gamma }_{r\theta }^{\theta }={\Gamma }_{r\varphi }^{\varphi }=1/r$, ${\Gamma }_{\varphi \varphi }^{\theta }=-\sin \theta \cos \theta$, and ${\Gamma }_{\theta \varphi }^{\varphi }=\cot \theta$. Thus, with primes on coordinates here denoting proper time derivatives,Initial conditions ${\theta }^{\prime }={\varphi }^{\prime }=0$ are preserved by these ordinary differential equations. The motion is therefore radial, with constant angular coordinates. Choose the polar axis so the apple is away from the angular coordinate singularities.

The exact remaining radial geodesic equation and four-velocity normalization areEliminating $dt/d\tau$ gives the particularly simple exact result $r^{\prime \prime }=-GM/r^2$. The supported observer's proper time $T$ differs from the apple's proper time by relative corrections of order $GM/(R_E{c}^2)$ and $v^2/c^2$. Consequently, to leading order in the weak-field approximation and $h/R_E$, $d^{2}r/d{T}^2\simeq -GM/R_E^2=-g$, with $r(0)=r_0$ and zero initial radial velocity. HenceThe proper height measured along a static radial ruler is ${\int }_{R_E}^{r}dr/\sqrt{f}\simeq r-R_E$ at this accuracy. The use of $T$ means the ground clock's elapsed time, with propagation delays corrected; raw visual arrival times include the changing light travel time.

If variation of $g$ over the fall is retained but relativistic corrections are neglected, a useful parametric solution isIts small-$\eta$ expansion gives the same constant-$g$ result. For an exact check on the choice of clock, the initial coordinate acceleration measured using $T$ is $-GMf(r_0)/[r_0^{2}f(R_E)]$, which reduces to the displayed approximation.

The time-translation Killing vector field of the Schwarzschild metric gives a conserved Killing energy per unit rest mass,For unit rest mass, this is the total energy relative to infinity, including rest energy. At release $dr/d\tau =0$, so the four-velocity normalization gives $dt/d\tau =f(r_0{)}^{-1/2}$ andSubstitution into the radial four-velocity normalization yieldsChoosing the inward square root and evaluating at the ground provesThis identity is exact within the assumed Schwarzschild metric, despite its familiar Newtonian mechanics form. The quantity $dr/d\tau$ is not exactly the locally measured speed. A Schwarzschild static observer measures $v_{\mathrm{loc}}=(dr/dt)/f=(dr/d\tau )/\sqrt{f(r_0)}$. Thus the distinction disappears in the stated weak-field approximation, giving $v_{\mathrm{loc}}^2\simeq 2gh$ for $h\ll R_E$.

Use metric signature $-+++$ and transverse-traceless gauge, $g_{ij}={\delta }_{ij}+h_{ij}^{\mathrm{TT}}$. To resolve the source's origin ambiguity, put the detector centre at $z=0$ and the source centre at $z=-r_0$, so the gravitational wave travels in the positive $z$ direction. Reversing that direction replaces $z$ by $-z$. Work in the weak-field approximation, with $\omega l\ll c$, $r_0\gg l,c/\omega$, and detector size $a\ll c/\omega ,r_0$. The radiation reaction is neglected, so the rod maintains the prescribed steady motion.

At the retarded source time $u=t-(r_0+z)/c$, take the source positions as ${\mathbf{x}}_{\pm }=\pm l(\cos (\omega u+\beta ),\sin (\omega u+\beta ),0)$. The arbitrary constant $\beta$ specifies the initial source phase. The second mass moment tensor $I_{ij}={\sum }_{A}M{x}_{Ai}{x}_{Aj}$ obeysThe quadrupole formula $h_{ij}^{\mathrm{TT}}=2G{\ddot{I}}_{ij}^{\mathrm{TT}}/(c^4{r}_0)$ projects away the transverse trace, which is already constant here. Differentiating twice therefore givesAll time and longitudinal metric perturbation components vanish in transverse-traceless gauge. The gravitational-wave frequency is $2\omega$ in angular units; $M$ is each source mass, not their sum, and the detector mass $m$ does not enter the strain. The plane wave approximation holds the amplitude fixed at $r_0$; retaining spherical spreading replaces its denominator by $r_0+z$ near the axis. The retarded quadrupole formula and the distinction between source and wave frequency are derived in ^[ref].

Use $x^0=ct$ and metric signature $(-,+,+,+)$. With the source's curvature sign convention, the Ricci tensor isIts time-time component is thereforeFor a slowly moving, pressure-negligible source, the stress-energy tensor is approximated by relativistic dust: $T^{\mu \nu }=\rho U^{\mu }{U}^{\nu }$, where $\rho$ is mass density and $U^{\mu }$ is the four-velocity. In its approximate rest local inertial frame, $U^{\mu }\simeq (c,0,0,0)$ and $U_0\simeq -c$. Thusand the trace-reversed stress-energy tensor satisfies

Independently, introduce the metric perturbation $g_{\mu \nu }={\eta }_{\mu \nu }+h_{\mu \nu }$, with $|h_{\mu \nu }|\ll 1$ and constant Minkowski metric ${\eta }_{\mu \nu }$. The Christoffel symbols are first order in $h$, so their products are discarded in the weak-field approximation. With a comma denoting a partial derivative, the linearized Riemann curvature tensor isTaking the tensor contraction gives, to this order,In the static Newtonian limit, all time partial derivatives vanish. The constant ${\eta }_{00}$ also has zero partial derivatives, leavingHere ${\nabla }^2={\sum }_i{\partial }_i^2$ is the spatial Laplace operator in the approximately Cartesian background coordinates. This sign follows the paper's curvature sign convention.

A photon follows a null geodesic, whose spatial path approaches straight lines far from the deflecting mass. Extend the incoming asymptote forward: this is the reference path the photon would follow if the mass were removed while its incoming direction and transverse offset were held fixed. The impact parameter $b$ is the perpendicular distance from the mass to that incoming asymptote. It is not, in general, the distance of closest approach.

The positive deflection angle $\delta$ is the angle between the outgoing propagation direction and the continued incoming direction. The photon bends toward the mass, so a ray passing above it has a downward outgoing slope in the figure. In the weak-field approximation for an isolated spherical mass, Schwarzschild light deflection gives $\delta \simeq 4GM/(b{c}^2)$ for distant endpoints. This angle differs from the apparent image angle measured from the observer's line of sight to the mass.

In the thin-lens approximation for gravitational lensing, consider a ray crossing the lens plane at transverse distance $b$ above the aligned source-observer axis. With the observer at axial position $0$, the black hole at $L$ and the source at $D$, the segment joining the lens plane to the source has slope $-b/(D-L)$, while the segment joining the observer to the lens plane has slope $b/L$. The observer sees the ray at apparent angle $\theta \simeq b/L$ to the axis. When following the photon toward the observer, the two small angles to the axis have magnitudes $b/(D-L)$ and $b/L$. The physical bend is their sum:Substitute the Schwarzschild light deflection, $\delta \simeq 4GM/(b{c}^2)$, and use $b\simeq L\theta$:Every azimuth around the axis is equivalent by rotational symmetry. Isotropic emission therefore supplies rays at all these azimuths, and the ideal image is an Einstein ring centered on the black hole, with Einstein angular radius $\theta$ and angular diameter $2\theta$. A finite source size or imperfect alignment broadens or breaks the ideal ring.

The stated expression uses small angles, distant endpoints and weak-field approximation $b\gg 2GM/c^2$, with ordinary Euclidean distances. For cosmological gravitational lensing, use angular diameter distances $D_l,D_s,D_{ls}$ in ${\theta }^2=4GM{D}_{ls}/(c^2{D}_l{D}_s)$; generally $D_{ls}\ne D_s-D_l$.

There are two first-order effects: an intrinsic temperature change from adiabatic compression, and the gravitational redshift of the emerging photons. Evaluate the excess Newtonian gravitational potential at the boundary of the spherical fluctuation, with the exterior reference potential subtracted:In the weak-field approximation, gravitational time dilation gives $d{\tau }_{\mathrm{local}}/dt\simeq 1+\delta \Phi /c^2$. A local emitter's successive wave crests therefore arrive farther apart at a reference observer outside the potential well. Their frequency changes by $\delta \nu /\nu =\delta \Phi /c^2<0$. Since a redshifted black-body radiation spectrum has its temperature scaled by the same factor as its frequency, the gravitational contribution is

For the intrinsic effect, compare the local scale factor $R+\Delta R$ with the unperturbed value $R$ at the reference epoch. An equilibrium photon gas has energy density $u=a_{\mathrm{rad}}{T}^4$ and pressure $p=u/3$. For adiabatic compression, the first law of thermodynamics applied to a comoving volume $V\propto R^3$ givesThus $TR$ is constant along this adiabatic process, and the initial intrinsic contribution is $\Delta T/T=-\Delta R/R$. The smaller local expansion associated with the fluctuation leaves its radiation hotter; this is the compression or local expansion-history contribution, distinct from the clock-rate effect on the escaping radiation.

Multiplying the two first-order factors, and cancelling the common subsequent background redshift, givesFor the overdense region produced by adiabatic compression, $\Delta R<0$, so the final term is positive. It competes with the negative gravitational redshift term. Their sum is the intrinsic-plus-endpoint part of the Sachs-Wolfe combination; computing a full observed Cosmic microwave background anisotropy can also require Doppler CMB anisotropy and the Integrated Sachs-Wolfe effect.^[ref]

General relativity models spacetime by a smooth four-dimensional Lorentzian manifold whose metric tensor governs clock readings, causal cones and free-particle motion. The Einstein equivalence principle motivates local special relativity and universal geodesic free fall for test bodies; it does not by itself select a unique gravitational field equation. We assume a Levi-Civita connection, a metric tensor as the gravitational field with no additional dynamical gravitational variables, and matter with a covariantly conserved stress-energy tensor. Units have $c=1$.

The existence of local inertial frames is a local statement. At any regular event choose an orthonormal frame and Riemann normal coordinates, so that $g_{ab}={\eta }_{ab}$ and ${\Gamma }^a{}_{bc}=0$ at the event, with metric signature $(+---)$. Metric compatibility then gives ${\partial }_c{g}_{ab}=0$ there. The geodesic equation at that point is the special-relativistic inertial equation. Along a timelike geodesic, a parallel-transported orthonormal frame gives Fermi normal coordinates, eliminating the connection along the central worldline. These constructions do not in general eliminate second derivatives of the metric tensor in a neighborhood. The Riemann curvature tensor is built from the derivatives of the connection coefficients as well as their quadratic terms; nonzero Riemann curvature tensor components can remain when all connection coefficients vanish at a chosen point. Conversely, Polar coordinates in Minkowski spacetime can have nonzero connection coefficients but zero Riemann curvature tensor. Connection coefficients alone are therefore not an invariant measure of gravity.

For the tensorial content, consider a two-parameter family of geodesics parametrized by an affine parameter, with tangent $U$ and connecting field $J$. Their coordinate construction gives $[U,J]=0$, and torsion-freeness gives ${\nabla }_{U}J={\nabla }_{J}U$. Since ${\nabla }_{U}U=0$, commuting derivatives in the convention of Question 1 yields geodesic deviation:The left side is covariant relative acceleration, rather than the second coordinate derivative of an arbitrary separation. In an orthonormal frame in free fall with $U=(1,0,0,0)$, its spatial components are $D^2{J}^i/d{\tau }^2=-R^i{}_{00j}{J}^j$. This is a measurable tidal force, and a coordinate transformation cannot make a nonzero Riemann curvature tensor disappear. It is the distinction between gravity at one event and the behavior of an extended freely falling laboratory.

In a static Newtonian limit, $g_{00}=1+2\Phi$ and ${\Gamma }^i{}_{00}={\partial }_i\Phi$, so the test-body acceleration is $-\nabla \Phi$. The curvature-sign convention in Killing derivative identities gives $R^i{}_{00j}\simeq {\partial }_i{\partial }_j\Phi$. Hence relative acceleration is the negative Hessian of the gravitational potential. For the Newtonian potential of a point mass $\Phi =-GM/r$, the Hessian has radial eigenvalue $-2GM/r^3$ and two tangential eigenvalues $GM/r^3$: radial separations stretch and tangential ones compress. Outside the source the Ricci tensor can vanish while the Riemann curvature tensor remains nonzero. Its vacuum tidal part is encoded by the Weyl tensor; vacuum does not mean flatness. Gravitational waves likewise carry Riemann curvature tensor disturbances without a local material source.

To obtain the field equations, make the additional dynamical assumptions explicit: locality and diffeomorphism invariance of general relativity; a symmetric metric tensor field equation using at most second derivatives of the metric tensor; a geometrical left-hand side linear in the Riemann curvature tensor with constant coefficients; and compatibility with local matter conservation. These restrictions exclude higher-curvature or extra-field alternatives. Under them the available rank-two curvature contractions give the ansatzThe contracted Bianchi identity and metric compatibility giveFor this to vanish identically for arbitrary metric tensors, $b=-a/2$. Assuming $a\ne 0$ and absorbing its normalization into the coupling givesThe permitted constant term is the cosmological constant. Its value is not fixed by the equivalence principle. Conservation of $T$ follows for generally covariant matter on its equations of motion: varying the matter action under a compactly supported infinitesimal diffeomorphism, then integrating its term $T^{ab}{\nabla }_a{\xi }_b$ by parts, gives ${\nabla }_a{T}^{ab}=0$. The divergence-free Einstein tensor is consistent with this identity.

The coupling sign must be fixed in the actual curvature-sign convention in Killing derivative identities, with $R_{ab}=R^c{}_{acb}$. For the static weak-field approximation, $R_{00}\simeq -\Delta \Phi$. Set $\Lambda =0$ for the local asymptotically flat Newtonian comparison. In four dimensions the trace equation gives $R=-\kappa T$, so $R_{00}=\kappa (T_{00}-\frac{1}{2}{g}_{00}T)\simeq \kappa \rho /2$ for slowly moving matter. Matching $\Delta \Phi =4\pi G\rho$ forces $\kappa =-8\pi G$. Thus, with this convention,Under the opposite curvature-sign convention in Killing derivative identities the coupling sign reverses, as in Einstein-equation coupling under reversed curvature. This sign statement is compatible with the supplied linearized handout. Local inertial frames remove the connection at an event; tidal Riemann curvature tensor effects remain tensorial, and the field equations require dynamical assumptions beyond the equivalence principle.

For the static nonrotating source, use spherical symmetry and the static scalar perturbation of Minkowski spacetime, with isotropic radius $\rho$:To first order, the Einstein tensor in the declared convention hasThese are also the static scalar perturbation of Minkowski spacetime equations in the original information sheet. The source has $T_{00}=M{\delta }^{(3)}(\mathbf{x})$ and zero spatial stress. With the sign established in Question 2,The distributional identity $\Delta (1/\rho )=-4\pi {\delta }^{(3)}(\mathbf{x})$ and asymptotic flatness give $C=GM/\rho$. For $F=A+C$, the spatial field equation has trace $-2\Delta F=0$ and therefore ${\partial }_i{\partial }_{j}F=0$. Its decaying spherical solution is $F=0$, so $A=-GM/\rho$. This determines both potentials; the spatial correction cannot be inferred from the Newtonian gravitational potential alone.

Now use the isotropic-to-areal weak-field radius change. The angular coefficient defines $r^2=(1+2GM/\rho ){\rho }^2$, hence $r=\rho +GM$ to first order. Thus $dr=d\rho$ to the same accuracy, and replacing $1/\rho$ by $1/r$ inside a perturbation changes only second-order terms. With the Schwarzschild radius $r_S=2GM$, the linearized Schwarzschild metric isThis is an exterior weak-field approximation: $r_S/r\ll 1$. The point source is used distributionally to fix the mass, not to claim a regular or weak field at the origin.

For a null geodesic choose its orbital plane as $\theta =\pi /2$, using rotational symmetry, and let a dot denote an affine parameter derivative. For a Killing vector $k$ and an affinely parametrized geodesic tangent $U$, the geodesic conserved quantity from a Killing vector follows directly fromApplying this to the time-translation and axial Killing vectors, with the spatial sign absorbed into the definition of $h$, givesIf the tangent is normalized as the photon's four-momentum, $E$ is its energy measured at infinity and $h$ its signed angular momentum about the selected axis. Rescaling an arbitrary affine parameter rescales both; the ratio $b=|h|/E$ is the invariant impact parameter. Future-directed orbits have $E>0$. Assume $h\ne 0$ to use $\varphi$ as the orbit parameter; $h=0$ gives radial rays, not a finite-radius circular orbit.

The null vector constraint givesExpand consistently to first order in the weak-field approximation correction: $(1-u^2{)}^{-1}=1+O(u^2)$ and $(1+u{)}^{-1}=1-u+O(u^2)$. Also $du/d\varphi =-r_S\dot{r}/h$. Therefore the first-order orbital equation isThe cubic term is the first correction relative to the flat inverse-radius term; the equation is not a controlled exact equation for large $u$.

A constant solution of a first integral is not by itself sufficient to establish a circular geodesic. For a circular trajectory, the radial Euler-Lagrange equation also requires $A^{\prime }(r){\dot{t}}^2/2=r{\dot{\varphi }}^2$, where now $A(r)=1-r_S/r$ is the metric's time coefficient. Combining this with the null vector constraint gives $r{A}^{\prime }=2A$, or $u=2/3$. Equivalently, the formal potential $V(u)=u^2(1-u)$ must have a stationary point. Apart from the infinite-radius endpoint $u=0$, its only stationary point is $u_c=2/3$, with $V(u_c)=4/27$. Thus the formal circular solution requiresThe reduced radial equation is $u^{\prime \prime }=-u+\frac{3}{2}{u}^2$, with prime denoting $d/d\varphi$. It follows by differentiating the first integral on moving segments and agrees with the regular Euler-Lagrange equation at turning points. Perturbing $u=u_c+ϵ$ givesThere is an exponentially growing linear instability mode; equivalently $V^{\prime \prime }(u_c)=-2<0$. The formal circular orbit is unstable.

The weak-field photon-orbit validity limitation is essential: $u_c=2/3$ is not small. There is therefore no finite-radius circular photon orbit established within the controlled weak-field region. The formal extrapolation happens to reproduce the radius and impact parameter of the exact Schwarzschild photon sphere, but that physical result needs the exact geometry. The calculation also cannot establish a Schwarzschild event horizon from a first-order expansion at $u\sim 1$.

Use $G=c=1$, select the retarded source field with no added homogeneous radiation, and work to leading order in the weak-field approximation and long-wavelength source approximation. The retarded fundamental solution of the Linearized Einstein equations in Lorenz gauge in linearized gravity isFor $r\gg d$, the denominator is $r$ to leading order. The delay is $t-r+\mathbf{n}\cdot {\mathbf{x}}^{\prime }+O(d^2/r)$, where $\mathbf{n}=\mathbf{x}/r$. In the usual slowly evolving source regime, its characteristic time $\tau$ obeys $d/\tau \ll 1$, so the source-size part of the delay is negligible:The stress-energy conservation equation ${\partial }_{\mu }{T}^{\mu \nu }=0$ is required at the approximation order used; it follows from the divergence of the gauge-fixed field equation. Compact support removes the integration by parts surface terms. Since $T^{00}=T_{00}$ and $T^{ij}=T_{ij}$ in this signature,Combining these identities gives the retarded quadrupole fieldFor an ordinary nonrelativistic bound source, $\tau \sim d/v$ makes $d/\tau \sim v\ll 1$. More generally the size relative to the variation timescale must also be small; speed alone does not exclude a rapidly varying small-amplitude motion.

The radiation boundary condition for linearized gravity is essential for a statement about the full field. Without it, the displayed implication is false: take $T_{\mu \nu }=0$ and add ${\bar{h}}_{11}=A\cos (k(t-z))$, ${\bar{h}}_{22}=-{\bar{h}}_{11}$, all other components zero. This weak plane gravitational wave satisfies both the homogeneous wave equation and Lorenz gauge in linearized gravity, while every $I_{ij}$ is zero. The proved formula is for the retarded source contribution, with the standard slow-source qualification. Its transverse–traceless projection gives the physical radiative strain.

Take the oscillation to be $z(t)=L\cos \omega t$; a different initial phase does not affect a period average. In the mass-density normalization of the quadrupole formula, the second mass moment tensor of the point mass has only $I_{33}=m{z}^2$ nonzero. Its trace-free mass quadrupole moment is thereforeThe constant part does not radiate. Since $d^3({\cos }^2\omega t)/d{t}^3=4{\omega }^3\sin 2\omega t$, the tensor contraction isUsing $⟨{\sin }^2(2\omega t)⟩=1/2$ in the quadrupole formula gives the averaged positive radiated powerThe mass quadrupole moment oscillates at twice the source's angular frequency, and the source energy loss has the opposite sign. The expression uses the leading slow-motion weak-field approximation, $\omega L\ll c$. With SI stress-energy tensor components, the mass-density integrand is $T_{00}/c^2$; the displayed $c^{-5}$ quadrupole formula uses that mass normalization. This is the point-mass contribution requested by the model: an apparatus maintaining an accelerating mass would also contribute its own mass quadrupole moment.