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Definitions of the Riemann curvature tensor can differ by an overall sign, independent of the metric signature. Reversing that definition reverses the Ricci tensor, Ricci scalar and Einstein tensor, as well as the curvature side of the geodesic deviation identity and the Palatini identity. A derivation must choose the contraction and derivative ordering explicitly before combining formulas from different conventions. Changing a metric signature alone is a separate operation.

For a geodesic variation $F(s,\tau )$, the deviation vector $S=F_{\ast }{\partial }_s$ describes infinitesimal separation at equal parameter values. With $T=F_{\ast }{\partial }_{\tau }$, commuting parameter derivatives give $[S,T]=0$. A torsion-free connection then gives ${\nabla }_{T}S={\nabla }_{S}T$. For an affinely parametrized variation, geodesic deviation states ${\nabla }_T^{2}S=R(T,S)T$ with the convention $R(X,Y)Z={\nabla }_X{\nabla }_{Y}Z-{\nabla }_Y{\nabla }_{X}Z-{\nabla }_{[X,Y]}Z$.

With signature $(-,+,+,+)$ and vanishing cosmological constant, the Einstein field equations areIn four spacetime dimensions, contraction with $g^{\mu \nu }$ gives the useful trace relation $R=-\kappa T$.

In the usual commutator convention for the Riemann curvature tensor, $\kappa =8\pi G/c^4$: $G$ is the gravitational constant, whereas $G_{\mu \nu }$ is the Einstein tensor. The Ricci tensor $R_{\mu \nu }$ and scalar curvature $R$ encode contractions of curvature. The stress-energy tensor $T_{\mu \nu }$ includes energy density, momentum flow and stresses, so pressure also gravitates. A cosmological constant adds $\Lambda g_{\mu \nu }$ to the left-hand side. See Tong's Einstein equations and matter coupling.

For a concrete consequence, set $T_{\mu \nu }=0$ and $\Lambda =0$. The trace gives $R=0$, and substitution gives $R_{\mu \nu }=0$. Yet the exterior Schwarzschild spacetime still has the tidal forces computed under geodesic deviation: vanishing Ricci tensor does not imply vanishing Riemann curvature tensor. Geometry carries gravitational effects through empty space. Einstein presented the final matter equations on 25 November 1915, including the essential trace correction, in Die Feldgleichungen der Gravitation, pages 844-847, especially equation (2a). His symbols and sign conventions differ from modern ones; the displayed formula uses modern notation.

In a parallel-propagated orthonormal frame, orthogonal geodesic deviation in constant sectional curvature reduces to three scalar equations ${\ddot{S}}^i=\kappa S^i$. For $\kappa >0$, $S=A{e}^{\sqrt{\kappa }\tau }+B{e}^{-\sqrt{\kappa }\tau }$: generic initial data grow exponentially, but the pure decaying mode is a counterexample to universal growth. Initially comoving neighbors have $S=S(0)\cosh (\sqrt{\kappa }\tau )$. For $\kappa <0$, $S=A\cos (\sqrt{-\kappa }\tau )+B\sin (\sqrt{-\kappa }\tau )$, so the deviation is bounded and oscillatory. Simultaneous focusing requires compatible vector initial data and is not automatic for every oscillatory solution.

In flat spacetime take a twist-free beam's two screen Jacobi factors to be $1+\lambda$ and $1-\lambda$. They solve the zero-curvature geodesic deviation equation. Their product is the displayed area, and subtracting the trace from their optical matrix gives the displayed nonnegative shear. Direct differentiation yields $(A^{1/2}{)}^{\prime \prime }=-(1-{\lambda }^2{)}^{-3/2}=-{\sigma }^2{A}^{1/2}$, an explicit check of the focusing sign.

General relativity describes gravity through the geometry of spacetime. Its metric tensor specifies distances, proper time measured by clocks, and the causal structure that limits communication. Matter and radiation affect this geometry through the Einstein field equations; freely falling test particles follow geodesics of the resulting geometry. A planet's orbit and a falling apple therefore belong to the same geometric story. The Newtonian limit recovers familiar Newtonian gravity when fields are weak and speeds are small compared with the speed of light. See David Tong's introduction to general relativity.

The most direct way to understand spacetime curvature is to compare neighboring freely falling bodies. Even when each body feels weightless, their separation can change: this is geodesic deviation, observed as a tidal force. Curvature concerns these relative measurements, rather than the shape of a drawing in an imagined surrounding space. A second accessible measurement is gravitational time dilation: identical clocks maintained at different heights accumulate different proper times. Together, tides and clocks turn an abstract geometry into laboratory questions.

The geodesic deviation equation makes the relative gravitational acceleration proportional to the separation $L$ and a typical Riemann curvature tensor component $\mathcal{R}$. Thus tidal acceleration vanishes as $L\to 0$ at fixed bounded curvature. A uniform electric field can give different accelerations $qE/m$ at the same point, independently of separation. Relative tidal changes accumulated over a fixed long time need not vanish as a fraction of the initial separation; a local spacetime experiment must control its duration as well as its spatial size.

Using $[{\nabla }_c,{\nabla }_d]V^a=R^a{}_{bcd}{V}^b$, the Newton–Cartan gravitational connection has only the displayed Hessian components and their antisymmetric partners in the last two indices. The geodesic deviation equation gives relative acceleration $-{\partial }_i{\partial }_j\Phi \delta x^j$. The Poisson equation for Newtonian gravity is therefore $R_{ab}=4\pi G\rho (dt{)}_a(dt{)}_b$. A spatially uniform gravitational acceleration has zero tidal curvature even if the connection coefficients do not vanish in the original coordinates.

Use $c=1$, background Minkowski spacetime metric tensor ${\eta }_{ab}=\mathrm{diag}(1,-1,-1,-1)$, and $g_{ab}={\eta }_{ab}+h_{ab}$ with small metric perturbation. All indices and derivatives in this linear approximation use $\eta$. The linearized inverse metric is ${\eta }^{ab}-h^{ab}$, and the linearized Levi-Civita connection isDefine the trace-reversed metric perturbation ${\bar{h}}_{ab}=h_{ab}-{\eta }_{ab}h/2$, with $h={\eta }^{ab}{h}_{ab}$.

An infinitesimal coordinate change $x^{\prime a}=x^a+{\xi }^a$ gives the linearized coordinate gauge transformationThe metric perturbation therefore has redundant components. The gauge invariance of the linearized Riemann tensor follows because its change contains third derivatives of $\xi$ cancelling by commutation of partial derivatives. Conversely, flat linearized metric perturbations are locally pure gauge on a contractible patch. Thus linearized curvature, rather than a particular coordinate value of $h$, detects the physical disturbance. Gauge freedom can also be restricted by global topology or boundary conditions.

Keeping the paper's curvature sign convention, its linearized Ricci tensor and scalar areThese are the negatives of the alternative frequently used curvature convention. Put $v_b={\partial }^a{\bar{h}}_{ab}$. The Linearized Einstein equations in vacuum areUnder a gauge change $v_b\mapsto v_b-\square {\xi }_b$, so solving $\square {\xi }_b=v_b$ imposes the Lorenz gauge in linearized gravity. The field equations then become $\square {\bar{h}}_{ab}=0$. The remaining residual gauge symmetry of linearized gravity has $\square {\xi }_b=0$; Lorenz gauge does not exhaust the coordinate freedom.

A Fourier mode ${\bar{h}}_{ab}=A_{ab}{e}^{i{k}_c{x}^c}$ obeys $k^a{k}_a=0$ and $k^a{A}_{ab}=0$. Hence the plane gravitational waves in linearized gravity propagate on the background light cones. Four transversality conditions and four residual gauge amplitudes leave $10-4-4=2$ physical degrees of freedom. The explicit plane-wave reduction to transverse-traceless gauge can set $h_{0a}=0$, spatial trace zero and $k^i{h}_{ij}=0$. For a wave in the $z$ direction, write its physical spatial strain as $H_{ij}=-h_{ij}$, so the spatial metric tensor is $-({\delta }_{ij}+H_{ij})$:There are two gravitational wave polarizations, plus polarization and cross polarization. Rotating transverse axes through $\theta$ rotates their amplitude pair through $2\theta$. Nonconstant profiles with nonzero second derivative have nonzero linearized Riemann curvature tensor and cannot be removed as pure gauge.

The TT coordinate separation and measured separation distinction explains a detector's response. Freely falling particles initially at rest can keep fixed TT coordinates, since ${\Gamma }^a{}_{00}=0$, while their proper separation changes. For a short arm along a unit vector $n$,This is the tidal response described by geodesic deviation; coordinate motion by itself is not the measured signal.

Finally, gravitational-wave energy is second order in the perturbation, so it is absent from a first-order vacuum equation. In a short-wavelength averaging regime the averaged stress-energy of transverse gravitational waves isFor a plane wave its energy flux is $(16\pi G{)}^{-1}⟨{\dot{H}}_{+}^2+{\dot{H}}_{\times }^2⟩$. The averaging scale and weak-field assumptions matter: this does not assign a coordinate-independent local gravitational energy density to an arbitrary first-order perturbation.

Interpret $A$ first as an infinitesimal screen area transported along generators of a future event horizon, not as the total area of an arbitrary trapped surface. Horizon generators form an affinely parametrized, hypersurface-orthogonal null geodesic congruence. The two-dimensional screen metric is positive definite. Its optical tensor has zero twist and decomposes asThe scalar here agrees with the supplied derivative definition: nullness and the affine geodesic equation remove the longitudinal terms in the contracted norm, leaving $B_{AB}{B}^{AB}-{\theta }^2/2$.

To derive the focusing equation, vary neighbouring geodesics and let $J$ be their screen Jacobi map. The geodesic deviation equation is $J^{\prime \prime }=-\mathcal{R}J$, where ${\mathcal{R}}_{AB}=R_{AcBd}{p}^c{p}^d$. Hence $B=J^{\prime }{J}^{-1}$ satisfies $B^{\prime }=-B^2-\mathcal{R}$. Taking its screen trace yields the Null Raychaudhuri equationSet $a=A^{1/2}$. Since $a^{\prime }/a=\theta /2$, the area-square-root optical focusing equation isThe PDF's plus sign before ${\sigma }^2$ is inconsistent with its own positive shear definition. This is an actual source error, not an antisymmetrization or curvature-sign convention.

For an explicit check, take a flat-space twist-free beam with screen Jacobi factors $1+\lambda$ and $1-\lambda$, for $|\lambda |<1$. ThenIts Ricci term is zero, so the result equals $-{\sigma }^2{A}^{1/2}$ and has the opposite sign to the printed equation. This is flat-space anisotropic beam shear focusing.

Under the null convergence condition $R_{ab}{p}^a{p}^b\ge 0$, the correct equation gives concavity of each local area square root. Equivalently ${\theta }^{\prime }\le -{\theta }^2/2$. If a horizon generator had $\theta ({\lambda }_0)={\theta }_0<0$, integration gives a focal point within affine distance at most $2/|{\theta }_0|$. Such a generator could not remain on an achronal boundary beyond the focal point. Under the usual global hypotheses that horizon generators remain regular and future complete, this contradicts their being generators of the event horizon. Hence $\theta \ge 0$ and $A^{\prime }=\theta A\ge 0$. Integrating over the horizon patches, and including newly joining generators which can add area, proves Hawking's area theorem. Global predictability/completeness assumptions are essential; the local differential inequality alone is not a theorem about arbitrary trapped-surface areas.

For the perfect fluid stress tensor, null contraction eliminates the pressure-metric term:A nonzero null vector cannot be orthogonal to a timelike fluid velocity. Thus the perfect-fluid null energy condition is preciselySeparate requirements $\rho \ge 0$ or $p_{\mathrm{fluid}}\ge 0$ are not needed for this null focusing argument. With the Einstein equation, $R_{ab}{p}^a{p}^b=8\pi T_{ab}{p}^a{p}^b$.

A cosmological constant contributes only a term proportional to $g_{ab}$, whose null contraction is zero:Therefore a nonzero cosmological constant does not change the fluid condition or the local area-focusing argument, though it can change the global asymptotics and which horizons and completeness hypotheses are appropriate.

The Pound-Rebka experiment measures a small gravitational redshift by comparing nuclear emission and resonant absorption at different heights in Earth's field. The $14.4\mathrm{keV}$ transition of iron-57 supplies the gamma-ray frequency standard. The Mössbauer effect permits recoil-free emission and absorption by nuclei bound in solids, giving a sufficiently narrow resonance. A source and absorber separated vertically by about $22.6\mathrm{m}$ have a predicted fractional frequency difference of order $2.5\times 10^{-15}$. A calibrated Doppler effect from slow source motion measures the shift through its effect on resonant absorption. Interchanging the upper and lower arrangements reverses the gravitational contribution and helps separate it from intrinsic source-absorber offsets; temperature-dependent frequency shifts must also be controlled. The observed change agrees with the weak-field prediction.

The prediction follows either from an accelerating laboratory and the equivalence principle, or directly from a static metric. With $x^0=ct$, write $g_{00}=N^2\simeq 1+2\Phi /c^2$. The time-translation Killing vector field gives a conserved photon energy $E_{\xi }=p_a{\xi }^a$, while a static observer has four-velocity $u^a={\xi }^a/N$ and measures $h\nu =E_{\xi }/N$. ThusFor an upward ray over height $H$, $\Delta \nu /\nu \simeq -gH/c^2$; downward propagation has the opposite sign. The equivalent compensating Doppler speed has magnitude $gH/c$, about $7.4\times 10^{-7}\mathrm{m}{\mathrm{s}}^{-1}$ for this baseline. Identical clocks at rest have $d\tau =Ndt$, so a clock at higher potential runs faster relative to the common static coordinate time. This does not mean that the nuclear transition changes its locally measured frequency with height: the comparison involves transporting a photon between different local proper-time standards.

The weak equivalence principle asserts universal free fall for test bodies. The Einstein equivalence principle adds local Lorentz invariance and local position invariance for nongravitational experiments in a freely falling laboratory. The redshift comparison tests the clock aspect of local position invariance and supports a metric description of gravity. The strong equivalence principle extends these requirements to self-gravitating bodies and local gravitational experiments, including the contribution of gravitational binding energy. Their outcomes should be independent of the external location and motion of a freely falling laboratory when tidal effects are negligible. A nuclear redshift experiment is therefore not a test of the whole strong equivalence principle, and its success does not uniquely establish the gravitational field equations.

The geometrical realization uses the Levi-Civita connection of a spacetime metric. At any event, normal coordinates make $g_{ab}={\eta }_{ab}$ and the Christoffel symbols vanish. Freely falling test particles follow geodesics; local nongravitational equations take their special-relativistic form at that event. The Riemann curvature tensor need not vanish, so finite laboratories can measure tidal geodesic deviation. The strong principle concerns removing a uniform external gravitational field locally, not removing curvature throughout an extended region.

For a perfect fluid, local rest-frame isotropy and the absence of heat flux or viscosity give $T^{\hat{a}\hat{b}}=\mathrm{diag}(\rho ,p,p,p)$, where $\rho$ is rest-frame energy density. With the PDF's signature $(+---)$, $u^a{u}_a=1$ and $h^{ab}=g^{ab}-u^a{u}^b$ projects spatially. Its covariant form isThe pressure sign follows from the negative spatial metric; it gives positive diagonal stresses in the rest frame. Local stress-energy conservation, ${\nabla }_a{T}^{ab}=0$, produces the two projectionsThe first is the energy equation, including pressure work; the second is the relativistic Euler equation. Its rest-frame spatial pressure force is $-\nabla p$.

Further assumptions are needed to derive the Einstein field equations. Take a local generally covariant metric equation, linear in curvature and of second differential order, and demand compatibility with stress-energy conservation. The available rank-two tensors are $R_{ab}$, $R{g}_{ab}$ and a constant multiple of $g_{ab}$. Defining $R_{ab}=R^c{}_{acb}$, the contracted Bianchi identity gives ${\nabla }^a{R}_{ab}=\frac{1}{2}{\nabla }_{b}R$. Hence the combination $R_{ab}+\alpha R{g}_{ab}$ is identically divergence-free only when $\alpha =-\frac{1}{2}$. This determines the Einstein tensor $G_{ab}=R_{ab}-R{g}_{ab}/2$; a cosmological constant term remains allowed.

The coupling sign must be matched to the actual curvature convention. Here $R^a{}_{bcd}$ represents $[{\nabla }_d,{\nabla }_c]$, and a static weak field has $R_{00}\simeq -{\nabla }^2\Phi /c^2$. With negligible pressure and cosmological term, a trial equation $G_{ab}=C{T}_{ab}$ implies $R_{00}\simeq C{\rho }_{\mathrm{mass}}{c}^2/2$. Comparing with the Poisson equation ${\nabla }^2\Phi =4\pi G{\rho }_{\mathrm{mass}}$ fixes $C=-8\pi G/c^4$. Thus the Einstein equation in the sequential-derivative curvature convention isEquivalently, $R_{ab}=\Lambda g_{ab}-(8\pi G/c^4)(T_{ab}-T{g}_{ab}/2)$, with $T=\rho -3p$. Pressure therefore gravitates as well as supplying a material force. The cosmological constant is not fixed by the local Newtonian matching.

Setting $c=1$ for the action normalization, the same metric equation follows from the Einstein-Hilbert action $S_g=(2\kappa {)}^{-1}\int \sqrt{-g}(R-2\Lambda )d^{4}x$, where $\kappa =8\pi G$, together with a matter action obeying $\delta S_m=\frac{1}{2}\int \sqrt{-g}{T}_{ab}\delta g^{ab}{d}^{4}x$. After the metric boundary term is removed or supplied separately, the bulk variation is proportional to $G_{ab}+\Lambda g_{ab}+\kappa T_{ab}$. This also makes the sign convention explicit. General covariance, the field equation and the contracted Bianchi identity are mutually consistent with the fluid conservation laws. The experiment motivates the local metric description; the curvature and conservation assumptions, plus the Newtonian limit, supply the additional route to the field equation.

Work to first order in a metric perturbation $g_{ab}={\eta }_{ab}+h_{ab}$, with $\eta =\mathrm{diag}(1,-1,-1,-1)$, raising perturbation indices with $\eta$ and discarding quadratic terms. The Christoffel symbols areUsing the printed curvature convention gives the linearized Riemann curvature operatorFor definiteness contract $R_{bd}=R^a{}_{bad}$. The corresponding first-order Ricci tensor isThis is the reversed sign convention specified in curvature-sign convention in Killing derivative identities; the vacuum equation is unaffected by an overall curvature sign.

An infinitesimal passive coordinate change $x^{\prime a}=x^a+{\xi }^a$ changes the perturbation by $\delta h_{ab}=-{\partial }_a{\xi }_b-{\partial }_b{\xi }_a$. Substitution into $R^{(1)}$ gives zero change, since partial derivatives commute. Equivalently the change in curvature is the Lie derivative of the background curvature, which vanishes in Minkowski spacetime. Thus linearized curvature, and the linearized Weyl tensor in vacuum, are gauge-invariant tidal observables, whereas individual components of $h$ are generally gauge dependent.

The trace-reversed metric perturbation ${\bar{h}}_{ab}=h_{ab}-{\eta }_{ab}h/2$ transforms by $\delta {\bar{h}}_{ab}=-{\partial }_a{\xi }_b-{\partial }_b{\xi }_a+{\eta }_{ab}{\partial }_c{\xi }^c$. Its divergence changes by $-\square {\xi }_b$. Solving $\square {\xi }_b={\partial }^a{\bar{h}}_{ab}$ therefore imposes Lorenz gauge in linearized gravity, ${\partial }^a{\bar{h}}_{ab}=0$. With the stated sign convention the Einstein tensor isso the linearized vacuum field equation becomes $\boxed{\square {\bar{h}}_{ab}=0}$. The residual gauge symmetry of linearized gravity has $\square {\xi }_a=0$. This makes a hyperbolic evolution problem after the gauge condition and its initial consistency constraints have been imposed.

To separate actual gauge invariants from a gauge choice, use gauge-invariant Minkowski perturbations. For spatial indices contracted with ${\delta }_{ij}$, decomposewhere ${\partial }_i{B}_i={\partial }_i{E}_i=0$, ${\partial }_i{h}_{ij}^{\mathrm{TT}}=0$ and $h_{ii}^{\mathrm{TT}}=0$. Spatial falloff or Fourier-mode conventions fix the decomposition. Write ${\xi }^0=\tau$ and ${\xi }^i={\partial }_{i}L+L_i$ with ${\partial }_i{L}_i=0$. Direct substitution in the gauge change givesHence the invariant combinations areA representative with $E=E_i=B=0$ has $h_{00}=2\Phi$, $h_{0i}=V_i$ and $h_{ij}=2\Psi {\delta }_{ij}+h_{ij}^{\mathrm{TT}}$. The explicit vacuum constraints for gauge-invariant Minkowski perturbations follow fromFor a nonzero spatial Fourier mode, or regular decaying fields on all space, $G_{00}=0$ forces $\Psi =0$, then $G_{0i}=0$ forces $V_i=0$, and the spatial trace gives $\Delta \Phi =0$, hence $\Phi =0$. Only $\square h_{ij}^{\mathrm{TT}}=0$ remains. This removal of the scalar and vector sectors does not exclude singular Coulomb perturbations outside a source, boundary-driven harmonic potentials or spatially homogeneous modes; those require separate boundary and gauge conventions.

For a propagating plane mode $h\propto e^{i{k}_a{x}^a}$, the wave equation gives $k^a{k}_a=0$. A transverse-traceless spatial tensor perpendicular to a fixed propagation direction has two independent components. For propagation along $z$, define the physical strain $H_{ij}=-h_{ij}^{\mathrm{TT}}$ in this mostly-minus convention, so the spatial line element is $({\delta }_{ij}+H_{ij})d{x}^{i}d{x}^j$. It has the formThese are the two gravitational wave polarizations, plus polarization and cross polarization. A rotation in the transverse plane rotates the polarization pair through twice the angle; cross is plus rotated through $45$ degrees. Lorenz constraints and residual gauge freedom give the same count: ten symmetric metric components minus four constraints and four gauge freedoms leave two radiative degrees of freedom.

In transverse-traceless gauge, particles initially at rest keep fixed spatial coordinates since ${\Gamma }^a{}_{00}=0$, but their measured separation changes. For a short separation of length $L$ in unit direction $n$, $\delta L/L=H_{ij}{n}^i{n}^j/2$. The gauge-invariant curvature gives the corresponding geodesic deviation:Here the last equation is in a freely falling orthonormal frame, with the printed reversed curvature convention. Coordinate separation and physical separation must not be confused. The linearized vacuum equations describe propagation and tidal measurements; wave energy transfer and an effective wave stress-energy tensor arise at quadratic order, beyond the linear approximation.

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.

Take $z$ positive outward and define the separation as upper minus lower position. The Newtonian potential of a point mass gives the acceleration $a(z)=-{\partial }_z\varphi =-GM/z^2$. Expanding its difference across a separation $\zeta \ll z$ gives the tidal force equationThe upper particle falls slightly less rapidly, so the separation increases. If the upper particle carries the electric charge and the electric field points outward, the extra relative acceleration is $qE/m$. Reversing the field or placing the charge on the lower particle changes its sign. Write this signed acceleration as $a_e$ and put $k=2GM/z^3$. With initial relative velocity zero and nearly constant $k$ and $a_e$, the ordinary differential equation has solutionFor $k{t}^2\ll 1$, the leading changes due to the tidal force and electric field are respectivelywith the stated sign convention. Thus, for a nonzero fixed electric acceleration,A laboratory of size $L$ has $|{\zeta }_0|\le L$, so this ratio tends to zero as $L\to 0$. This is local tidal acceleration scaling: gravitational relative acceleration is proportional to separation, whereas the electric difference persists even at coincident positions.

The duration also matters. At fixed time, $\Delta {\zeta }_g\to 0$ in absolute size, but $\Delta {\zeta }_g/{\zeta }_0\simeq (GM/z^3)t^2$ need not tend to zero. To keep the electrically displaced particles in the shrinking laboratory, take $|a_e|t^2\lesssim L$; then $|\Delta {\zeta }_g|/L\lesssim GML/(z^3|a_e|)\to 0$. Alternatively a stationary terrestrial laboratory has a free-fall time of order $\sqrt{L/g}$, giving $|\Delta {\zeta }_g|/|{\zeta }_0|=O(L/z)$. Both make precise the small spacetime-region limit. An extended, long-running experiment can still measure arbitrarily small tidal forces.

The weak equivalence principle says that freely falling test particles with the same initial data have composition-independent motion. In particular, a freely falling frame removes the common gravitational acceleration of both particles. The electric accelerations depend on charge and cannot be removed for charged and uncharged particles simultaneously by the same change of frame.

The Einstein equivalence principle extends the freely falling laboratory statement to all local nongravitational experiments: their outcomes obey special relativity, with local Lorentz invariance and local position invariance. The strong equivalence principle also includes local gravitational experiments and bodies with significant gravitational binding energy. It asserts universality of their free fall and independence of local experimental outcomes from the laboratory's location and velocity. The two-particle experiment illustrates the local distinction between gravity and a nongravitational force; by itself it does not establish this stronger claim about self-gravitating bodies.

In a metric theory, the common freely falling trajectories are timelike geodesics of a Lorentzian metric. Normal coordinates make $g_{ab}={\eta }_{ab}$ and the Christoffel symbols zero at one point. Fermi normal coordinates do the same along a reference timelike geodesic. The residual effects are controlled by the Riemann curvature tensor through geodesic deviation, and enter the local metric at quadratic order in distance. Curvature is a physical tidal field, so no coordinate choice removes it over a finite region.

A relativistic theory should consequently couple matter universally to the metric tensor, recover nongravitational laws in local inertial frames and recover the Newtonian limit. General relativity implements this with the Einstein field equations; the contracted Bianchi identity is consistent with stress-energy conservation. The equivalence principle motivates this geometric description but does not alone uniquely determine the field equations. Extra gravitational fields may satisfy the Einstein equivalence principle for ordinary matter while failing the strong equivalence principle because local gravitational measurements or self-gravitating trajectories depend on those fields.

Let $x^a(\lambda ,s)$ be a smooth family of affinely parametrized null geodesics. The tangent is $U=\partial x/\partial \lambda$, and the deviation vector is $Z=\partial x/\partial s$, comparing nearby rays at the same affine parameter. The two parameter derivatives commute, soThus the separation vector is Lie-transported along the null geodesic congruence; it is not required to be parallel-transported. Differentiating once more gives the geodesic deviation equation. With curvature convention $[{\nabla }_c,{\nabla }_d]V^a=R^a{}_{bcd}{V}^b$,For optical measurements one uses transverse deviations with $U\cdot Z=0$ and identifies longitudinal additions proportional to $U$. The scalar $U\cdot Z$ is constant along affine null rays by the displayed transport relation, so an initially transverse deviation remains transverse.

An affinely parametrized geodesic satisfies $d^2{x}^a/d{\lambda }^2+{\Gamma }^a{}_{bc}{T}^b{T}^c=0$, which is exactly ${\nabla }_{T}T=0$. The Levi-Civita connection is metric-compatible, sowhen both vectors are parallel transported. Their inner product is constant, including their lengths; this preservation in fact holds along any path with parallel vectors, not only a geodesic.

For the two-parameter family, mixed partial derivatives commute and ${\Gamma }^a{}_{bc}$ is symmetric in its lower indices. ConsequentlyUsing this equality and the supplied curvature-commutator convention,In components the geodesic deviation equation isIt describes the relative acceleration of nearby freely falling worldlines. A freely falling observer can remove the connection at an event by choosing a local inertial frame, but cannot remove the curvature-dependent tidal acceleration of neighboring freely falling particles. Measuring their separation and relative acceleration therefore measures the local gravitational tidal field.

The strong equivalence principle asserts universality of free fall even for bodies with gravitational self-binding, together with independence of local experimental outcomes, including gravitational experiments, from the location and velocity of a freely falling laboratory. “Local” means that external tidal variation across the apparatus can be neglected. It does not mean that all curvature, or a laboratory's own gravitational field, can be removed. The weak principle concerns freely falling test bodies without appreciable self-gravity; laboratory composition tests alone therefore do not establish the strong version.

Lunar laser ranging supplies evidence specifically sensitive to the strong version. The Earth and Moon have different fractions of gravitational self-energy. If those fractions affected gravitational and inertial mass differently, their accelerations toward the Sun would differ and the lunar orbit would acquire a periodically varying polarization, the Nordtvedt effect. Combining lunar ranging with laboratory bounds on composition-dependent differential acceleration separates this self-energy effect. A determination available before this examination found the self-energy contribution to the Earth–Moon difference in $m_G/m_I$ to be $(-2.0\pm 2.0)\times 10^{-13}$, consistent with zero; see the 2004 experimental analysis. Universality-of-free-fall experiments, gravitational redshift measurements and tests of local Lorentz invariance support the weaker constituent principles, but should not be confused with a direct self-gravity test.

In general relativity, all matter couples to one metric tensor, and freely falling test particles follow its geodesics. At any event, Riemann normal coordinates give $g_{ab}={\eta }_{ab}$ and ${\Gamma }^a{}_{bc}=0$. Nongravitational local equations reduce to their special-relativistic forms in that frame. Gravitational binding is governed by the same metric field equations rather than a separate composition-dependent force, and the local gravitational coupling is universal. These features implement the strong equivalence principle. Curvature still governs geodesic deviation, so this construction removes uniform acceleration, not tidal gravity.

To geometrize Newtonian mechanics, use a four-dimensional manifold with coordinates $(t,x^1,x^2,x^3)$, absolute time ${\tau }_a=(dt{)}_a$ and a degenerate spatial inverse metric $h^{ij}={\delta }^{ij}$, $h^{0a}=0$. For a gravitational potential $\Phi (t,x)$ define the torsion-free Newton–Cartan gravitational connection byIt satisfies ${\nabla }_a{\tau }_b=0$ and ${\nabla }_a{h}^{bc}=0$. An affine geodesic has $d^{2}t/d{\lambda }^2=0$; choosing $\lambda =t$ makes its spatial equationsThis is Newton's free-fall law. The geometry is Newton–Cartan theory, with an affine connection and degenerate metric data, rather than a nondegenerate four-dimensional Lorentzian metric. The spatial slices themselves remain Euclidean.

Use the curvature conventionIn these coordinates the quadratic connection terms vanish, and the only nonzero curvature components, apart from antisymmetry in $c,d$, areIn particular $R_{00}={\nabla }^2\Phi$ and all other components of the Ricci tensor vanish. This Newtonian tidal curvature gives relative acceleration $\delta {\ddot{x}}^i=-{\partial }_i{\partial }_j\Phi \delta x^j$. Spatially constant gravitational acceleration can be removed by an accelerating frame because its Hessian vanishes. The Poisson equation for Newtonian gravity becomes $R_{ab}=4\pi G\rho {\tau }_a{\tau }_b$. Fluid mechanics is compatible with the same connection: the covariant continuity equation is the usual mass-conservation equation, and the spatial Euler equation is $u^b{\nabla }_b{u}^i=-{\rho }^{-1}{\partial }_{i}p$, which includes the gravitational force through the connection.