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A tidal force is a difference of gravitational acceleration across an extended region. In Newtonian gravity, $g_i=-{\partial }_i\Phi$. Two neighboring freely falling particles separated by ${\xi }^j$ have, to first order,For a point mass, $\Phi =-GM/r$. In a local orthonormal radial/transverse frame,A small freely falling cloud stretches radially and compresses transversely. Outside the source the three eigenvalues sum to zero, consistently with ${\nabla }^2\Phi =0$; within matter the trace is $-4\pi G\rho$. These are accelerations of separations, not the much larger common acceleration of the cloud's centre.

In general relativity, freely falling test particles follow timelike geodesics. A freely falling frame can eliminate the connection at one event, but not the Riemann curvature tensor throughout a finite laboratory. Let $u$ be the geodesic tangent and $\xi$ a connecting vector for neighboring geodesics with $[u,\xi ]=0$. In the explicitly chosen curvature conventionthe geodesic deviation equation isIt follows by commuting ${\nabla }_u{\nabla }_{\xi }u$, using ${\nabla }_{u}u=0$ and zero torsion. In a parallel-propagated orthonormal frame, its spatial components directly describe measured relative acceleration. The weak-field, slow-motion limit recovers the Newtonian Hessian above. Reversing the curvature sign convention reverses the displayed curvature side, without changing the physical stretching. The equivalence principle removes locally uniform gravity, while geodesic deviation measures the residual tidal field. The connection between parallel transport, curvature and neighboring geodesics is developed systematically in the standard geometric construction.^[ref]

The weak equivalence principle states that freely falling, sufficiently small test bodies have the same motion in a given gravitational field irrespective of composition or mass, neglecting their own gravity and nongravitational forces. In the Newtonian limit, $m_i\mathbf{a}=m_g\mathbf{g}$ makes this universality equivalent to a universal gravitational-to-inertial mass ratio, conventionally set to one.

The Einstein equivalence principle adds that local nongravitational experiments in a freely falling, nonrotating laboratory have the laws of special relativity, independent of the laboratory's position and velocity. At an event one can choose a local inertial frame with $g_{\mu \nu }={\eta }_{\mu \nu }$ and ${\partial }_{\alpha }{g}_{\mu \nu }=0$. A freely falling observer feels no supporting force, while a supported observer can model a locally uniform gravitational field by an accelerated laboratory. This explains why freely moving particles follow timelike geodesics and light follows null geodesics when other interactions are negligible.

The locality qualification is essential. The Riemann curvature tensor generally remains nonzero; separated freely falling bodies have tidal relative acceleration, measured by geodesic deviation. A finite-size experiment can therefore distinguish a curved gravitational field from a merely accelerated coordinate description of flat spacetime. Equivalence alone also does not select the complete Einstein field equations.

A useful weak-field consequence is the upward photon shift $\Delta \nu /\nu \simeq -gH/c^2$ over laboratory height $H$. In an upward-accelerating laboratory the receiver gains recession velocity during the light flight, giving the same leading redshift. The Pound-Rebka experiment tested this with resonant absorption of 14.4-keV gamma rays over about 22.5 m. A controlled source velocity calibrated the Doppler sensitivity; interchanging source and absorber reversed the gravitational contribution while retaining an intrinsic source–absorber offset. Monitoring their temperature difference was essential because lattice-motion shifts could mimic the small signal.^[ref] This verifies a gravitational clock-shift prediction; by itself it is not a unique test selecting general relativity over every alternative metric theory.

Nearby freely falling particles can accelerate relative to one another even though each follows a geodesic. A local inertial frame removes the common gravitational acceleration at one event; it cannot remove the variation across an extended laboratory. That variation is a tidal force, measured covariantly by geodesic deviation.

Let $u^{\mu }$ be the reference four-velocity and ${\xi }^{\mu }$ an infinitesimal separation to a neighboring geodesic, with matching affine parameters. With the convention $[{\nabla }_{\beta },{\nabla }_{\gamma }]V^{\mu }=R^{\mu }{}_{\nu \beta \gamma }{V}^{\nu }$,This relates measured relative acceleration to the Riemann curvature tensor. Reversing the curvature-tensor convention reverses the displayed curvature sign, without changing the physical relative motion.

In a small freely falling frame outside a spherical mass, the weak-field tidal tensor has radial and transverse components proportional to $2,-1,-1$. ThusAn initially spherical cloud stretches radially and compresses in both transverse directions. Two separated test masses, with their changing separation measured by light, form the basic idea of a gravity gradiometer; the common falling motion drops out. A uniform apparent acceleration in an accelerated laboratory can occur even in flat Minkowski spacetime, so it is the separation-dependent tidal response that diagnoses curvature.

Local equivalence and tides. In Newtonian mechanics, inertial mass $m_{\mathrm{i}}$ measures the response to a force, whereas passive gravitational mass $m_{\mathrm{g}}$ couples to an imposed Newtonian gravitational field. A freely falling body obeysThe observed composition independence of free fall says that this ratio is universal. With a common choice of units it is one. This is the Newtonian content of the weak equivalence principle for test particles; it does not by itself specify all nongravitational physics.

The Einstein equivalence principle further says that local nongravitational experiments in a sufficiently small freely falling laboratory have the form predicted by special relativity, independently of the laboratory's location and motion. In such a local inertial frame, the metric tensor is locally the Minkowski metric and the Christoffel symbols vanish at the selected event. The strong equivalence principle extends the universality claim to local gravitational experiments and bodies with significant gravitational self-energy. None of these statements removes measurable tidal forces across a finite apparatus.

In an ideal small freely falling lift, a released ball initially floats relative to the cabin, because both undergo the same free fall. Two separated balls, however, need not retain their separation: they sample different parts of the gravitational field. For a point source, linearizing $\mathbf{g}=-GM\mathbf{r}/r^3$ across a small separation $\mathbf{\xi }$ gives the tidal tensor predictionA radial separation stretches with coefficient $2GM/r^3$, whereas a transverse separation is compressed with coefficient $-GM/r^3$. Initially parallel velocities therefore develop relative motion even though the individual bodies experience zero proper acceleration.

The relativistic statement is geodesic deviation. With the Riemann curvature tensor conventiona connecting vector between neighboring freely falling worldlines satisfiesHere $D/d\tau$ is the covariant derivative along the reference geodesic, and $U^{\mu }$ its four-velocity. Curvature can be nonzero where the Christoffel symbols have been made zero. In the weak, static, slow-motion limit this equation reduces to ${\ddot{\xi }}^i=-{\partial }_i{\partial }_j\Phi {\xi }^j$, recovering the tidal tensor above. A finite falling lift can detect gravity through these relative accelerations; local equivalence does not imply a globally uniform field.

Weak-field connection and potential. Take $x^0=ct$ and the $+---$ metric signature. There is an index error in the printed formula for the Christoffel symbols: its first differentiated metric component must be $g_{\nu 0}$, not $g_{\mu 0}$. The correctly contracted expression isThe offending term vanishes under the stipulated time independence, so this correction does not change the intended calculation. Write $g_{\mu \nu }={\eta }_{\mu \nu }+h_{\mu \nu }$ and treat both $h$ and its derivatives as first order. To that order $g^{00}=1-h_{00}$, $g^{0i}=O(h)$, and $g^{ij}=-{\delta }^{ij}+O(h)$. A stationary metric tensor givesFor the temporal component the inverse-metric coefficient is already first order, soFor a spatial component the zeroth-order inverse metric contributes:The time component of the approximate geodesic equation says that $dt/d\tau$ is constant to the retained order. In its spatial component, convert the proper time derivatives to coordinate-time derivatives and cancel $(dt/d\tau {)}^2$:Comparison with the Newtonian equation $d^2\mathbf{x}/d{t}^2=-\nabla \Phi$ identifies $h_{00}=2\Phi /c^2+C$. For an isolated source, choose the gravitational potential to tend to zero at infinity and normalize the time coordinate there, giving $C=0$. ThusThe quantity $\Phi$ is the Newtonian gravitational potential, or gravitational potential energy per unit mass, in $\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}={\mathrm{m}}^2{\mathrm{s}}^{-2}$. For a point source, $\Phi =-GM/r$. A static clock has $d\tau =\sqrt{g_{00}}dt\approx (1+\Phi /c^2)dt$, connecting the Newtonian limit with gravitational time dilation.

Frequency of the orbiting source. Treat the gas blob as a test particle on a timelike circular Schwarzschild geodesic, with pressure and other nongravitational forces negligible. Treat our very distant location as infinity, so the observer's own gravitational potential is negligible. These are the assumptions that make a unique answer in terms of $R_S/r$ possible.

Put $f=1-R_S/r$ and use a local static observer at the emission point as an intermediate measuring frame. The remote observer lies on the symmetry axis of the orbit. A ray arriving on that axis has zero conserved axial angular momentum, so its local direction is perpendicular to the emitter's azimuthal velocity. Bending within a meridional plane does not alter this orthogonality. The transverse relativistic Doppler effect therefore giveswhere $v$ is the orbital speed measured by that local static observer. The gravitational redshift between static Schwarzschild observers then givesA gravitational circular orbit has $v^2=GM/r+O((GM/r{)}^2/c^2)$ in the weak-field limit, so $v^2/c^2=R_S/(2r)+O((R_S/r{)}^2)$. Expanding both factors,Solving for the emitted frequency givesThe gravitational part is $R_S/(2r)$ and the transverse kinematic time dilation contributes another $R_S/(4r)$; the absence of line-of-sight motion does not remove this second effect.

As an exact check, the local speed on a timelike circular Schwarzschild geodesic satisfies $v^2/c^2=\mu /(r-2\mu )$, where $\mu =GM/c^2=R_S/2$. Therefore the face-on frequency shift of a circular Schwarzschild source iswhose expansion agrees with the boxed answer. If the gas is supported or accelerated by pressure or magnetic stresses rather than following a geodesic, its orbital speed is not fixed by the mass and radius alone; the appropriate first-order result is instead ${\nu }_{\mathrm{em}}\approx {\nu }_{\mathrm{ob}}[1+GM/(r{c}^2)+v^2/(2c^2)]$, with $v$ supplied independently.

A gravitational wave of plus polarization has opposite transverse strain or geodesic deviation eigenvalues along two fixed perpendicular axes. For a sinusoidal periodic function profile, the stretching and compressing roles alternate every half-period.

The orthonormal-frame components of Schwarzschild curvature scale as $M/r^3$. The geodesic deviation equation therefore gives radial stretching and transverse compression whose magnitude diverges as $r\to 0$.

A short rigid nonrotating guide constrains two freely sliding beads to one axis $n$. In its parallel-propagated orthonormal frame, the axial geodesic deviation is $\ddot{L}=-R_{n0n0}L$. For a transverse gravitational wave, $R_{n0n0}=-ϵ{\ddot{h}}_{nn}/2$, so a wave incident after the beads are at rest changes the measured separation by $ϵh_{nn}{L}_0/2$ to first order. Transverse reaction forces do not supply an axial restoring force. Alignment with the propagation direction gives no first-order length change.

The relative acceleration of neighboring freely falling bodies is measured by geodesic deviation, comparing their infinitesimal separation $\xi$ in a parallel transport frame along a reference timelike geodesic with tangent $U$. With $R^a{}_{bcd}$ defined by the covariant derivative commutator, it has the displayed form. In the Newtonian limit the corresponding tidal tensor is the spatial gradient of the gravitational acceleration. A single freely falling body can remove local gravitational acceleration by its frame choice, whereas relative tidal acceleration measures curvature.

In transverse-traceless gauge, particles initially at rest have fixed spatial coordinates because ${\Gamma }^{\mu }{}_{00}=0$. Their separation measured in a parallel-propagated orthonormal frame changes: to first order $S^{\hat{A}}=S^A+h_{AB}{S}^B/2$. Hence the physical geodesic deviation is ${\ddot{\xi }}^{\hat{A}}={\ddot{h}}_{AB}{L}^B/2$, whereas the coordinate displacement obeys ${\ddot{\xi }}_{\mathrm{coord}}^A=0$. The extra ${\partial }_0{\Gamma }^A{}_{0B}{L}^B$ term in the coordinate expansion of the second covariant derivative reconciles the two results. This applies to infinitesimal separations and linear metric perturbations.

Use signature $(-,+,+,+)$, $x^0=ct$ and $R^{\rho }{}_{\sigma \mu \nu }={\partial }_{\mu }{\Gamma }^{\rho }{}_{\nu \sigma }-{\partial }_{\nu }{\Gamma }^{\rho }{}_{\mu \sigma }+\cdots$. For a weak stationary metric, the Christoffel symbol satisfies ${\Gamma }^i{}_{00}=-\frac{1}{2}{\partial }_i{g}_{00}$ to first order. Differentiation gives the displayed Riemann curvature tensor component; products of connections are second order. For $g_{00}=-1+2GM/(c^{2}r)$ it is $GM({\delta }_{ij}-3x_i{x}_j/r^2)/(c^2{r}^3)$. Its radial eigenvalue is negative and its transverse eigenvalues positive, giving radial stretching and transverse compression in the geodesic deviation equation with a positive curvature term on the left.