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A spacetime is geodesically complete if every maximal geodesic has an affine parameter ranging over all of $\mathbb{R}$. For timelike geodesics this is equivalent to unbounded proper time in both directions. An extendible geodesic segment can be prolonged in the same spacetime; an inextendible geodesic cannot. A finite coordinate endpoint need not imply finite affine parameter.

For the Kruskal spacetime, use $U,V$ withIn the right exterior $U<0<V$. A truncated ray $U=U_0<0$, $V_1<V<V_2$ is an extendible geodesic of radial null type: neither artificial endpoint is a spacetime boundary. A future ray $U=U_0>0$ in the black hole reaches $UV=1$, hence $r=0$, and is inextendible geodesic and future null-geodesically incomplete. Its Killing energy $E\ne 0$ gives $dr/d\lambda =-|E|$, so the Schwarzschild singularity occurs at finite affine parameter. The maximal continuation toward the past supplies the other half of this same null geodesic.

There is no inextendible, complete radial timelike geodesic in positive-mass Kruskal spacetime. This requested example is impossible as printed. For a radial timelike geodesic, the conserved Killing energy and normalization giveIf $E^2<1$, $r$ has at most one turning point, a maximum $r_{\max }=2M/(1-E^2)$; a maximal trajectory runs from the white hole Schwarzschild singularity to the Schwarzschild singularity. If $E^2\ge 1$, there is no finite turning point; one end can lie at infinity but the other reaches $r=0$. The exceptional $E=0$ trajectory through the bifurcation surface also reaches $r=0$ in both time directions. Near $r=0$,whose integral is finite. Constant-$r$ radial timelike curves are accelerated, not geodesics.

Two plausible repairs have different meanings. Removing “radial” permits a complete circular timelike geodesic at $r=R>6M$, with nonzero angular momentum and proper time ranging over $\mathbb{R}$. Replacing “timelike” by “null” permits a complete horizon null geodesic: $U=0$, $V\in \mathbb{R}$, with $V$ an affine parameter. The Penrose diagram shows both repairs explicitly, together with the two valid requested examples; the circular trajectory is only a radial projection and is labelled as nonradial.

The horizontal boundaries are the Schwarzschild singularities, diagonal dashed lines are the Killing horizons, and outer diagonal edges are null infinity.

For radial free fall, let $E=f{\dot{t}}_A>0$ be Alice's conserved Killing energy. The inward branch of the timelike geodesic has $\dot{r}=-\sqrt{E^2-f}$, henceWith $\delta =r-2M$, the near-horizon geodesic equations giveThe constants in the logarithms are understood to make their arguments dimensionless. Therefore $u_A=t_A-r_{\ast }=2t_A+O(1)$. For an outgoing signal reaching the fixed-radius Bob, $t_B=u_A+r_{\ast }(r_B)$, so $t_B=2t_A+O(1)$. The redshift consequently behaves asThis is the answer when $t$ is Bob's Schwarzschild time, as appropriate to the stated observation. It is the Schwarzschild surface gravity $a=\kappa =1/(4M)$. If $t$ instead means the emission Schwarzschild time $t_A$, the same redshift is proportional to $e^{t_A/(2M)}$ and gives $M=1/(2a)$. The two answers use different clocks, not different dynamics. If the measured exponential uses Bob's proper time ${\tau }_B$, its rate is $1/(4M\sqrt{f_B})$. In SI units the reception-time result is $M=c^3/(4Ga)$.

The reception-time exponent also holds for smooth radial infall crossing the future Schwarzschild event horizon with finite nonzero $dr/d\tau$: regular ingoing Eddington-Finkelstein coordinates give finite $v$ there, $u=v-2r_{\ast }\sim -4M\log ({\tau }_H-\tau )$. Thus no special value of $E$ is needed.

tidal forces measure differences in gravitational acceleration across an extended region. Neighboring freely falling particles each have zero four-acceleration, but their separation need not accelerate uniformly. Let $u^{\mu }$ be the tangent to a reference timelike geodesic and ${\xi }^{\mu }$ its infinitesimal separation from a neighboring one, chosen from an affinely parametrized geodesic congruence. Using the Riemann curvature tensor convention specified in Solution 3,Changing the overall curvature convention changes this displayed sign. The covariant derivatives compare vectors by parallel transport, avoiding a coordinate-dependent subtraction of accelerations.

In the Newtonian limit, expand the force field at two nearby points:For $\Phi =-GM/r$, a local orthonormal frame aligned with the radius givesThere is radial stretching and transverse compression; the trace vanishes outside the mass. Two freely falling test masses and a measurement of their changing separation implement gravity gradiometry. Their common acceleration drops out, while a relative acceleration proportional to separation remains. A local inertial frame removes Christoffel symbols at a point, but cannot remove nonzero curvature over a finite region.

The Einstein tensor and its contractions areTo match the supplied Ricci tensor components, use the Riemann curvature tensor conventionThis convention also fixes the sign used for geodesic deviation above. In vacuum, tracing the Einstein field equations gives $-R+4\Lambda =0$. Substitution therefore yields $R_{\mu \nu }=\Lambda g_{\mu \nu }$.

Differentiate the angular field equation to obtainConsequently $R_{00}=\Lambda A=\Lambda g_{00}$ and $R_{11}=-\Lambda /A=\Lambda g_{11}$. The other angular component follows from $R_{33}={\sin }^2\theta R_{22}$. Thus the single angular equation implies all the displayed vacuum equations wherever this coordinate chart is regular.

Integrating $(rA{)}^{\prime }=1-\Lambda r^2$ gives $A=1+C/r-\Lambda r^2/3$. Match $g_{tt}\simeq 1+2\Phi$ with the Newtonian gravitational potential $\Phi =-GM/r$ in the weak-field overlap region to identify $C=-2GM$. Hence the Schwarzschild-de Sitter spacetime hasNegative $\Lambda$ instead gives the Schwarzschild--anti-de Sitter black hole case. This matching fixes the normalization locally; a spacetime with nonzero $\Lambda$ is not asymptotically flat.

For timelike geodesics in the orbital plane, take proper time as parameter and use $L=(A{\dot{t}}^2-{\dot{r}}^2/A-r^2{\dot{\varphi }}^2)/2$. The cyclic coordinates give $A\dot{t}=k$ and $r^2\dot{\varphi }=h$; the angular canonical momentum is $-h$ in this signature. The four-velocity normalization $2L=1$ then givesHere $k$ is conserved Killing energy per unit rest mass in the chosen time normalization, and $h$ is conserved angular momentum per unit rest mass. There is no general interpretation of $k$ as energy measured at an asymptotically flat infinity when $\Lambda \ne 0$.

For radial motion $h=0$, differentiation gives $2\dot{r}\ddot{r}=-A^{\prime }\dot{r}$. The radial Euler-Lagrange equation extends the resulting expression through a turning point where division by $\dot{r}$ would fail. ThusThis is the proper-time second derivative of the areal radius, not a nonzero four-acceleration of the freely falling particle. Positive $\Lambda$ contributes an outward term; negative $\Lambda$ reinforces inward attraction. Restoring units in the slow-motion, weak-field Newtonian limit givesThe extra term describes the gravitational effect of a cosmological constant, not drag or a contact force. The approximation requires $GM/(r{c}^2)\ll 1$, $|\Lambda |r^2\ll 1$ and speed much less than $c$.

For geodesic motion of uncharged dust, use the pressureless stress-energy form, $T^{\mu \nu }=\rho u^{\mu }{u}^{\nu }$ and $u_{\nu }{u}^{\nu }=c^2$. stress-energy conservation expands toContracting with $u_{\nu }$ kills the second term, since $u_{\nu }{\nabla }_{\mu }{u}^{\nu }=\frac{1}{2}{\nabla }_{\mu }(c^2)=0$. Hence ${\nabla }_{\mu }(\rho u^{\mu })=0$, and wherever $\rho >0$,On a flow line $u^{\mu }=d{x}^{\mu }/d\tau$, so this becomes $d^2{x}^{\nu }/d{\tau }^2+{\Gamma }^{\nu }{}_{\alpha \beta }(d{x}^{\alpha }/d\tau )(d{x}^{\beta }/d\tau )=0$. Each relativistic dust element therefore follows a timelike geodesic with proper time as an affine parameter. pressure gradients or external nongravitational forces would add an acceleration term.

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.

Interpret the orbit in the Schwarzschild metric as an unforced circular timelike geodesic and choose its plane as $\theta =\pi /2$. Put $f=1-2\mu /r$. The radial geodesic equation at constant radius uses the Christoffel symbolsand gives the Angular velocity of a circular Schwarzschild geodesicThe proper-time normalization then yieldsOver one revolution, the Proper period of a circular Schwarzschild geodesic is thereforeBoth the gravitational and orbital kinematic contributions to time dilation are included. Circular timelike geodesics exist for $r>3\mu$; they are stable for $r>6\mu$ and unstable for $3\mu <r<6\mu$, with marginal stability at $6\mu$.

If “circular orbit” instead means a forced trajectory with an externally chosen angular velocity, the radius alone does not fix the answer. Its elapsed time is $\Delta \tau =\Delta t\sqrt{1-2\mu /r-r^2{\Omega }^2/c^2}$, or $2\pi \sqrt{1-2\mu /r-r^2{\Omega }^2/c^2}/|\Omega |$ per revolution. The geodesic interpretation supplies the missing angular velocity.

This is the past endpoint of complete inertial timelike geodesics in the Minkowski conformal compactification, at $(T,R)=(-\pi ,0)$. It is the time reversal of future timelike infinity and is reached at infinite physical proper time to the past.

An affine parameter on a timelike geodesic can be scaled to proper time. With metric signature $(+---)$ this fixes the tangent-vector norm to $c^2$. A further constant rescaling preserves the affine geodesic equation but changes that norm; conserved energy and angular-momentum constants must be rescaled consistently. The unit term in a proper-time-normalized radial energy equation is consequently a normalization choice, not a property of every affine parameter.

Every maximal radial timelike geodesic in positive-mass Kruskal spacetime is incomplete in at least one direction. Its equations are ${\dot{r}}^2=E^2-1+2M/r$ and $\ddot{r}=-M/r^2$. A finite turning point is a maximum, and each trajectory reaches $r=0$ at one or both ends. The remaining proper time is finite because $d\tau \sim \sqrt{r/(2M)}dr$. Complete circular timelike geodesics exist, but they have nonzero angular momentum.

For two equal-mass inward timelike geodesics of unit specific Killing energy, their invariant center-of-mass energy has this finite horizon limit. If both are captured from infinity, Schwarzschild marginally bound capture gives $E_{\mathrm{com}}<m\sqrt{20}$, with $m\sqrt{20}$ a supremum. No such universal bound follows for particles prepared inside the angular-momentum barrier.

For a small radial excursion about a stable circular timelike geodesic in Schwarzschild spacetime, the inverse radius oscillates with azimuthal frequency $\sqrt{1-6GM/(c^{2}R)}$. A satellite launched tangentially returns to its launch radius once per radial period. A later exact encounter with a point station at the launch azimuth requires the exact radial-to-azimuthal rotation ratio to be rational. An irrational ratio instead gives dense, arbitrarily close recurrence, which still threatens objects of finite size. The linearized rotation ratio is an approximation; exact orbital closure uses the actual orbit's radial advance.

A timelike geodesic with unit specific Killing energy can plunge from infinity into a Schwarzschild black hole only if its specific angular momentum satisfies $|h|<4M$. The limiting case $|h|=4M$ asymptotes to the unstable orbit at $r=4M$. A geodesic already inside the angular-momentum barrier can cross the event horizon with larger $|h|$, so the origin-at-infinity condition matters.

In synchronous coordinates, fixed spatial-coordinate worldlines are timelike geodesics, and $t$ is their proper time. The metric has unit lapse function and zero shift vector in time units consistent with the displayed factor $c^2$. Indeed $g_{tt}=c^2$ and $g_{ti}=0$ give ${\Gamma }^{\mu }{}_{tt}=0$. Locally, an irrotational freely falling congruence permits this construction before crossings of the coordinate worldlines. Comoving observers in a Friedmann-Lemaître-Robertson-Walker metric provide the standard cosmological example.

A test particle is an idealized body whose own fields and effect on its environment are negligible. Its motion probes an externally prescribed field. A gravitational test particle follows a timelike geodesic when nongravitational forces are absent; appreciable self-gravity requires a more complete treatment.

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.

For the Schwarzschild metric with $\mu =GM/c^2$, let $k=(1-2\mu /r)\dot{t}$ and $h=r^2\dot{\varphi }$ on an equatorial timelike geodesic. The normalization of the four-velocity gives ${\dot{r}}^2+V(r)=c^2(k^2-1)$, where $V(r)=(1-2\mu /r)h^2/r^2-2\mu c^2/r$. A circular geodesic obeys $V^{\prime }(R)=0$, giving the displayed constants and requiring $R>3\mu$. Its coordinate angular velocity is ${\Omega }^2=GM/R^3$, and its proper clock rate is $d\tau /dt=\sqrt{1-3\mu /R}$. Differentiating $V$ twice at the circular solution gives stability for $R>6\mu$.

A timelike geodesic has timelike tangent and vanishing covariant acceleration in an affine parameter. Choosing proper time gives $g(\dot{\gamma },\dot{\gamma })=-1$.

Physically, an ideal massive test particle in free fall follows a timelike geodesic when nongravitational forces and appreciable self-gravity can be neglected. The normalization above uses $c=1$; in SI units it is $g(u,u)=-c^2$. Zero covariant acceleration does not mean constant coordinate velocity: a dropped object accelerates relative to the ground. Conversely, a person standing on the ground is pushed away from free fall by the supporting surface. A photon follows a null geodesic, whose parameter cannot be proper time. See Tong's geodesics in spacetime.

In a positive-time patch, proper-time first integrals give $\dot{x}=p{t}^2$ and ${\dot{t}}^2=t^2+p^2{t}^4$. If $p\ne 0$, the unparametrized timelike geodesics are $t^2=(x-x_c{)}^2-p^{-2}$ on an appropriate branch; in the expanded quadratic form their discriminant is positive. If $p=0$, there is the separate family $x=x_c$, $t=t_{\ast }{e}^{\pm \tau }$. A description using only the nonvertical quadratic family omits genuine timelike geodesics.