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For smooth spatial fields $E^i,C^i{}_j$, define a torsion-free connection on $\mathbb{R}\times {\mathbb{R}}^n$ by ${\Gamma }^i{}_{00}=-E^i$, ${\Gamma }^i{}_{0j}={\Gamma }^i{}_{j0}=-C^i{}_j/2$ and all other coefficients zero. The time geodesic equation is $\ddot{t}=0$. For $\dot{t}\ne 0$, use $t$ as affine parameter; the spatial geodesic equation becomes $d^2{x}^i/d{t}^2=E^i+C^i{}_{j}d{x}^j/dt$. In dimension three, $C^i{}_j=2{ϵ}^i{}_{kj}{B}^k$ realizes $\ddot{\mathbf{x}}=\mathbf{E}+2\mathbf{B}\times \dot{\mathbf{x}}$. Such a connection need not preserve a metric; a spacetime description of these trajectories only needs affine geometry.

A smooth nondegenerate component of the fixed-point set of an isometry is a totally geodesic submanifold. If a geodesic starts tangent to the fixed component, its image under the isometry has identical position and tangent. Uniqueness of the geodesic equation makes the two geodesics coincide throughout their common domain, so the original geodesic stays in the fixed set.

Differentiating the spherical-geodesic first integral givesThe same equation follows from the affine geodesic equation after eliminating the arc-length derivatives. The squared first-derivative equation also fixes the normalization constant; a second-order solution must retain that first integral.

For coordinates $(t,x,y,z)$ and line element $d{s}^2=c^{2}d{t}^2-a^2(t)d{\mathbf{x}}^2$, these are the nonzero Christoffel symbols. The prime means differentiation with respect to $t$, and coordinate zero is $t$, not $ct$. They follow either from the metric-derivative formula or the Euler-Lagrange equations for the affine geodesic Lagrangian. Both mixed lower-index orders must be counted to obtain the factor of two in the spatial geodesic equation.

In geodesic normal coordinates at $p$, radial geodesics have coordinates $x(t)=tv$. The geodesic equation at $t=0$ gives ${\Gamma }_{jk}^i(p)v^j{v}^k=0$ for every $v$. The Levi-Civita connection is torsion free, making these coefficients symmetric in $j,k$; the polarization identity then gives ${\Gamma }_{jk}^i(p)=0$. This does not imply vanishing curvature, which depends on derivatives of the coefficients.

Let $X$ be a Conformal Killing vector field with conformal Killing factor $\lambda$, and let $U$ be the tangent vector of a geodesic with affine parameter $s$. The product rule, metric compatibility and geodesic equation giveEvery null geodesic therefore has the conserved quantity $g(X,U)$. On a timelike geodesic with proper time as parameter, the derivative is $-\lambda$ in metric signature $(-,+,+,+)$. With a nonaffine parameter satisfying ${\nabla }_{U}U=fU$, an extra term $fg(X,U)$ appears; the conserved expression for a null geodesic is tied to its affine parameter normalization.

For the Riemannian exponential map ${\exp }_p(v)={\gamma }_v(1)$, uniqueness of the geodesic equation gives ${\exp }_p(tv)={\gamma }_v(t)$. Differentiation at $t=0$ yields $(d{\exp }_p{)}_{0}v=v$. Smooth ODE dependence and the inverse function theorem therefore make ${\exp }_p$ a diffeomorphism on a neighborhood of zero, producing geodesic normal coordinates. This is a local assertion and requires no geodesic completeness.

A mechanical Lagrangian quadratic in velocities can be represented by geodesics of a Lorentzian metric with an extra null coordinate. For $L=\frac{1}{2}|\dot{\mathbf{x}}{|}^2-2\mathbf{A}\cdot \dot{\mathbf{x}}-U$, use $g=d{\mathbf{x}}^2+2dt(du-2\mathbf{A}\cdot d\mathbf{x}-Udt)$. The null Killing vector field ${\partial }_u$ gives the conserved momentum $p_u=t\prime$. Fixing $p_u=1$ projects the geodesic equation to the mechanical Euler-Lagrange equations. This is useful for relating mechanical forces to spacetime geometry.

A fixed-endpoint variation of a curve $\gamma :[a,b]\to S$ is a smooth family $\Gamma (s,t)$ with $\Gamma (0,t)=\gamma (t)$, $\Gamma (s,a)=\gamma (a)$ and $\Gamma (s,b)=\gamma (b)$. Its variation vector field $V(t)={\partial }_s\Gamma (0,t)$ vanishes at the endpoints. Consequently integration by parts in the first variation of energy of a curve has no boundary contribution, and stationarity is characterized by the geodesic equation.

A geodesic is a curve whose tangent vector is parallel along itself under the Levi-Civita connection. With an affine parameter it has zero covariant acceleration:This is the Riemannian manifold version of moving straight ahead with constant speed. Its coordinate form is the geodesic equation. In Euclidean space it becomes $\ddot{\gamma }=0$, giving straight lines. On an embedded surface with its induced metric, the ordinary acceleration of such a curve is normal to the surface: all acceleration is used to follow the surface, rather than to turn within it.

On a unit round sphere, the great circle $\gamma (t)=(\cos t,\sin t,0)$ has $\ddot{\gamma }(t)=-\gamma (t)$, which is normal to the sphere. Two points at $t=0$ and $t=\pi /3$ can be joined along that circle by the short arc of length $\pi /3$ or the long arc of length $5\pi /3$. Each admits a constant-speed geodesic parametrization, yet only the short arc realizes their great-circle distance.

Every geodesic minimizes arc length on sufficiently short subarcs. A long segment can cease to be globally minimizing, as the sphere example shows; the first loss from a starting point is described by the cut locus. Thus “locally straight” and “shortest route between these endpoints” are distinct assertions. On a circular cylinder, different winding gives the same distinction, made explicit by Riemannian distance on a circular cylinder.

This provides a useful division of work. The geodesic equation predicts a path from an initial point and tangent vector; comparing arc lengths decides which candidate solves a global shortest-route problem. Variations of energy of a curve give the equation as an Euler-Lagrange equation, offering a proof outline and a connection with classical mechanics. ^[ref].

Each Killing vector field supplies a conserved quantity for freely moving test particles: it turns a geometric symmetry into a first integral of the geodesic equation. Let $\gamma (\lambda )$ be a geodesic with affine parameter $\lambda$, tangent vector $U^{\mu }=d{x}^{\mu }/d\lambda$, and ${\nabla }_{U}U=0$. Wherever a Killing vector field $K$ is defined along the geodesic,The statement applies to Riemannian manifolds and to timelike geodesics, null geodesics and spacelike geodesics in a Lorentzian manifold. It requires an affine parameter; for a massive test particle, proper time is one choice. See Tong, section 4.3.2, equation (4.32).

For a worked example in the Euclidean plane, take $\gamma (t)=(1+t,2-t)$ with affine parameter $t$ and velocity $U=(1,-1)$. The Euclidean Killing vector fields for translation and rotation giveThe rotational field along the geodesic is $K_{\mathrm{rot}}(\gamma (t))=(t-2,1+t)$: it changes from $(-2,1)$ at $t=0$ to $(-1,2)$ at $t=1$, yet its pairing with $U$ stays $-3$. With toy mass $m=1$ and $t$ interpreted as time, the first constant is the $x$ component of linear momentum, and the second is the signed $z$ component of angular momentum about the origin. Their numerical values use toy units. The geodesic is straight; the rotation flow orbits are circles.

This conservation law follows from metric compatibility and the Killing equation:The geodesic equation kills the acceleration term, and the symmetric product $U^{\mu }{U}^{\nu }$ selects the vanishing symmetric derivative. This tensor contraction is the whole proof outline; see Carroll, equation (5.43).

In a stationary spacetime, a chosen time-translation Killing vector field $K$ gives Killing energy. With metric signature $(-+++)$, $c=1$, and a massive test particle's four-momentum $p^{\mu }=m{U}^{\mu }$, the physical normalization isHere $\Phi$ generates the axial rotational symmetry when one exists. In coordinates with $K={\partial }_t$ and $\Phi ={\partial }_{\varphi }$, these are $E=-p_t$ and $L_z=p_{\varphi }$. Both are conserved quantities along the geodesic, even if the metric has a time-angle cross term. A future timelike $K$ makes $E$ positive for a future massive particle; the scale of $E$ depends on the normalization of $K$. Not every spacetime has a global timelike Killing vector field, and a stationary field can become spacelike in an ergoregion of a stationary spacetime. Conservation on the field's domain does not imply a universal global positive energy; see Tong, sections 4.3.1-4.3.2 on the timelike-domain qualification and conserved charges.

For $g_{\mu \nu }={\phi }^2{\eta }_{\mu \nu }$,For a null curve, the last term drops from the geodesic equation; the remaining acceleration is parallel to the tangent and can be removed by reparametrization.

The geodesic flow sends a tangent vector $v$ to the velocity at time $t$ of the unique geodesic with initial velocity $v$. Smooth dependence of solutions of the geodesic equation makes it a smooth flow on the tangent bundle wherever defined.

Suppose reflection in a plane is an isometry of a smooth surface, and its intersection with the plane is a regular curve. That curve is a geodesic when parametrized by arclength. At any intersection point, reflection fixes both the point and every tangent vector to the intersection. It carries the geodesic with these initial data to one with the same data. Local uniqueness of the geodesic equations forces it to remain in the fixed plane. This is the one-dimensional instance of fixed-point sets of Riemannian isometries being totally geodesic.

For the arc-length meridian metric $d{s}^2+f(s{)}^{2}d{\theta }^2$, a unit-speed parallel has $\dot{s}=0$ and $\dot{\theta }=\pm 1/f(s_0)$. Its radial geodesic equation is satisfied exactly when $f\prime (s_0)=0$. Thus extrema of the radius, and any other stationary radii, give geodesic circles. If every parallel is geodesic, the surface is a circular cylinder.

In coordinates $(\theta ,z)$ on a circular cylinder of radius $a$, the induced metric is $a^{2}d{\theta }^2+d{z}^2$. Its coefficients are constant, so the Christoffel symbols vanish and the affinely parametrized geodesic equation is ${\theta }^{\prime \prime }=z^{\prime \prime }=0$. The nonconstant solutions are generators when $c=0$, circular parallels when $d=0$, and helices when both are nonzero. Unrolling the cylinder with coordinate $u=a\theta$ makes all these curves straight lines in the Euclidean metric. Unit-speed parametrization additionally requires $a^2{c}^2+d^2=1$.

For a worked helix on a circular cylinder of radius $1$, choose $\theta (t)=2\pi t,z(t)=2t$ for $0\le t\le 1$. Unrolling gives the straight segment $(u,z)=(2\pi t,2t)$, so its arc length is $\sqrt{4{\pi }^2+4}$. Its acceleration in space points normally to the surface, consistent with the geodesic equation.

This geodesic is not globally shortest between its endpoints: they share the same angular position, and the vertical generator has length $2$. The helix minimizes length among paths with the same endpoints and one full winding, because their unrolled lifts join $(0,0)$ to $(2\pi ,2)$. In general, geodesics minimize sufficiently short subarcs; changing winding can produce a shorter route. Keeping this local/global distinction makes unrolling useful without mistaking every straight lift for an unrestricted shortest path.

For smooth stationary fields on ${\mathbb{R}}^3$, set ${\Gamma }^i{}_{tt}=-E^i$ and ${\Gamma }^i{}_{tj}={\Gamma }^i{}_{jt}={ϵ}^i{}_{jk}{B}^k$, with all other Christoffel symbols zero. This torsion-free connection makes the equation $\ddot{\mathbf{x}}=\mathbf{E}+2\mathbf{B}\times \dot{\mathbf{x}}$ a geodesic equation with time as affine parameter. Arbitrary smooth fields suffice; a metric realization imposes additional potential conditions.

Let $K={\partial }_v$ generate a Killing horizon with constant surface gravity $\kappa$, so ${\nabla }_{K}K=\kappa K$. In geometric optics, an affinely transported light wavevector on the same generator is $k=q(v)K$. The geodesic equation gives $q^{\prime }+\kappa q=0$. Observers related by the isometry generated by $K$ have the same $-K\cdot W$ at their crossings, where $W$ is their four-velocity, so their measured angular frequencies $\omega =-k\cdot W$ obey the displayed relation. A degenerate Killing horizon gives ratio one for equally prepared observers; changing their crossing velocities can produce a separate Doppler effect.

The velocity of a geodesic can be extended to one smooth ambient vector field on a sufficiently short neighbourhood of any parameter value. If its initial velocity is zero, uniqueness of the geodesic equation makes it locally constant, and the zero field works. Otherwise one coordinate has nonzero velocity; the inverse function theorem supplies coordinates straightening the small embedded arc to $(t,0,\dots ,0)$. The coordinate field $\partial /\partial t$ is the required extension. Shrinking the interval is essential: arbitrary fields along self-intersecting curves need not have one ambient extension.

A geodesic $\gamma :I\to M$ is maximal when it has no geodesic extension to a strictly larger interval. Every initial position and velocity determine a unique maximal geodesic through the geodesic equation.