Geodesic equation

A geodesic has zero covariant acceleration and in coordinates satisfies ${\ddot{u}}^k+{\Gamma }_{ij}^k{\dot{u}}^i{\dot{u}}^j=0$.

A regular curve with tangent $v=dx/du$ is an unparametrized geodesic of an affine connection if ${\nabla }_{v}v=f(u)v$. Choose a new parameter $s$ with $ds/du=r(u)>0$ and $r^{\prime }/r=f$. Then $w=dx/ds=v/r$ obeys ${\nabla }_{w}w=r^{-2}({\nabla }_{v}v-(r^{\prime }/r)v)=0$. Conversely a reparametrized affine geodesic has acceleration proportional to its tangent. This criterion concerns the geometric curve, not a fixed parametrization.

For a Lorentzian metric and timelike $v=dx/du$, put $L=\sqrt{-g(v,v)}>0$ and $\mathcal{L}=L^2/2$. The Euler-Lagrange equations of $\mathcal{L}$ give ${\nabla }_{v}v=0$, while those of $L$ give ${\nabla }_{v}v=(\dot{L}/L)v$. Taking $ds/du=L$ removes the latter tangential acceleration and gives a unit timelike tangent. Thus the two actions have the same geodesic images; the quadratic action singles out an affine parameter. They do not have the same solutions for an arbitrary fixed parameter: $t=u^2$, $\mathbf{x}=0$, $u>0$, in Minkowski spacetime solves the length equation but not the energy equation. Null curves are excluded since $L=0$.

For a metric $g_{ij}$, the Lagrangian $L=g_{ij}{\dot{x}}^i{\dot{x}}^j/2$ has the affinely parametrized geodesic equations as its Euler-Lagrange equations. Its value is half the constant squared norm of the tangent.

For a smoothly parametrized curve on an embedded surface in Euclidean space, the geodesic equations hold exactly when the ambient acceleration is normal to the surface.

The velocity of a surface curve is tangent. If its acceleration is normal, thenso every affinely parametrized geodesic has constant speed.

A constant-speed intersection with a plane containing every surface normal along the curve is a geodesic.

An affine parameter on a geodesic is one for which the coordinate equation has the form ${\ddot{x}}^k+{\Gamma }_{ij}^k{\dot{x}}^i{\dot{x}}^j=0$. Its affine changes $\lambda \mapsto A\lambda +B$, with $A\ne 0$, preserve this form.