Geodesic equation

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

On a quadrant $xy\ne 0$, the geodesic Lagrangian gives $y^2\ddot{x}+2y\dot{y}\dot{x}-x{\dot{y}}^2=0$ and $x^2\ddot{y}+2x\dot{x}\dot{y}-y{\dot{x}}^2=0$. Rays $y=mx$ reduce these to $(x\dot{x}{)}^{\prime }=0$ and hence $x^2=a\lambda +b$. The coordinate change $\rho =|xy|/\sqrt{2}$, $\vartheta =\log |y/x|$ gives the locally flat metric tensor $d{\rho }^2+{\rho }^{2}d{\vartheta }^2$. Here $\vartheta$ ranges over the real line, so the polar-plane representation is local, rather than a globally one-to-one map onto the punctured plane. The axes are excluded by the degenerate metric tensor; these rays do not establish continuation through the origin as a regular geodesic.

For a surface metric $Ed{u}^2+2Fdudv+Gd{v}^2$, all coordinate curves are unparametrized geodesics precisely when the displayed identities hold. Parametrize $u=c$ by arc length, so $\dot{v}=G^{-1/2}$, and substitute into the transverse Euler-Lagrange equation for energy of a curve. This gives the first identity; $v=d$ gives the second. These identities concern geodesic images and do not assert that the coordinate variable itself is an affine parameter.

If the oriented coordinate tangent vectors meet at a constant angle with cosine $k$, the first fundamental form has $F=k\sqrt{EG}$. The geodesic coordinate grid identities reduce to $k(\sqrt{E}{)}_v=(\sqrt{G}{)}_u$ and $k(\sqrt{G}{)}_u=(\sqrt{E}{)}_v$. Since positive definiteness gives $|k|<1$, both derivatives vanish. Thus $E=E(u)$ and $G=G(v)$ locally. Changing coordinates to $U=\int \sqrt{E}du$ and $V=\int \sqrt{G}dv$ gives the constant metric $d{U}^2+2kdUdV+d{V}^2$. It is locally Euclidean after a linear change of coordinates, hence has zero Gaussian curvature.

// Target: geometry-and-topology.bigb

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.

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.

A change $\lambda \mapsto A\lambda +B$ rescales a geodesic tangent and all its linear Killing constants by $1/A$. A null norm remains zero and cannot fix $A$, unlike proper-time normalization of a material particle. Ratios of the constants, such as the photon impact parameter $b=|h|/(ck)$ in Schwarzschild spacetime, are independent of the affine scale. A physically normalized four-momentum fixes the photon energy and hence selects a tangent scale.

For physical four-momentum $P^{\mu }=md{x}^{\mu }/d\tau$, the covariant-component equation is $d{P}_{\mu }/d\tau =({\partial }_{\mu }{g}_{\nu \sigma })P^{\nu }{P}^{\sigma }/(2m)$. The equation without $1/m$ instead uses a tangent parametrization, or the affine parameter $\lambda =\tau /m$ when the tangent is the physical momentum.