Free massive and null particle geodesics

On a prescribed smooth time-oriented Lorentzian background, stationarity for the massive action of Massive spinless test-particle action is defined on a finite segment $[a,b]$ with the stated fixed-endpoint $C^2$ variations. A stationary massive path on that segment satisfies $D_{\tau }U=0$ in proper-time parameter, with $P=mU$ and $m>0$ constant. In the separately adopted null sector of Massless test-particle action and momentum scale, stationarity for the einbein action is likewise defined on a finite segment, using independent fixed-endpoint $C^2$ variations of the curve and smooth interior variations of the einbein. Its nonzero future-null branch has momentum $P$ satisfying $D_{\lambda }P=0$ in an affine parameter $\lambda$ with $dx\mathrm{/}d\lambda =P$. These equations hold locally on the interiors of the finite segments; their ODE solutions extend uniquely to maximal solutions on open intervals, with no action integrated over an infinite interval. For any ${\tau }_0\in \mathbb{R}$ and $p_0\in M$, an initial vector $U_0\in T_{p_0}M$ that is future timelike and satisfies $g_{p_0}(U_0,U_0)=-c^2$ determines a unique local massive solution with $x({\tau }_0)=p_0$, $U({\tau }_0)=U_0$, and a unique maximal solution on an open interval containing ${\tau }_0$. For any ${\lambda }_0\in \mathbb{R}$ and $p_0\in M$, a nonzero future-null vector $P_0\in T_{p_0}M$ determines a unique local null solution with $x({\lambda }_0)=p_0$, $P({\lambda }_0)=P_0$, and a unique maximal solution on an open interval containing ${\lambda }_0$. Neither maximal interval is asserted to be all of $\mathbb{R}$.

More explicitly, there are open intervals $J\ni {\tau }_0$ and $J^{\mathrm{\prime }}\ni {\lambda }_0$ such that the massive local solution is a $C^2$ curve $x:J\to M$ with $U=dx\mathrm{/}d\tau$, and the null local solution is a $C^2$ curve $x:J^{\mathrm{\prime }}\to M$ with $P=dx\mathrm{/}d\lambda$; their stated initial values are imposed at the indicated parameter values.