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The globally hyperbolic maximal Cauchy development of appropriate Reissner-Nordstrom spacetime data obeys the Penrose singularity theorem when it contains a closed trapped surface. Its resulting null geodesic incompleteness can include generators reaching a smoothly extendible inner Cauchy horizon in finite affine parameter. The full analytic extension is not globally hyperbolic and cannot be substituted into that version of the theorem. Incompleteness alone does not establish curvature blowup at every endpoint.

A pregeodesic is a curve whose tangent obeys ${\nabla }_{U}U=\kappa U$ for some scalar $\kappa$. A change of parameter makes it a geodesic with an affine parameter. A varying speed along a geodesic changes its parameter, not its image.

The two radial null geodesic families are $v=\text{constant}$ and $dr/dv=(1-2M(v)/r)/2$. The first has affine radius, while the second has tangent $k={\partial }_v+(f/2){\partial }_r$ obeying ${\nabla }_{k}k=(M/r^2)k$, so advanced time is generally a nonaffine parameter. An affine parameter $\lambda$ for the outgoing family satisfies $d\log (dv/d\lambda )/dv=-M/r^2$.

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

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 $L=0$.