Perhaps you know the question what the captain of a spaceship trapped inside the black hole event horizon shall do in order to maximize the time left to being sucked into the singularity. I know the following idea (and I used to believe it over years :-) The geodesic equation of "free-fall" motion and the proper time calculation are both based on the same integral, namely [tex]S[C] = \int_C ds[/tex] As geodesics maxime S and as the question is to maxime proper time, the answer is simply: "free-fall"! So the caption should not start the engine and try to accelerate outwards, try to reach an orbit around the singularity or something like that. This reasoning is well-know from the twin paradox, too. But thinking about it in more detail shows that the two scenarios are not equivalent. For the twin paradox the situation is as follows: both twins start from one spacetime point [tex]C_1(x^0=0) = C_2(x^0=0)[/tex] and they will meet again after some time (coordinate time T) in the same spacetime point [tex]C_1(x^0=T) = C_2(x^0=T)[/tex] [tex]x^0[/tex] is the coordinate time in one reference frame. For the spaceship the situation is different: one compares curves (labelled by an index a) all starting start from one spacetime point [tex]C_a(x^0=0)[/tex] but the end of the curves is not always in the same spacetime point! Instead one has [tex]C_a(x^0=T_a) \to x^\mu = (x^0=T_a, 0, \theta, \phi)[/tex] where I have used polar coordinates; the singularity is located at radial coordinate 0, angles are unspecified. Any other coordinate system will do as well. Now the problem is that we have to check this larger class of curves, namely all curves which eventually meet the singularity. But this applies to all physically allowed curves. That means that we can classify the set of all those curves by the coordinate time when they will meet the singularity. Of course we can calculate the proper time and the coordinate time when a geodesic (free-fall curve) will meet the singularity; let's call this coordinate time [tex]T^0[/tex]. And we know that for all other curves meeting the singularity at the same coordinate time [tex]T_a = T^0[/tex], the geodesic will be the one which maximizes the proper time for this specific subset of curves. But how can we prove that all other curves (resulting from some acceleration of the spaceship) meeting the singularity at a different coordinate time [tex]T_a \neq T^0[/tex] have smaller proper time?