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I've started studying general relativity course and now I have a question about proper time. Consider functional $$S[x]=-\int_A^B ds,$$ where $A$, $B$ are fixed points of the space-time and $ds^2=dt^2-dx^2$ (let our space-time be two-dimensional without loss of generality). Finding its minimum is equivalent to maximizing proper time $s=\int_A^B ds$ and it a well-known fact that maximizers of proper time are straight lines, as it can be easily checked by writing Lagrange equation for $\mathcal{L}(t,x,\dot{x})=-\sqrt{1-\dot{x}^2}$. So my question is: what are minimizers of proper time?

I've heard that there are, hm, lots of them, but I can't write down any. If one wants to minimize $s$, he takes lagrangian $\mathcal{L}=\sqrt{1-\dot{x}^2}$ and he obtains that $$\ddot{x}=0,$$ so minimizers necessarily are straight lines. But we have already proved that they are maximizers, so I come to contradiction. Where am I wrong?

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  1. Given two (timelike separated) points in Minkowski space, there is a unique timelike curve that maximizes the proper time, namely the straight line, as OP already mentions. However there is no timelike curve that minimizes the proper time. Nevertheless proper time does have an infimum, namely zero. This is essentially because a massive point particle can always fly a bit closer to the speed of light without reaching it.

  2. For a general action functional, the Euler-Lagrange (EL) equations yield stationary configurations. Extremal configurations might not exist.

    Example. For a differentiable function $f:I\to \mathbb{R}$ on an (open or closed) interval $I$, recall that a stationary point is neither a necessary nor a sufficient condition for an extremum for $f$. Similar statements are true in calculus of variations.

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