Maximization of proper time between two timelike events in Minkowski space So it is accepted that the path that maximizes the proper time between two timelike separated events in Minkowski space is a straight line (in Minkowski space). I am having trouble deriving this from the expression of the proper time. The idea I have is that one should try to find the extremum of the proper time by solving an Euler-Lagrange type equation in Minkowski space (but I have no idea about how to do this) and arrive at an equation of motion for a straight line in Minkowski space. I also don't know what the equation of a straight line might look like in Minkowski space. (I know what a differential line element looks like, but not what a straight line looks like.)
So I tried solving the Euler-Lagrange equations for this action in one-dimensional Euclidean space:
$$\int_{t_{0}}^{t_{1}}\sqrt{1-\dot{x}^{2}}\,{\rm d}t$$
and arrived at an equation of a straight line in Euclidean space, which I am sure isn't a straight line in Minkowski space.
Help on how to do this would be appreciated.
 A: Here is the proof. Consider two events $p$ and $q$ connected by a timelike segment, i.e., a timelike geodesic of Minkowski spacetime:
$$\gamma(\tau) = p + \tau{\bf n}\:, \quad {\bf n}:= \frac{\vec{pq}}{\sqrt{- g(\vec{pq}, \vec{pq})}}\:.$$
Since the constant tangent unit vector ${\bf n}$ to the segment is future directed and timelike, we can fix a Minkowski reference frame $t,x$, where  $x\in \mathbb{R}^3$,  such that the segment is a portion of the $t$ axis from $t_p$ to $t_q$. In particular, its proper-time length turns out to be
$$\sqrt{- g(\vec{pq}, \vec{pq})} = \sqrt{(t_q-t_p)^2 -0^2-0^2-0^2} = t_q-t_p\:.$$ With this choice, exploiting the said coordinates,  the length of every timelike curve  joining the two events takes the form
$$\int_{t_{p}}^{t_{q}}\sqrt{1-\dot{x}^{2}}\,{\rm d}t\leq \int_{t_{p}}^{t_{q}} 1\,{\rm d}t= t_q-t_p\tag{1}$$
where I used the fact that the path is timelike so that  its velocity satisfies $|\dot{x}|\leq 1$ and thus
$$\sqrt{1-\dot{x}^{2}}\leq 1.$$
Inequality (1) implies that the proper time interval  (the length of a curve joining two timelike connected events) attains its maximum along timelike geodesics, i.e., timelike segments in Minkowski spacetime.
