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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.

Qmechanic
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Bedabrata Choudhury
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1 Answers1

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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.

Valter Moretti
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