Why does 'proper length' exist as a notion?

Question

$$\text{proper time}= \tau= \sqrt{dt^2-d\mathbf{s}^2}$$

$$\text{proper length}= L= \sqrt{-dt^2+d\mathbf{s}^2}$$

What tangible benefit is brought about by calling $i \tau$ 'proper length' (applying when $\Im(L)=0$ (the spacetime intervals are spacelike))?

Could one extend the notion of the interval to simulatneusly cover both proper time and distance? Call this $\text{Interval}=dt^2-d\mathbf{s}^2$, which can be imaginary. Is it impossible to merge proper time and length because they are fundamentally different things, as proper time is a explicitly measurable quantity, but proper distance must be determined implicitly?

My problem is that I don't understand the rationale behind having proper length, rather than a cheap mathematical trick to make the interval real when $ dt^2-d\mathbf{s}^2<0$.

Answer 1

This isn't complex analysis. There is no reason to complexify anything. These notions do not both exist for a single pair of events. Either two events are timelike separated from one another, and there is a proper time between them that all observers agree on, or the two events are spacelike separated, and there is a proper distance between them that all observers agree on.

This is just a fundamental result of having a metric isn't positive definite. The invariant interval between the two events tells us whether they are spacelike or timelike separated, sure, and that's why proper time and proper distance are closely related, but they are not the same thing, and I think a complexification just confuses things.

Edit: the proper distance between two events is the distance that would be measured between them in the frame in which they are simultaneous. This is also called proper length because it measures the length of an object in its own rest frame.