Could anyone explain how Δs is related to the proper time interval?
There's no real difference between proper time and [itex]ds[/itex] except for (maybe) a factor of [itex]\pm c[/itex] to make sure that a real trajectory has a positive proper time and that the units are right. Either way, proper time is the analogue of arc length in Euclidean spaces, and for a curved trajectory, one integrates to get the right result (the same way you would in 3D).
Δs is really c*ΔTau where ΔTau is the proper time interval. In Minkowski space and using the (+---) sign convention, when Δs2 is positive, then the proper time interval is real and represent the proper time of a clock that moves inertially between the two events. If Δs2 is zero then it represents a light like interval. (i.e. ΔTau is zero). If Δs2 is negative, the proper time interval is imaginary and in that case, no real particle or physical clock can physically travel between those two events and the interval is said to be spacelike and after reversing the signature to (-+++) represents the proper distance (ruler) measurement between the two events.
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