As I understand it, the(adsbygoogle = window.adsbygoogle || []).push({}); proper time, ##\tau##, between to events in spacetime is defined in terms of the spacetime interval ##ds^{2}=\eta_{\mu\nu}dx^{\mu}dx^{\nu}##, such that $$d\tau =\sqrt{-ds^{2}}$$ (where we are using the "mostly +" signature with ##c=1##).

Now, fortime-likeintervals, for which ##ds^{2}<0##, it is clear that proper time is well-defined since the quantity ##\sqrt{-ds^{2}}## ispositive, and furthermore, one can always find a frame in which the two events occur at the same point in space, such that one can construct a worldline connecting the two events, along which an observer can travel, at rest with respect to both events, such that ##d\tau =\sqrt{-ds^{2}}=dt##.

However, why is it the case that forspace-like, ##ds^{2}>0##, andlight-likeintervals, ##ds^{2}=0##, the notion of proper time is undefined (or perhaps ill-defined)?

For thespace-likecase, I get that heuristically, one cannot construct a path between the two events along which an observer can travel and so in this sense proper time is meaningless, since a worldline connecting the events does not exist and so no clock can pass through both events. However, can this be seen purely by examining the definition of proper time in terms of the spacetime interval? Is it simply that the quantity ##\sqrt{-ds^{2}}## will become imaginary and so clearly cannot be used to represent any physical time interval?

Likewise, for alight-likeinterval, only a beam of light can pass between both events and since there is no rest frame for light one cannot construct a frame in which a clock is at rest with respect to the beam and passes through both events. However, purely in terms of the spacetime interval, is it simply because the quantity ##\sqrt{-ds^{2}}## equals ##0##, and so the notion of proper time is ill-defined since there is no invertible map between reference frames (here I'm thinking in terms of time dilation, ##t =\gamma\tau## and so for a light-like interval, ##\gamma\rightarrow\infty## meaning that the inverse relation ##\tau =\frac{t}{\gamma}## is ill-defined)?!

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# I Why is proper time undefined for spacelike/lightlike paths?

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