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In the definition (in one spatial dimension) of $\Delta \tau$ there is the relation:

$(\Delta \tau)^2 = (\Delta t)^2 - (\Delta x)^2$ which is invariant. If $(\Delta x)^2 > (\Delta t)^2$ then there is the characterization "spacelike."

In this case $\Delta \tau$ will be an imaginary number.

My question is: what is the intuitive physical or geometric meaning of imaginary in this context?

(Just a guess; is it related to hyperbolic functions and the Lorentz transformations?)

Thanks

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  • $\begingroup$ @Qmechanic Moments ago I sent you a comment asking about the causality tag. After a quick look at wikipedia, I hastily deleted it. Quite an interesting distinction between spacelike and timelike. Thanks for the pointer. Regards, $\endgroup$ – user41976 Apr 14 '14 at 22:19
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As you have discovered proper time, $\Delta\tau$, can be either real or imaginary. However, this means that it does not necessarily reflect something measurable with a clock.

When it is imaginary, as in the case of a space-like relation of two events, then there is no single clock that can be present at both events. To do so would require having a velocity $v>c$. This is, I suppose, a fancy way of saying that the two events are not causal (event 1 doesn't cause event 2 and vice versa).

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