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[tex]\Delta {s^2} = \Delta {x^2} + \Delta {y^2} + \Delta {z^2} - {c^2}\Delta {t^2}[/tex]

and despite which inertial frame we are in, [tex]\Delta s[/tex] for two particular events would be the same.

If I use Lorentz transformation, this can be proved easily. But is there any more "intuitive" way to verify the invariance? Like when (x,y,z,t) describes propagation of light, it'll be trivially true,e.g. [tex]\Delta s=0[/tex] because of the principle of constancy of light velocity.

But what about other cases, when two events don't lie on the same light cone? Of course [tex]\Delta s[/tex] is not 0 but still remains invariant, but how to convince myself it's true without doing the arithmetic manipulation of Lorentz transformation?