I was just wondering if two events $a$ and $b$ are time-like separated it means that one happens before the other. However, does this guarantee that $a$ always happens before $b$ or $b$ always happens before $a$ in every inertial reference frame or is that not an invariant?
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You may find this animated diagram helpful. The line of black dots that are initially vertical have timelike separation. Their order is invariant. – PM 2Ring Nov 20 '18 at 05:03
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In flat (Minkowski) spacetime, we can choose a time orientation that consistently fixes the distinction between future and past for all timelike intervals, and this ordering is invariant. (It is still not invariant for spacelike intervals.)
For some other spacetime metrics, the presence of closed timelike curves means that such a distinction cannot be made consistently for all timelike intervals — because some timelike intervals loop back into themselves! This occurs, for example, behind the event horizon of a spinning (Kerr) black hole:
Closed timelike curves in the Kerr metric
I think it's fair to say that such situations are usually considered to be unphysical, an artifact of pushing general relativity beyond the domain where it is known to be valid, where quantum effects become important. But until we understand better how to reconcile general relativity with quantum physics, such statements remain speculative.
Chiral Anomaly
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