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Can one consider the fact that the partial order among a set of time-like separated events is the same in all inertial frames of reference as a criterion to precisely identify causally connected events? Is this the only physical criterion for identifying causality(I neglect the other philosophical arguments and criterion about causality like the notion of 'constant conjunction' provided by David Hume) ?

p.s- By 'Physical Criterion' I mean a precise mathematical condition ,like the in-variance of partial order among time-like separated events under Lorentz Transformation .

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The answer is essentially "yes" to both questions, although to be precise, two events don't have to be timelike separated to be causally connected; they can instead be null (lightlike) separated. More precisely, for events $x$ and $y$:

  • $x$ chronologically precedes $y$ if there exists a future-directed timelike curve from $x$ to $y$.

  • $x$ strictly causally precedes $y$ if there exists a future-directed causal (non-spacelike) curve from $x$ to $y$.

  • $x$ causally precedes $y$ if $x$ strictly causally precedes $y$ or $x$=$y$.

  • $x$ horizmos $y$ if $x$ causally precedes $y$, but $x$ doesn't chronologically precede $y$. (I.e., if $y$ is on $x$'s future light cone.)

For more information, see Wikipedia's Causal structure article.

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