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It is well known that all causally connected pairs of events are either time-like or light-like separated. Is the inverse -'All time-like or light like separated events are causally connected'- also true? It seems that the inverse can be true but not necessarily true. This seems to be the case because the set of points in the future light cone of an event represents the set of events that can possibly be influenced by a signal of velocity less than $c$. It does not represent the set of points that are all 'caused' by the event at the origin of the Minkowski diagram

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The logical converse (not inverse) of your statement is not necessarily true, as you suggest. My deciding to not eat my apple at lunch today has no causal influence on the eventual fate of the sun.

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