See Rigid Designators :
Kripke famously argues that because a rigid designator designates the same object in all possible worlds, an identity statement in which both designators are rigid must be necessarily true if it is true at all.
‘Hesperus = Phosphorus’ is necessarily true if true at all because ‘Hesperus’ and ‘Phosphorus’ are proper names for the same object. Like other names, Kripke maintains, they are rigid: each designates just the object it actually designates in all possible worlds in which that object exists, and it designates nothing else in any possible world. The object that ‘Hesperus’ and ‘Phosphorus’ name in all possible worlds is Venus.
a rigid designator designates its designatum in every possible world containing the designatum and in other possible worlds the designator fails to designate. In places, Kripke suggests that this is his idea:
When I use the notion of a rigid designator, I do not imply that the object referred to necessarily exists. All I mean is that in any possible world where the object in question does exist, in any situation where the object would exist, we use the designator in question to designate that object. In a situation where the object does not exist, then we should say that the designator has no referent and that the object in question so designated does not exist (Kripke, “Identity and Necessity”, 1971, p. 146).
