It is the properties of the meter stick that maintain its rigidity in all possible worlds where the same accessibility relations (laws of nature) are the same. — Wallows

The metre stick is not rigid. It might be different lengths in other possible situations. But the metre is rigidly designated by "one metre"

And accessibility is not about laws of nature. It's more about which individuals are in which world. But the explanation needs some decent formal logic. Someone else might like to explain it for us.

Then it is instantiated or obtains through something that cannot be disputed, such as the laws of physics or nature. — Wallows

We can't dispute the las of nature? Shoot the physicist then, we don't need them anymore.

The metre stick is not rigid. It might be different lengths in other possible situations. But the metre is rigidly designated by "one metre" — Banno

Let's at least try and not be obtuse here. The length of the meter stick is rigid in all possible worlds where the accessibility relation of one world also guarantees nomological necessity in another. Those accessibility relations, or if you prefer, "properties" of said object are outlined in the OP's quoted text.

See:

https://en.wikipedia.org/wiki/Accessibility_relation#Philosophical_applications

One of the applications of 'possible worlds' semantics and the 'accessibility relation' is to physics. Instead of just talking generically about 'necessity (or logical necessity),' the relation in physics deals with 'nomological necessity.' The fundamental translational schema (TS) described earlier can be exemplified as follows for physics:

(TSN) P is nomologically necessary means that P is true at all possible worlds that are nomologically accessible from the actual world. In other words, P is true at all possible worlds that obey the physical laws of the actual world.

So, that is to say, that the stipulation that the meter stick is the same as a meter, is true in all possible worlds, and talking otherwise would invoke some nonsense or senseless declarative statement.

Let's at least try and not be obtuse here. The length of the meter stick is rigid in all possible worlds where the accessibility relation of one world also guarantees nomological necessity in another. — Wallows

Very droll.

The length of the meter stick is rigid in all possible worlds — Wallows

But it isn't. The length of the stick might have be other than it was.

Did you mean "The length of the meter is rigid in all possible worlds"? That works.

Well, yes. Here's is what I'm referencing again:

Today, a meter is defined as the length of the path travelled by light in vacuum during a time interval of 1/299,792,458 of a second. A second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom. So in effect, we have substituted the caesium-133 atom for the standard meter bar. The same points could still be made, but we’ll stick with the meter bar for simplicity. — John MacFarlane

Had the speed of light been any different in another possible world, then the above would simply not be true or nonsensical wrt. to define the length of a meter in that world. I don't see how you can quibble over that.

But it isn't. The length of the stick might have been other than it was.

Did you mean "The length of the meter is rigid in all possible worlds"? That works. — Banno

My point seems to be that there is no difference between the two. If we are to assume a possible world where the nomological necessity of the length of the meter stick is defined by the conditions quoted text in the OP, which obtains due to being the same as our world, then stating otherwise is nonsense.

"Metre" is a rigid designator for a certain length. — Banno

I'm not being stubborn here, so help me out. What is a rigid designator other than a certain type of precise definition, where a definition of the form X is Y is reducible to X is X because there's perfect synonymity?

Could you provide an example of a flexible designator? My point will be (spoiler alert) that all designators are flexible.

What am I missing?

It's simple, really, but folk over-think it, or confuse it with definite descriptions.

So the man who was first on the moon was Armstrong.

But someone else, perhaps Aldrin, might have been first, had things gone differently. Had things been different in this way, then the phrase "The first man on the moon" would have referred to Aldrin, and not to Armstrong.

But notice that in this alternate story, the name "Armstrong" still refers to Armstrong, and the name "Aldrin" still refers to Aldrin.

Kripke calls a designator (in this case a proper namer) rigid if it refers to the same thing in all such stories - that is, in all other possible worlds.

SO, a rigid designator refers to the very same item or individual in every possible world in which that individual exists.

In contrast, "The first man on the moon" refers to Armstrong in this world, but in some other possible world it refers to Aldrin. SO it is not a rigid designator.

An important side note: This is not to say that Armstrong could not have had a different name. In some possible world, Armstrong might have been named "Fred". But notice - and this is a point that seems to escape many folk - that Armstrong might have been named "Fred" is about Armstrong. So he is still rigidly designated, even if his name has been changed in another possible world.

Notice also that a possible world is just a story about what might have happened, had things been different. Some folk think that somehow these worlds blink into existence as a part of the logic of possibility. They don't.