Can there be causality when there is spacelike separation?

[entropy1]

If two events A and B are spacelike separated, is it then true that there can be no causal relationship between the two?

 

[martinbn]

If two events A and B are spacelike separated, is it then true that there can be no causal relationship between the two?

Isn't this the definition?

 

[entropy1]

Isn't this the definition?

I didn't succeed in even finding a definition on Wikipedia.

 

[Ibix]

I'd suggest using a more reliable source than Wikipedia for learning relativity. I like Taylor and Wheeler, and others here recommend Morin. The first chapters of both are available online if you want to have a look.

The point about spacelike separation is that $\Delta s^2=c^2\Delta t^2-\Delta x^2-\Delta y^2-\Delta z^2<0$. That means that $c^2\Delta t^2<\Delta x^2+\Delta y^2+\Delta z^2$, which means that not even something traveling at the speed of light can get from one event to the other. Thus neither one can cause the other.

 

[Nugatory]

If two events A and B are spacelike separated, is it then true that there can be no causal relationship between the two?

Yes. Whether “no causal relationship” is the definition of “spacelike separated” or the other way around is mostly a matter of taste, as either one implies the other.

 

[Rahzan]

I suggest you adopt the verbiage "locality" which is stricter term than causality. The corollaries to the postulate of relativity (aka the "special" theory of relativity) only stipulates that it is not possible to accelerate from under to above the speed of light in a finite time. Similarly, anything already going faster than c (aka a Tachyon) is allowed but must stay faster than c.

Since experiment has so far confirmed that some information must travel (at most) at the speed of light, (or better, speed of information c) for interaction to occur, all observed phenomena in classical physics are presumed to conform to locality. There is no fundamental requirement on causality (or locality) other than what is seen in observation.

Seemingly-semi-classical violation of causality do exist for example in pre-acceleration of electrons.In fact it is possible to show (in a very general way) that a certain form of uncertainty kind-of-like we see in quantum mechanics arises when considering communication faster than c over space-like distances. This too violates causality but not locality (this time on the upside of c!).

Look for the book by Andrzej Dragan and his extremely cool youtube videos.

 

[PeterDonis]

Isn't this the definition?

I don't think it's quite that simple. See below.

not even something traveling at the speed of light can get from one event to the other. Thus neither one can cause the other.

I think it's worth expanding on this a bit.

Strictly speaking, the concept that has a simple definition in relativity is "spacelike separated"; the definition is the one @Ibix gave (or its appropriate generalization in GR when spacetime is curved). We then observe that the time ordering of spacelike separated events is frame-dependent in relativity, and we therefore infer that, since "causal connection" is a directed relationship--the cause must come before the effect--spacelike separated events cannot be causally connected.

Whether “no causal relationship” is the definition of “spacelike separated” or the other way around is mostly a matter of taste, as either one implies the other.

I don't think this is true. While the implication from "spacelike separated" to "not causally connected" is straightforward, as given above, the reverse implication is not, since if all you know is that two events aren't causally connected, that in itself tells you nothing about the type of spacetime interval between them. You have to already know what "spacelike separated" means in order to make the connection from "not causally connected" to "spacelike separated".