Why strings have to be spacelike? What does it mean? I don't understand the following statement from Zwiebach's book:

"We want strings to be spacelike objects. More precisely, the interval between any two points on a string should be spacelike, perhaps null in some limit, but certainly never timelike."

References:


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*Zwiebach, B. (2004). A First Course in String Theory. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511841682

 A: Here is one argument: Zwiebach wants to use the Nambu-Goto square root action, and hence needs a generic worldsheet to have fixed signature, in this case signature (1,1). This forces the string at a fixed time $\tau$ [in whatever coordinates $(\tau,\sigma)$] to be generically spacelike separated. 
A: String theory introduces fundamental extended entities. This is in stark contrast with previous theories introducing either fundamental point like entities (like electron, quarks, etc.) or non fundamental extended objects (whatever objects around us, a shoe-string for instance.)
Since the string is fundamental, there should not be privileged points inside the string (in fact there is not even a concept of something inside the string in the full theory, but sometimes is useful to model things as such), therefore in a relativistic light cone every point on the string should be space-like separated from each other.

You can picture the string as spread on the plane intersecting the present and past cones at the tip.
Notice that the whole thing is just an analogy, an interpretation. You are trying to talk about string theory in the language of a less fundamental theory. The correct way is the opposite one, but of course it's not available to us yet.
A: Here's another argument : consider the classical string. The worldsheet in string theory is just a $2$-dimensional submanifold $\Sigma$. The condition asked of it is that $\Sigma$ is a timelike submanifold (that is, any normal vector is spacelike), this leads to any intersection of the worldsheet with a Cauchy surface is a string.
If the worldsheet was not timelike, this would lead to the same causality issues as tachyons would. I don't mean the string theory kind of tachyon but the much worse classical SR type of tachyon. In fact, if you consider some foliation of the worldsheet by geodesics, you'll see that a timelike worldsheet is foliated by timelike curves (hence roughly equivalent to a one-parameter family of timelike geodesics), while if you had chosen a spacelike worldsheet, I don't think such a foliation is likely. You can probably check easily enough that given such a worldsheet, the momentum modes will be spacelike, and therefore not terribly physical.
Also remember that in the limit of infinite string tension, the action reduces to that of a point particle. This limit will be a timelike curve for a timelike worldsheet, and a spacelike curve for a spacelike worldsheet.
