Not that I will understand the implications of the answers, I have two related questions about the nature of so-called “compact dimensions” which are postulated to exist in most (?all) models of string theory.
Are compact dimensions all space-like, or can they be time-like?
Need all space-like compact dimensions be of the same “length” (i.e. are they all the same size?)
Thanks!
The answer to 2 is no they need not be. I’m not positive about the answer to 1, but I’m pretty sure they must be space like. I can’t recall the cite, but I recall reading that in n dimenional space time our physics makes sense only if 1 or n-1 dimensions are time-like. I can’t imagine what that would be like.
One way to thikn about it, is a compact dimension is closed – that is if you travel in a straight line, you return to your stating point. That would be odd for a time-like dimension, though there is research into closed time-like curves in relativity.
So far as I know, all versions of string theory invoke multiple dimensions.
There was bosonic string theory, which had 26 total dimensions . . . but it could only describe one kind of particle, bosons, not the other kind, fermions, so no one thinks it describes the real world. It’s still taught to students learning string theory, though.
Then there’s superstring theory, which has 10 dimensions and can describe fermions. By means of an idea called supersymmetry, this can also be used to describe bosons. So that’s more like the real world.
There are several versions of superstring theory, all of which are understood to be special cases of a more general theory called M-Theory, which brings in one extra dimension, bringing the total to 11. This is the one string theorists really think describes our world, but it’s still not very well understood.
All of the dimensions are space-like except the good old time-dimension we all know and love.
The dimensions can be curled up (“compactified”) in lots of different ways, so no, they don’t have to all be the same length.
Disclaimer: I’m not a string theorist . . . I did do a Ph.D. in atomic physics, but that no more qualifies me to do string theory than to perform surgery. However, I’m sure someone will be along soon to correct me if I’ve misled you.
It’s not necessarily even well-defined to say how long an individual dimension is, since when you have multiple compact dimensions, they can be arranged in all sorts of different ways. For instance, if you have two compact dimensions, they could be wrapped up like a torus, in which case each one has a well-defined length, but they could also be wrapped up like a sphere, in which case the length of one dimension would depend on your coordinate in the other dimension (latitude lines are not all the same length). For that matter, you couldn’t even necessarily say what each of the dimensions is: There’s a natural way to break up a torus into two coordinates, but on a sphere, you can stick a pole anywhere you want, and draw “latitude” and “longitude” lines according to that pole.
I think that all of the various ways of arranging the extra dimensions are what are referred to as the Calabi-Yau manifolds, and one of the reasons why there are many different versions of string theory is that there are many different possible Calabi-Yau manifolds.