Can we go where? The extra dimensions in string theory are curled up tiny, they aren't like Dr Who. No holidays there I'm afraid. A sceptic would even say that these extra dimensions were invented simply to keep the idea of grand unification alive by having somewhere for the protons predicted by other experiments to conveniently fall into. The problem is that string theory says nothing at all about the sizes of these extra dimensions, they could be anything from infinitely large to infinitely small, so there’s no real prediction. If we ever do see extra dimensions, there’s no particular reason to believe these have anything to do with string theory. It's always assumed that the 'extra dimensions' will be somehow similar to the familiar spatial dimensions 'but smaller'.

It's not, though, really. It might be easier to think of the extra dimensions of string theory as additional degrees of freedom at very small scales/very high energies. Rather like a molecule has three translational degrees of freedom, plus rotational, vibrational and so on. So at very small scales strings have access to additional translational degrees of freedom, but they're not exactly like the three macroscopic dimensions we're familiar with.

As for whether extra dimensions can be timelike, there are theories which include that (they're beyond me, though). At the macro scale, our space time signature of 3+1 seems to be uniquely privileged- i.e. universes just don't work very well at other signatures, you get tachyons, negative probabilities, all sorts of chaos. Minkowski space (ours, basically) has a signature of -,+,+,+ where time has a negative sign to make the maths work, and the mathematics works nicely when you calculate space time intervals - a lot of special relativity, for instance, is basic geometry not much more complicated than Pythagoras' theorem, and with a signature of -,+,+,+ it's very neat, and you can work out how events in spacetime relate to one another. Throw in another -ve sign and the whole thing falls apart. I think there might be models with extra time-like dimensions "off the brane" but it's all very speculative.

Dammit, hope they help prove string theory is correct, cos while I'm not sure I understand all of it despite having read at least one fat book on the subject, it's been fabulous to go around saying: 'It's/We're all just vibes man'.

General relativity was falsifiable. To be a good theory it is not necessary that a test can be carried out using current technology but a falsifiable test should at least in theory be possible. GR made various predictions that could be tested. As you point out Eddington performed an early test. Other predictions flowing from GR have been made and have been tested including gravitational lensing and gravitational waves. It has taken decades for the technology to evolve to start to test for gravitational waves but GR makes a prediction and now it can be tested.

The problem with string theory is not that a test cannot be carried out with current technology but that no test is even theoretically possible because string theory makes no prediction beyond there being a massive variety of possible universes. Whatever results future experiments provide, string theorist will just say that we happen to be living in a universe which produces those results.

As Peter Woit pointed out "speculative scientific ideas fail not just when they make incorrect predictions, but also when they turn out to be vacuous and incapable of predicting anything". Please, this was all solved long ago: It's turtles all the way down.

No-one uses tesseract in a sentence any more. It inspired me ...

I wish my house was a tesseract
a place where I could tell fiction from fact
I'd invite politicians to sit in the middle
Then I'd focus the heat so it's hot as a griddle
I'd make then elucidate policies at length
And keeping them talking to sap all their strength
And right at the end I would shout and declare
"Your lies and deceit are now totally clear
My house has deciphered your thoughts and your words
And showed them as nothing but bright polished turds
I'm leaving you now and I'll never come back
This part of my house is now fading to black .... ( )

A really like the first 12 chapters!