If the relativistic speeds inside the bubble are kept approximate to what we currently are experiencing on Earth, how does that affect time dilation and the like?
It would have the potential to be matched. The time differences between the piece of spacetime carried inside a Alcubierre warp would be the same as they'd be before the drive was started up.
So if it snatched a piece of spacetime near the moon and carried it and the ship inside to Vega 25 light years away, the clocks would be in relative agreement on the macro-scale that's important to humans. And in theory, were the warp constructed right, the only difference would have been the original difference between the observer at the bottom of Earth's gravity well, and the piece of spacetime near the moon.
The kind of thing our current satellites and things like the GPS system already take into account.
I know the mass-energy constraints may have been reduced for creating an Alcubierre warp down to just a few hundred kilograms of energy, and some other tricks that may or may not work out might get the exotic matter/negative energy needed down to just a few grams. Although these schemes run into trying to squish things darn close to the Planck length, being several orders of magnitude smaller than a single proton is... tricky, to say the least.
And that of course requires we prove that one of the theoretical exceptions to the Standard Model that provide for Tachyonic/exotic matter is correct rather than one of the one's that don't. So I'm not sure if the LHC will help us out there or not.
Then, there's a kind of chicken-n-egg/Catch-22 problem that may or may not be an issue, that to create the warp, you have to constantly superluminally position the positive/normal mass energy for the gravity well in front of the bubble, but you'd need an Alcubierre drive to do it to start your Alcubierre drive. I'm not sure if the newer scheme to make the drive/field into a torus shape gets around this or not...
Then there are those who posit that operating an Alcubierre drive will sweep up Hawking radiation by separating the virtual particle pairs of the universal quantum foam into "real" ones, much like a black hole's event horizon does, and release a blast of nasty radiation at the destination when it arrives.
Although that might be handy if you don't like the people/things where you're going.
We were talking about this a few weeks back, and a few of us were wondering if maybe it would be a good candidate for non-radio SETI, that if such warp drives are possible, and advanced species use them, we might be able to see the gamma bursts of the collected Hawking radiation from the arrivals.