The simplest one to understand is basically a modification of the Lorentz Ether Theory, the key being that the "ether" is the locally dominant electromagnetic or gravity field (depending on which version of the theory you're going by.) So the Michelson–Morley didn't detect any such effects because the test was done solely within the Earth's gravity well/magnetosphere.
These theories don't seem to be especially likely to be true, but it's nice to pretend otherwise because although identical in most aspects they _do_ allow for the possibility of FTL. (There is a privileged reference frame in any given location and/or time dilation is a physical rather than temporal effect, so there are no causality violations in getting from one spot to another faster than light could make the trip.)
This of course got me thinking about relativity in general of course, and i've run into a problem i'm stuck on, i'm sure it must have been covered in class at some point but i can't remember now and i'm hoping someone can give me an answer that doesn't go completely over my head. (I'm looking at you steuard ;)
So we have a ship wandering about in the depths of space, it's one of those stereotypical relativity example ships which can accelerate up to very significant fractions of c.
They've been doing various maneuvers for quite some time and have lost track of their relationship to whatever their original point of origin was, not that it should technically matter anyways.
So they're drifting along in an inertial frame, and suddenly see another ship, identical in all the important elements, heading towards them in its own inertial frame. The two ships are approaching each other at a rate of 0.866 c, which if my notes are right means a time dilation factor of 2.
Of course since they're both in an inertial frame each sees the other as being dilated and perceives time as going at half the normal rate on the other ship. Just for ease of example each ship has a giant clock mounted on the outside that measures time in 24 hour days (but makes no indication of what day it currently is.)
The two ships decide this is the perfect opportunity to perform a little experiment and work out the details on approach. When they pass each other they synchronize their clocks (mainly the internal ones that keep track of days and years, but might as well synchronize the external hour/minute/second display too) and the captains secretly determine which of them is going to turn around at a given point of the future in order to meet up again.
So they continue on their ways directly away from each other, still in inertial frames, still at a relative velocity of 0.866c. Crewperson #1 on Ship A watches the clock on the other ship through a very good telescope. He sees it moving at half the normal rate, and knows that thanks to relativity the crew on the other ship is actually aging at half the normal rate currently. His reference frame is just as good as anyone else's after all.
Come turnabout day, which is t time after they two ships passed each other and traveled a total distance of d apart from each other (in this case d shall be nameless value x,) everyone rushes up to the main conference room to find out which ship is going to turn around. In the hurry Crewperson #1 slips and hits his head.
When he wakes up he finds that the ships are now heading towards each other at 0.866c and he has no idea which of them actually changed course. His buddies think this is hilarious and tell him to figure it out for himself. Of course there's no way for him to figure it out because they're still in an inertial frame. (Let's pretend that he can't do some celestial navigation or check the fuel levels to figure it out.) He still sees the clock on the other ship going at half the normal rate.
So they ships finally pull together and Ship A decelerates. This seems to indicate to Crewperson #1 that their ship was probably the one that changed course, though of course they could have agreed to some other arrangement than one ship doing both course changes.
Still, he's rather surprised that after having watched t time pass on the clock of the other ship during the coasting parts of the voyage that took their own ship 2t time, that the crew of the other ship are actually 4t older than when they passed each other the first time!!
That's okay though, general relativity comes to the rescue since clearly Ship A did a lot of accelerating at both ends of the trip. (A grand total change of 2.598 c.) In order for things to work out during the course changes the clock on the other ship must have sped up dramatically, right? Enough for 1.5x time to pass, presumably most of it during the course change of 1.732 c at the far end.
The only problem is, the exact same course changes could be used to perform a trip of any length, 0.5x, 1x, 2x, 5x, yet somehow despite the coasting part of the trip always appearing to take half as long for Ship B, the exact same course change for Ship A is supposed to make up a variable amount of time so that they always arrive back at 4t after the initial meeting going by Ship B's clock!
So what am i missing?