Not long after the Star Trek reboot premiered in 2009 did the fanfare turn to criticism. One of the crucial geekhilisms thrown at Star Trek was the Transwarp Teleport calculations given to a young Scotty by our aging Spock.
For many fans, it was seen as a means of defeating the purpose to ships or that it was a bad plot device that isn't canon to the Star Trek universe. Yet there is a technical side to this teleportation feat which I think many novice Trekies missed and many veteran fans disregarded.
So let's delve into this equation and its functionality and why it is neither a bad plot device nor an end to the need for ships.
Understanding the Star Trek teleport.
In the simplest of terms the teleport or transporter dematerializes matter and stores it as a digital pattern and then transmits this pattern to a remote location where it rematerializes. Besides the long list of things that would prevent this in reality, the simple method through which it works allows for a plethora of possibilities. The range of a transmission for the transporter is limited only by available power. The greater the distance the more power is needed to execute the teleport. This is seen throughout the Star Trek universe, be it the TV series or films. What prevents long distance transportation then, besides power, is the ability to accurately pinpoint the location for which rematerialization would occur. This has to take into account very precise location relative to movement which takes time for a distant observer to know. If one can predict with certainty where the point of rematerialization will be at the time of rematerialization and if one had enough power, long distant teleport is possible.
Understanding Star Trek Warp Drive.
The Warp Drive, is in fact, a real thing, at least, in theory. The Alcubierre Drive is a form of warp drive being studied and experimented upon by NASA currently. It and the Star Trek Warp Drive work on the same principle which allows for us to side-step the speed of light barrier.
Faster than light travel is conventionally impossible. The universe prevents objects with mass to travel at the speed of light let alone faster than it. But a work around does exist. If you can isolate space-time from the greater universe, the relative speed found within a pocket of space-time is zero, while the isolated bubble of space-time can move at any speed in relation to the rest of space-time. This is what the Warp Drive achieves in the Star Trek universe.
Any bubble of isolated Space-Time, however, is also intrinsically unique in that while it moves it is in a state of quantum flux. What I mean by this is that it can exist in two or more places at once. The very principle that allows for the isolated pocket of space-time to move faster than the speed of light without breaking the mass barrier also causes quantum tunneling. Until such time that the isolated space has shifted enough from its starting point to its end point, it can be seen as both moving and stationary.
Why the Transwarp Teleport works.
There is a crucial moment when Scotty, upon examining the equations given to him by Spock, exclaims, "I've never thought of viewing it as though Universe was moving." This statement right here definitively explains why the Transwarp Teleport they execute would work.
In a nutshell, the Enterprise was still in orbit around the ice-planet where they dropped off Kirk while also in transit within a warp bubble. The state of quantum uncertainty (Shrodinger's Space Ship if you will) meant that the Enterprise, with some subspace jiggery-pokery, could be targeted while in transit. The relative distance between the Enterprise and the ice-planet hadn't actually changed. Nor would it effectively change until the warp bubble had fully transited from point A to point B. The universe, from the relative position of the isolated bubble of space-time would be the object that was moving not the Enterprise which, within the bubble, would have had a relative speed of zero.
Even then the transport wasn't precise enough, while Scotty had aimed for a cargo-bay they ended up in engineering giving us the scenes of a wet Scotsman. This is because while the Enterprise wasn't technically moving, it was still in transit held within the warp-bubble. The relative position changed within the time span of his calculating the Enterprise's location within the bubble and executing the transport. Uncertainty principle tells us that we can either know the speed or the position of an object in a quantum state (which usually only applies to quantum particles granted). This played a huge part into why this scene worked.
You should really be up-in-arms about this bit of continuity error:
There is a moment in Into Darkness when it is exclaimed that the new Dreadnaught couldn't catch the Enterprise during warp and then it did. For most fans this might seem like a feat of speed of one ship versus another. The problem is, that isn't how Warp Drives work. A Warp Drive isolates a pocket of Space-Time. This isn't like hyperspace where a ship is using a wormhole to tunnel from place to place. This is a matter of creating a pocket-universe in which only that found within the warp field exists in this universe and then transiting this pocket from place to place. That means a second ship couldn't enter into the same isolated pocket as another without starting in the same occupied space. You can have a faster warp drive that can apply more energy and transit the pocket faster through space, but what we see in Into Darkness when the Dreadnaught overtakes the Enterprise isn't possible with the Star Trek Warp Drive, at least not without a lot of work on behalf of both ships to merge their warp fields.