Is warp 10 the same speed as trans-warp? Does it mean infinite speed in both cases?
In the episode Threshold, Tom Paris is experimenting with warp 10 speed and the effects of this on him are quite poor: an accelerated form of natural human evolution, two hearts, etc.
At the same time, Borg drones created out of humans do not have any negative effect of traveling with trans-warp speeds (or any at all). How can this be?
It's definitely important to note that Threshold is really bad.
Having said that, Warp 10 and transwarp are not the same term, but they aren't unrelated either. Warp and transwarp are different ways of measuring a ship's velocity relative to the outside world (since a ship in a warp bubble is theoretically stationary, it just warps the space around it to actually break the light barrier, insert technobabble as you see fit).
Think of it like this -- the sound barrier, right? If something is traveling at subsonic speeds, it's fair to measure its speed in MPH or KPH or whatever the preferred metric is. Upon breaking the sound barrier and attaining velocities several times the speed of sound, one measures their speed as a factor of Mach; Mach 2, 3, 4 and so on. It becomes a bit cumbersome to think of the speed in terms of MPH or KPH and ultimately less useful under most circumstances, in particular if the craft is regularly traveling several times faster than sound.
The light barrier is another such breakpoint in terms of measuring speed; one would not clock a ship which is travelling roughly 11c as Mach 9593023; better to just call it "Warp 2"!
In Star Trek, the light barrier has been smashed and now uses the idea of Warp 10 as being unattainable, citing it as being 'infinitely fast'. Whether or not that is literally the case, or an artifact of their perception due to their inability to actually get there (barring Threshold, aiee) could be speculated on; however, Star Trek does lend us the ill-explored idea of transwarp, which is touted as significantly faster than plain old vanilla warp. It is simply another scale for measuring the speed of a ship moving at superluminal speeds.
The original Excelsior experiment having famously failed in Star Trek III, the assumption has been that they simply abandoned the project and instead moved on to making their own warp drives more efficient; indeed, whereas old warp speed can be commonly calculated as lightspeed times the warp factor to the power of 3 (Warp 2 = 8c, Warp 3 =27c, and so on), the TNG warp scale is lightspeed times the warp factor to the power of 10/3 (3.333...), which means that Warp 2 is something like 11c, Warp 3 is about 29c, and so on. The math breaks during the transition from Warp 9 (1516.3c) to Warp 10(apparently infinite) and so the speed starts to climb towards infinity, which is why it is A Big Deal when the Enterprise-D, the Borg, and other vessels are capable of pushing Warp 9.5 and higher - even fractions of warp speeds at that point are significantly faster than the previous increments.
Speaking somewhat non-canonically; the TNG Technical Manual hints at the idea that warp factors exist because there are certain warp fields which are more or less "easier" to maintain; it's more energy efficient to travel right at Warp 3 than Warp 2.9, which is why these seemingly arbitrary Warp speeds exist at all. What transwarp would offer, then, is a completely new scale that would be to warp speed what lightspeed is to Mach, and Mach to MPH, etc.
It's important to recognise that the episode that you're referring to is about as close as a TV episode can get to becoming "non-canon". As well as being recognised as the single worst episode of Voyager (and frequently voted the "worst of trek"), even the writers accept that there were many technical failings with the description of how Transwarp works in the episode.
TNG's Producer Brannon Braga describes it thusly;
"It's a terrible episode. People are very unforgiving about that episode. I've written well over a hundred episodes of Star Trek, yet it seems to be the only episode anyone brings up, you know? 'Brannon Braga, who wrote 'Threshold'!' Out of a hundred and some episodes, you're gonna have some stinkers! Unfortunately, that was a royal, steaming stinker."
and TNG's 'Senior Technical Consultant' Rick Sternbach basically tried to handwave the whole episode away;
""I think what may have happened with the silly Warp 10 episode was that there was a coupling of the energy from the shuttle to all of the energy and matter of the universe (which might be possible if we're looking at a finite system), and the shuttle was able to access any point anywhere by some amazing tunnelling phenomenon which shrank the normal 3D distances to points, much like all the universe being squished into a pinpoint at the big bang because it was all energy with no need for elbow room. Whew."
Warp 10 is not the same as transwarp. Warp 10 is the fastest a drive can possibly go, and as is seen in Threshold, has negative effects on living tissue. Transwarp technology, on the other hand, first theorised in The Search for Spock is a way of travelling faster than traditional warp technologies allow. Transwarp doesn't seem to have any physical effect.
'Threshold' episode was apparently kicked out of canon... but to me that episode is not as bad as people would imagine. Given what we know of epigenetics and neuroscience from the real world, it is possible that by travelling at 'Warp 10' (or Infinite velocity - which would be different than Transwarp as used by the Borg or the Voth), Paris was exposed to a number of phenomena throughout the entire UNIVERSE of which SF would have little to no knowledge of. We've seen several instances of anomalies and certain forms of radiation penetrating shields and affecting the crew.
It's not impossible that while being everywhere in the universe at once, Paris might have been exposed to a trillion different phenomena at the same time that would have affected his DNA against which his shields might not be effective - human DNA (or specific genes) from what we know of epigenetics can become activated or deactivated given lifestyle choice, diet, etc (essentially, environmental influence)... and I would imagine that something as strange as subspace alone would affect his DNA in addition to those anomalies I mentioned).
Yes, SF uses subspace for Warp (and has been using it for centuries), but, it stands to reason that different layers of subspace might exist which allow for say Slipstream, Soliton Waves, and Borg/Voth style Transwarp. Or Paris simply used subspace differently to access Warp 10.
As for 'Infinite Velocity requiring infinite fuel' problem... that could be explained with Bussard collectors having access to tremendous amount of subspace energy/particles which could have been used to keep the engines running for as long as you need them. Once you get close to the threshold, and if you attenuate the Warp field and hull structural integrity correctly, it might be possible that at those velocities, subspace itself ends up powering the engines.
So it wasn't that 'infinite velocity' was a cause of Paris DNA mutations... but rather that his exposure to unknown amount of phenomena in the universe at the same time, coupled with subspace radiation in the mix might have affected his DNA in such a manner.
As we've noted, the Voth and The Borg don't travel at 'infinite velocity'. The Voth have really fast engines allowing them to travel about 90 lightyears in about 15 seconds.
The Borg TW conduits may operate similarly... or they have comparable speeds, just their TW operates differently than Voth's.
So, I personally wouldn't kick Threshold out of canon... I'd simply say that SF needs to develop adequate shielding that would protect the user from all possible anomalies in the universe combined... or a simpler method would be to not use 'Warp 10' but simply use slightly slower speeds that would be comparable to TW as used by the Borg and Voth (and I'd imagine would be easier to develop shielding for).
Paris : 'We're trying to break the maximum warp barrier.'
Kim : 'Nothing in the universe can go warp ten. It's a theoretical impossibility. In principle, if you were ever to reach warp ten you'd be travelling at infinite velocity.'
Neelix : 'Infinite velocity. Got it, so that means... very fast?'
Paris : 'It means that you would occupy every point in the universe simultaneously. In theory you could go anyplace in the wink of an eye, time and distance would have no meaning.'
Paris : 'Warp 9.92... the pylons are secure, everything looks good. 9.97... 8... 9... warp ten!'
B'Elanna : 'You've crossed the threshold. You've done it! And there's been no damage to the nacelles.'
