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I am not going to say what I believe is impossible. About two thousand years ago, a writer wrote of a mission to the moon. In that time, this would have been thought impossible. Eventually, transporters and FTL travel may be possible. However, these technologies may take a form that is unrecognizable to us. (The writer I spoke about earlier imagined a wooden ship carrying the astronauts to the moon. The space ship of the Apollo design would be alien to him.)
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I believe Arthur C. Clarke said that any sufficiently advanced technology is indistinguishable from magic. Given enough time, energy, and resources, you can accomplish anything.
Registered: Mar 1999
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Clarke's Third Law, yes. But that suggests that the "sufficiently advanced" technology is being viewed by someone of a lower technology, not by the user.
Pacific islanders thought that the aircraft of the outside world were gods, or sent by the gods, but to the pilots who flew them they were nothing special.
Likewise, a transporter that is magic to us may be ho-hum boring to the people who use them daily. Moon-landing would be impossible to the ancients, but history to us. Still, if the ancients had certain understanding of how things worked (not the least being thrust, Newtonian reactions, etc) they might have envisioned a Moonlanding.
We can envision a transporter, but aren't likely to built one. The problem can be seen, but not necessarily overcome.
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The Universal Translator as depicted on the show.
That an the amazing shrinking device installed on the Defiant.
------------------ "But compared with Star Wars, Star Trek, for all its obnoxious spin-off "make it so" durability, is Hamlet and Lear alongside Saved by the Bell[i]." "Good old Liam as Qui-Gon Jinn, the hero in this film, is represented as fighting against the forces of greed. A [i]Star Wars picture that preaches against greed is a little like Bill Clinton in the pulpit for a chastity-begins-at-home campaign." -Rex Murphy on The Phantom Menace
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The transporter: It will never work the way shown in Star Trek. Even if the transmission of a matter stream were possible, it would not work without a receiver.
The universal translator: Besides all "small" technical obstacles (filtering out the original voice, simultaneous translation), it is impossible to analyze a new language with only a few words available.
Warp drive & subspace: Might be impossible, but the explanation is plausible enough for a sci-fi series.
Time travel: The fact that it is connected with paradoxes does not necessarily mean it's not possible.
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According to General Relativity, faster than light travel, anti-gravity and time travel are all the same thing and are all impossible.
Transporters are completely impossible as depicted in Trek.
My personal bugbear: life form sensors. How can you detect that there are five life forms, three of which are Vulcans and two of which are Romulans on a starship 1000s of kilometers away? What strange particles are living beings constantly emitting that can penetrate a starship hull travel great distances across a vacuum and which can carry information about the race, gender and health of the life form?
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Personally I think that in time we'll be able to accomplish any task not expressly forbidden by the laws of Physics.
Transporters It's the smegging Heisenberg Uncertainty Principle. You cannot know exactly where a particle is AND simultaneously know what its momentum is. No way around it. Add to this the problem of the vast amount of data required to store a set of coords for every atom in your body, and the requirement of either "moving" or "copying" all of that and you get a horrendous amount of energy. All that with no apparatus at one end?! Much easier to pop in and out like a Q. Perhaps via personal wormholes. At no time would you be "disassembled".
Gravity nets are a problem too, but we know so little about gravitons it'd be risky to say too much against the concept.
TranslatorsI agree with the ridiculousness of the "instant" Universal translator as outlined by others above.
FTL TRAVEL I have a hunch this will turn out to be possible after all. As Frank says, it's a very big universe out there, which to me implies there is a way to access it. (Not involving giant gyroscopes I trust - not much a "Star Trek" that!)
------------------ "It seems strange that I, Kudos, a doubter, should be given this luxurious window seat whilst you.... AGEING with age, rot away in that disgrace of an aisle seat. Ha, Hah! Where is your God now old woman?!" "Jesus, I'm sorry I asked...!"
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I'm not so pessimistic about artificial gravity or antigravity. Maybe it will be possible to create gravitons without the according mass being present.
FTL travel is impossible according to Special Relativity, but I wouldn't say that there is no way to overcome it at least hypothetically.
The Uncertainty Principle applies to elementary particles, while atoms or molecules as a whole are exactly to determine. I'm not sure to what degree living organisms are really dependent on elementary particles. Most microelectronic circuitry is not (yet).
Sensors: Hmmm. I always tend to forget these nifty devices, although I'm amused every time they yield a precise view of the interior of a starship.
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I think it was the January '99 issue of Scientific American that had a big section on various new theories about the workings of the universe, and I think one of them had to do w/ antigravity. FTL and time travel are intertwined, but I don't think AG is part of that...
------------------ "Although I'm so tired, I'll have another cigarette. And curse Sir Walter Raleigh; he was such a stupid git." -the Beatles, "I'm So Tired"