It depends on what time travel logic they are using.
Star Trek has bounced all over the place and I don't think we know what they're using, except that it is not the
Lost logic, which is:
1. Timelines are preordained. If you try to go back in time and kill your own grandfather, the cosmos will conspire to stop you, even if implausible things need to occur, like your gun jams 200 times in a row.
So
Star Trek might be doing other things:
2. Timelines change all the time, but people don't realize it. If you try to go back in time and kill your own grandfather, you can do it, but then you vanish from the timeline. Nobody remembers you in your own time because you didn't exist.
2a. Sometimes people are "outside the timeline" and can perceive these changes that nobody else realizes.
3. You can't actually time travel within your own timeline. If you try to go back in time and kill your own grandfather, you'll kill someone else's grandfather, who is almost exactly like you, and now that person doesn't exist in that timeline. This timeline logic comes in one of two flavors:
3a. The time travel and/or attempt to change the timeline (killing grandad) causes the timeline to split and create a new timeline.
3b. Infinite parallel timelines already exist, and since they are infinite, anything that could happen does happen in at least one of them. So whatever you want to do in the past, there's a timeline for that to happen in, and you can kill that other guy's grandfather, in fact, you
have to, in at least one timeline. This isn't actually time travel so much as parallel reality travel.
And then there's
4. Time traveller as ghost. You can't materially interact with the past or future, but only observe. No grandfather paradox possible.
These all have serious problems when used in fiction.
1. Is a problem in ongoing stories - you can get away with it in a 2-hr movie (
Twelve Monkeys) but to try to use it in a serialized TV show raises the so-what issue. Why do we care what the characters do if we know they are straightjacketed by the cosmos that is forcing them into certain paths.
Lost has been clever in building up the characters so interestingly that this isn't a big objection - I watch the show just to see how the characters react. Also,
Lost has set up the implausible-coincidences notion from the start of the show, before there was a whiff of time travel, so it doesn't seem like it's cobbled on for the sake of the time travel now.
2. Creates total chaos. How can you write a story where nobody knows what's really going on? (A very creative writer might be able to do something good with this, but I'd expect to see that in a novel and not in a TV series, where the audience would be baffled.) So writers usually fall back on 2a. which is a cheat - what if someone "outside the timeline" tries the grandfather thing?
3. If it isn't your own past, or even your own universe, who gives a crap? This logic gets used a lot anyway, because at least it makes internal sense, and even parallel doppelgangers of Our Heroes can be interesting characters. I'd be up for watching a few episodes on the adventures of Mirror Universe Spock for instance.
4. Not usually interesting as the basis of a story, though sometimes okay as a device for giving the characters vital information.
Another way time travel problems are obscured is to turn the whole thing into a comedy. Then illogic and so-what problems can be buried under jokes.
What makes one reality more valid than another?
Under 1, Temporal Investigations has nothing to worry about and the organization does not need to exist.
Under 2, if you assume someone somewhere (and some
time) has the ability to time travel, and in
Star Trek that's a given, they are already changing the cosmos out from under you at any given moment, so you better start fighting back and shaping your present to be what you want it to be. You may be making things worse, but the other guys might be making it worser. You can't afford not to time travel.
Under 3, you're playing in somebody else's sandbox, so it's the same as 1 - don't bother.
So this all really heavily depends on what logic is being employed.
Trek seems to be using 2, but with the caveat that the timeline can be returned to its original state, which is like being able to unscramble an egg.
Trek may actually be using 3, dressed up as 2. The key is that actions propel you between parallel realities - trying to kill your grandfather creates a new reality or kicks you over into one that already exists.
So when
Trek characters try to change a timeline, they aren't changing anything. Their apparently timeline-changing actions are like pushing a button that opens a portal to another reality where that action makes sense. But they can't see the portal or perceive their motion between realities, so to them it's as though their reality has changed.
They start in Reality A, which is how they like things. They get transported to Reality B, which is not to their tastes. They do something to change B so it will become A, but their action merely returns them to A rather than changing B, and they are fooled into thinking they've accomplished something, or that Reality A was ever in danger. And Temporal Investigations is the galaxy's biggest boondoggle.
