Do you remember the scary moment when Ashton Cutcher alias Evan recognizes suddenly that his arms are crippled – and that they have been crippled since many years?
This is a scene from The Butterfly Effect: Evan discovers that he is capable of time-travelling to his troubled childhood. He tries to fix the present by changing the course of tragic events in this past – and most of the times the outcome in the present is somewhat unexpected.
I am finally living up to my promise on Dr. Dan’s(*) blog to come up with my musings on the logic of time travelling in movies. [Insert your favorite disclaimer about spoilers here.]
(*)As the future of this post is going to unfold below the reason for this formal address will become clear. Mind the Douglas Adams quote.
Based on thorough and extensive research I came to the conclusion that there are three distinct kinds of time travelling (in movies):
The Butterfly Effect is an example of what I denote as  The Immediately Changing Past as Remembered or Not by Conscious Beings, and so is Back to the Future:
Our heroes travel to the past, fuss about the timeline, and travel back to the present: And the present it different from what it had been before starting the round trip. Even more, the present indicates that all the events prior to this moment must have been changed as well. In The Butterfly Effect the reactions of his friends indicate that handicapped Evan has got along well, probably until he ‘suddenly’ re-discovers that he is crippled.
Marty McFly – stuck in 1955 – needs to make sure that his mother and father become a couple ‘again’ despite the mess he had introduced to the old timeline; otherwise Marty would cease to exist. When he is finally back to the future his family has ‘suddenly’ changed from cliché lower class losers to smart cheery middle class high-flyers.
How and if the main characters’ memories have been (will be) aligned with the new timeline is typically not explained in a satisfactory way. Do they remember the other timeline forever (in the future of the future)? Did the old timeline cease to exist or is it preserved merely as a representation in the main character’s mind? Does the representation in mind count as a version of a multiverse?
We need to distinguish this theory about time travel from  Twisted and Warped but Yet Fully Causal Timelines which denotes the infamous branching of times while indicating that the previous timeline is still preserved in history.
It can be illustrated by the first Star Trek movie released 2009, featuring the new old crew which is simply called Star Trek:
At the end of the movie we can start the history of the old enterprise crew all over which is a clever move of the movie producers from a commercial perspective. The new timeline can and will never be in line with the ‘original’ plots of the episodes and movies. The Romulan villain as well as Spock had travelled to the past and created an alternate timeline. Young James T. Kirk asks Spock, the elder, if his (Kirk’s) father was alive in the original timeline (which he was).
As with  there are two (or more) distinct timelines but the important distinction is: At any point of time any human being (or alien for that matter) experiences a smooth unfolding of his / her / its personal story. Neither the elder nor the young Spock’s mind is subject to a sudden disruption.
I believe the underlying reason is that the physical and mental identities of all persons involved in  remain intact. Time travellers travel as physical beings, but you do not jump right back or forth into your younger self’s body which requires your minds to merge. Evan jumps mentally only – both forward and backwards. Marty jumps to the past physically, but on fast-forwading to the future he sort of merges with the self he left back (in the future) when travelling to the past.
I tag Minority Report as class  Fully Causal: John Anderton shows up as a future criminal in the precogs’ crystal ball – yet finally he decides not to commit a crime. But there was no mind-disrupting time travelling. Note that my categorizations does not care about the direction of time-travelling, and I do not distinguish between viewing the alternate timeline (only) in contrast to actively being part of it – as viewing it may be sufficient to introduce a change in the other timeline you are part of.
As you have noted, it is incredibly difficult to speak about time ordering in time travelling scenarios. (By the way Time Ordering is a scientifically rigorously term used in quantum field theory, applied to the quantum mechanical operators that dictate the time evolution of systems.)
Of course it was Douglas Adams who noted wisely in The Hitchhiker’s Guide to the Galaxy Vol.2 – The Restaurant at the End of the Universe:
One of the major problems encountered in time travel is not that of accidentally becoming your father or mother … There is also no problem about changing the course of history … All the important changes have happened before the things they were supposed to change and it all sorts itself out at the end.
The major problem is quite simply one of grammar, and the main work to consult in this is Dr. Dan Streetmentioner’s Time Traveller’s Handbook of 1001 Tense Formations. It will tell you for instance how to describe something that was about to happen to you in the past before you avoided it by time-jumping forward two days in order to avoid it.
I would conclude from this that Douglas Adams adheres to the  Predetermined and Eternal Timeline Theory: Even if or exactly because the characters know something about their futures and try to avoid the outcomes at any cost everything turns out exactly as it has been foreseen. This has been demonstrated in the TV series FlashForward.
As you noted, this post is about movies, not about physics. In physics, special relativity demands that theories are causal which basically means that events in the future can only be influenced by events in the past (More precisely only events in the past lightcone can signal to events in the future lightcone in 4-dimensional space-time). However, a  Predetermined Timeline might imply that nothing causes anything else really – events just are and have always been ingrained into the fabric of space-time. It would not matter if we could signal from the future to the past as this is futile anyway.
But only an excessively warped space-time – not a very friendly environment for living beings are we know them, probably close to blackholes – might give a chance of influencing the timeline.
A direct link to the past via such a warped channel is proposed in Deja Vu starring Denzel Washington. We are able to watch the past unfolding real-time in parallel to the present time elapsing. A piece of paper with a note is sent backwards in time and past and present are constantly influencing each other. But I would still argue these are  Twisted But Yet Fully Causal Timelines.
Things get messy when Doug travels to the past himself, changes the timeline, rescues the future victim Claire, and dies heroically. But there is still another instance of Doug’s in the ‘present’ in the final scene that now is not aware his is alternative time traveller’s self’s history.
Believe it or not, but there is already an infographic on Wikimedia to explain this by proposing four different timelines – you need to enlarge the full-sized image at the Wikimedia page:
This is very similar to the Star Trek movie analyzed above – complication arises due to multiple interactions between the timelines:
Describing the unfolding of event in causal terms, Timeline 1 was the first, and the interaction with the past at the ‘end’ (in the ‘present’) of Timeline 1 triggered Timeline 2 – and onward to Timeline 3 and 4 in the same way. So the whole universe is walking through these four timelines – or the observer is made aware of four parallel universes, each of them equipped with an eternal timeline that cannot be modified. If the latter is true, you should really not meet yourself in an alternate timeline or space-time will collapse. So the causal consistency of  in contrast to  comes with its risks.
In a similar way to the twisted plot of Deja Vu, the old enterprise crew has boldly gone where no man has gone before as we witnessed in the 1970s on black and white TV sets. Then the Romulan Nero and the elder Spock travelled from to past (from the perspective of bold old episodes and the 2009 movie this was / is the future) to meet the elder Kirk and the younger Kirk – both time travellers have interacted with the past at different times – very similar to the Deja Vu approach.
Every timeline-crossing event kicks off a new timeline. In the most ‘causally recent’ Timeline 3 both entities of Spock do coexist of did two versions of Doug in Deja Vu – thus I tag Deja Vu as a class  movie, but the categorization should maybe be enriched with tags for complexity.
I owe to the members of the IMDB discussion board who started to discuss this movie – released in 2006 – a few days ago. What a coincidence. Even scarier than the spacefolding surveillance technology is the fact that members discuss the applicability of this tool to current investigations in Boston and that they watch the movie again right now because of the terrorist attack.
Most likely we need to do more research and setup a global project in order to come up with better categories and standards for time travel movies: This collection of the Top 10 Time Travel Movies is a starting point.
And I did not even mention the movie that has the closest resemblance to our daily lives. Groundhog Day does probably not qualify as an SF / time travel movie because looped structure of space-time is too realistic and too familiar.
Edit: I have just stumbled upon this on Google Plus – the definitive chart including many popular time travel movies:
All Time Travel Movies Explained in One Awesome Infographic and here is an update by the creator of that said chart.