I am a history buff. Less from school and more for just stuff that just interest me.
WARNING HEAVY NERD CONTENT that is not 100% supported by science:
As for the actual physics behind the time travel? Space-Time, gravity, and QM (Quantum Mechanics) are linked and what the Grand Unification Theory are about
I'd have to go with some sort of string manipulation (via string/membrane theory). You should be able to manipulate gravity with it. One flavor of M-theory has 26 dimensions... the math for it is way beyond me. I can grasp a lot of Relativity & QM but not M-theory.
There was a few time travel episodes of the Outer Limits (the one that was on ShowTime), a lot of the people who later became involved in Stargate:SG1 were on board.
I sort of like the idea for a device like the Star Gate (movie and TV) or a point that is weak in space/time (say that rose bush in the neighbor's yard) as the only way back and forth between the two times. It's good as a plot point in gaming. A young couple are making out on the park bench when all of a sudden there is some special effects and the people come through. One the other end of it you can have the only portal to a certain time be in a military base.... Or you're heading back to the spot to head back home and there are a ton of cops, police tape, and a dead body just a few feet from where you need to be.
Along that lines something I have had problems trying to pin down for a setting has been how to time the return home. Do you have to be the back door of the diner at midnight? Do you have a secret button on your coat that sends a pulse back to HQ and tells them to activate the time travel door? Can you communicate with your own time via a radio? In 12 Monkeys they left a message on an answering machine that was found.
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For a game setting I think it might best be handled like an old UPN show called "7 Days". Premise is that the NSA had a time machine that could only go back 7 days. Their chrononaut was a military intelligence agent who had a nervous break down and was in a top security mental ward, can you say plausible deniability?
The team would gather all the intel they could in a few days and send the chrononaut back 7 days. hopefully with enough intel and time to find and stop the event.
I mean that would sure help the GM when the players want to go back and say kill Hitler or pour bleach in the first tidal pool where life was formed.
As for the grandfather paradox I think that when you remove yourself from your natural time line you are immune to the effects created by dickering with time. You can go back and kill your grandfether. You will still be alive. However no one in your native time line would know you, your bank account, friends, and house would be gone since you never existed.
You start thinking about temporal mechanics long enough it drives you bonkers... whic might explain my mental state
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If I had a working time machine I think I might sell tours. I have two tickets to see Harry Chapin at the Bottom Line Theater before he died. I have a a few tickets to see Hendrix.
For a solo trip, on my own? I'd like to maybe go back to the 1930s with a digital camera that looks like a period camera. Ride the rail and take pictures and gather stories of folks during the Great Depression and the Dust Bowl. Same deal with the D-Day Invasion & American Civil War.
If I seriously wanted to frak with the time line and didn't want to come back I would do something that would change the world. I'd grab a bunch of horses, pigs, chickens, cows, and other livestock as well as seeds and head back to about 700 CE by the Great Lakes of North America. The Lakota adapted to horses in under 100 years. Dairy Cows have cow pox which offers a bit of immunity to small pox. Living in close proximity to animals (not mention having draft animals) helped fuel some of teh diseases that killed off the Native Americans.
In 1500 there was a city outside of Toronto (Canada) that had more people than London. The Mound Builders had large cities like that as well, not to mention some of the MesoAmericans). With draft animals, bronze or iron tools, and food crops that were not available in Pre-Columbian America this place would have been a hell of a lot different by the time 1492 rolled around.
I've learned enough about Pre-Columbian America to know all that "at one with nature peaceful people" hippy new age crap is bull.
That time travel bit I would give up my life in the now for in a heart beat.