If you could travel back in time it should be possible to meet yourself.
With most writings involving time travel this is nearly always avoided at all costs because of the major effects it has on the Space Time Continuum. Also it is avoided as the same matter cannot occupy the same piece of space at different times. Although it could be different matter occupying the same piece of space at the same time. Either way a reason is to be found as things begin to get really complicated as we are about to see.
As discussed in - Holiday To The Past - when a Time Traveller goes back in time the Time Traveller's World Line simply moves to a different position on the Cosmic World Line. The example given was a trip that was deliberately back to a time before the Time Traveller was born. Also the length of time in the past was the same as the length of time in the present so that the Time Traveller's World Line was still in sync with the Cosmic World Line. The Time Traveller's World Line clock was running at the same rate as the Cosmic Time clock.
Now imagine a time travel trip where the Time Traveller returns to a time earlier in the Time Traveller's life. There is now a time where the Time Traveller exists twice and if the short trip in Time is also a short trip in Space there is the possibility for the two to meet. Taking a trip for a 40 year old Time Traveller in 1998 to travel back 20 years in time to 1978 for only a few days before returning to 1998 the Time Traveller's World Line will be as shown in the following diagram.
We can see from the diagram that a vertical line in 1978 clearly shows that there are two versions of the Time Traveller - one that is 20 years old and one that is 40 years old.
So let us see what happens when they meet. As far as the 20 year old is concerned he is meeting a stranger for the first time. The 40 year old however has been here before and presumably should be able to remember the discussion that took place. But how can this be? The discussion in 1978 involves a person from 1998 which is 20 years in the future and hasn't happened yet. So the meeting cannot take place.
But if it is happening then the 40 year old will be able to remember exactly what took place. This would mean as far as the 20 year old was concerned that the future was predetermined. In effect the 40 year old could present the 20 year old with a script of the whole discussion. As far as the 40 year old is concerned the meeting is part of the Past
So could the 20 year old then decide to say something else and therefore change the Future?
Or could the 40 year old do the same and therefore change the Past?
This simple meeting throws up many questions and apparent paradoxes. Is the Future predetermined? If so then everything that can happen in the Universe has already happened. It is only the position of Now that changes. Is the Past unalterable? If so then Time Travel if possible is safe. With all these questions perhaps it is easier if Time Travellers do not return to meet themselves after all!
However let's take these ideas further with a fuller, deeper example.....
A Strange Meeting -
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Anthony Edwards - December 1996