Here's an example. Say you start on world line Beta where the world is like what it is now. Then you go back and kill your grandfather***, then after you kill your grandfather, the world lines will diverge, and you will be on world line Alpha. It will be a wholly different world line where your grandfather got mysteriously killed by someone, and a world where 'you' don't exist, but you can still be on the world line because you are from world line Beta, and therefore, is a different person because your grandfather still exists on world line Beta. No paradoxes necessary.
*** Connor Stein also stated that you can only send information back in time, so you can only send a memory (send information back in time) of trying to kill your grandfather back in time in order to create a virtual body, which uses that memory to execute the task.
*** Connor Stein also stated that you can only send information back in time, so you can only send a memory (send information back in time) of trying to kill your grandfather back in time in order to create a virtual body, which uses that memory to execute the task.