Lesson 5 - Faster Than Light travel yields Time Travel

page1 For this lesson I want to show how faster than light (FTL) travel could be used to do time travel. Even if we just send an instantaneous message, it would allow us to send a message back in time.

page2 Let's start with two people, $A_1$ and $A_2$ , in the same reference frame. At this instant, $A_1$ sends a message to $A_2$ . I'm going to start adding events in red that happen in order: $P_1$ is when $A_1$ creates the message, $P_2$ is when the message is sent by $A_1$ . $P_3$ is when $A_2$ receives the message.

page3 At $P_4$ , let's say $A_2$ transmits the message to another observer who is traveling relative to the $A$ s.

page4 I'm adjusting my velocity and adding two new observers, $B_1$ and $B_2$ . I'll make them green. Notice that in this reference frame, $P_4$ happens before $P_1$ . I'm going to pick an instant between these two events. Now we have $B_2$ pass an instantaneous message to $B_1$ .

page5 I'll add two new events: $P_5$ is when $B_2$ sends the message, and $P_6$ is when $B_1$ recieves it.

page6 We just need to go back to the first event, $P_1$ and see that $B_1$ can send a message to $A_1$ because they both pass through the same event. And now we have a causal loop: $A_1$ can send a message through $A_2$ , $B_2$ and $B_1$ , back in time to himself, by following the events $P_1$ , $P_2$ , $P_3$ , $P_4$ , $P_5$ , $P_6$ , and back to $P_1$ .