Help me understand this example about FTL = time travel

Question

Searching for clear explanations of why FTL equals time travel I found this http://www.physicsmatt.com/blog/2016/8/25/why-ftl-implies-time-travel Go at the bottom, when he draws the blue lines. He says that those are the lines of costant time and they are parallel to the space line of the relativistic ship. This imply that the time line is perpendicular to the space line. But before he had said that the time line of the ship was the one named "space ship time" that is skewed towards the space line. So it seems to me that there are two time lines for the ship: the "space ship time" line and the one perpendicular to the space line that isn't drawn.

To say it in other words: if we have a relativistic ship its space and time lines should both be skewed towards the light line, right ? Because light speed is costant for any observer. This means that the light line should always be at the same distance from both the time and the space line. But when the blue lines are drawn the time line is said to be perpendicular to the space line. This means that the light line would be closer to the space line that the time line. Thus meaning that light would be faster for that observer.

The problem is that if we consider the "space ship time" line as the correct time line there is no time travel.

EDIT: I think I'm starting to understand. In other examples I found online they suggest to rotate the graph, so that the time line of the space ship is pointing towards high. That makes evident the FTL communications between Earth and AC are distorted from the point of view of the ship, with the ones sent from AC to Earth arriving before they are sent. If the ship can sent messages at near istantaneous speed those messages would be a line perpendicular to their time line, but these lines woldn't be perpendicular to the time lines of Earth and AC, arriving after or even before they are sent. So to avoid time paradoxes you have time lines of different reference frames to always make an acute angle. If they were to make a dull angle (by going FTL) that would mean they are going in the past of each other. In fact you could rotate the graph to make on of them your y axis and see that the other is going down (towards the past). Is this correct ?

EDIT 2: Graph I made In the horrbible graph above we have Earth and AC time lines (black), the realtivistic ship time line (blue) and near istantaneous FTL communications (purple). The first message is sent from AC to Earth, from their point of view is istantaneous, but from the point of view of the ship is going back in the past. When they receive it they sent a message back to AC (purple line perpendicular to blue line). Again this message is istantaneous from their POV, but from AC POV it arrives before it is sent. So AC receive an answer to a message that they haven't sent yet. What did I do wrong ?

Answer 1

I guess the source of your confusion is that you suppose that "time lines" (dots on diagram having same coordinate in ship's frame of reference) and "space lines" (dots on diagram having same time in ship's frame of reference) must be perpendicular on diagram. This is not true. Actually both sets of lines are skewed towards the direction of "light line". Here are several examples: https://en.wikipedia.org/wiki/Minkowski_diagram

It was quite difficult for me to follow the article. It seems that actually author did NOT show that FTL travel must mean time machine is possible.

What author showed is that if FTL message transmission is possible than in some frame of reference the message would be received before the message was sent. It seems impossible (all the results of Relativity Theory seem impossible at first), but this is not a time machine yet.

Time-travel paradoxes would happen if someone would be able to receive the message before he sends it out (this someone send a message to himself in the past). If arbitrary fast message transmission is possible than it is quite easy to send messages to the past. In the example described in the article observer sees:

1) message from Earth received on Alpha Centauri

2) later on this message is sent from Earth

and the observer has some time to get the contents of message received on Alpha Centauri and transmit it to Earth so that it would be received before it is sent. But to do it would be necessary to transmit information between AC, observer and Earth quite fast: faster than light and faster than the original message from Earth to AC was sent.

So, if Relativity Theory is correct and you can send messages arbitrary fast, than you have time-travel paradoxes.

Actually if one can send signals just a little bit faster than light you would be able to send messages to yourself in the past. But the article does not explain why and how.