I am looking to show mathematically (i.e. with derivations and algebraic relationships) that FTL implies time travel. Here's what I got:
Suppose we have 2 ships $O_1, O_2$ traveling at $0.9c$ in opposite directions and originating at a point called $0$ at time $0$.
Suppose now 1 year has passed in the frame of reference $O_2$ where $O_2$ is still. In this frame of reference $O_1$ has been travelling at $ \frac{0.9c + 0.9c}{1 + \frac{0.81c^2}{c^2}} = .9944c $
So according to $O_2$, a total of $(1) \times \sqrt{1 - \frac{(.9944c)^2}{c^2} } $ years have passed on $O_1$'s clocked.
So now suppose $O_2$ instantaneously sends a "visitor" to $O_1$. That visitor when it arrives at $O_1$ finds that year is $(1) \times \sqrt{1 - \frac{(.9944c)^2}{c^2} } $ in $O_1$ and therefore the year is
$(1) \times \sqrt{1 - \frac{(.9944c)^2}{c^2} } \sqrt{1 - \frac{(.9944c)^2}{c^2} } $ in $O_2$ (since from $O_1$'s frame of reference O_2 is doing the moving). So if the visitor instantenously teleports back they return to $O_2$ at time $1 - \frac{(.9944c)^2}{c^2} $ years whereas they started at time $1$ years.
Does this capture the essence of the FTL implies time travel argument? I made my life easy by letting the visitor teleport. Suppose I had the visitor travelling at some speed $v >> c$, I can still use some algebra and perhaps elementary calc (since I'm too lazy to do this elegantly) to argue how much time passes on $O_1$ before the visitor arrives (consistent with $O_2$'s interpretation of $O_1$'s clock).
Why this wasn't answered for me earlier:
I checked:
Can FTL-Communication between two points in the same frame of reference break causality?
What spacelike, timelike and lightlike spacetime interval really mean?
What's the problem with light traveling at speed higher than $c$?
with the terms "Special Relativity FTL implies Causality" and really couldn't find a plain and simple mathematical proof.
I came across:
How does faster than light travel violate causality?
But I couldn't understand the where the accepted answer got their $2796$ from in their last bullet point of the FTL implies Causality thought experiment. So I decided I had no choice but to set it up from scratch myself and have someone comment.
