# Can someone review this mathematical derivation that in Special Relativity FTL implies Time Travel

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:

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.

• Please note that check-my-work questions are generally considered off-topic here. We intend our questions to be potentially useful to a broader set of users than just the one asking, and prefer conceptual questions over those just asking for a specific computation. – ZeroTheHero May 27 '18 at 2:56
• That is fair, but there is a conceptual question here in what I’m asking. Never on the site was a clear, down to the algebra explanation of how faster than light implies time travel given. So I decided to ask based one that. Hence my mention of the links in the second half – frogeyedpeas May 27 '18 at 2:58
• instantenously teleports This immediately looks like a problem as the one thing you can't do in SR or GR is instantaneously travel anywhere. – StephenG May 27 '18 at 4:01
• So my thought process: you can’t FTL either so I assumed if we break that faster than c travel we might as well, make it into travel that is arbitrarily faster to keep the math simple. If the question suffers substantially from that I’ll edit it to have a body traveling at 10c and re-calculate the numbers. I might have taken too much of a liberty there – frogeyedpeas May 27 '18 at 4:03
• Keeping the maths simple is not the goal. When the maths is simplified you can't do that by breaking a rule that's so fundamental to the model. – StephenG May 27 '18 at 4:26