# At what rate would I travel back in time at $2c$? [closed]

I was thinking about Einstein's equations and the solutions using imaginary mass and imaginary time.
How fast would I have to travel over $c$ to be traveling back in time at minus one meter per second? At what rate would I be traveling back in time at twice the speed of light?

(I thought this could be a problem if it takes me 50 years to go back 50 years. I could die before I arrive to kill my grandfather.)

• At no rate. You can't travel back in time. Boring, isn't it? – CuriousOne Sep 15 '15 at 5:28
• Yes I know that:-) In the solution that was derived from GR, what speed would you need for -1sec per +1sec. What does the maths say? – Jitter Sep 15 '15 at 5:41
• It says that you can't go back in time. :-) – CuriousOne Sep 15 '15 at 5:56
• Imaginary time has nothing to do with FTL. – Carl Witthoft Sep 15 '15 at 15:00
• Can the topic be removed and placed on sci-fi stack exchange or something if there are no valid peer reviewed papers on traveling back in time. – Jitter Sep 15 '15 at 18:25

You and your friends line up. You sit down, you spend some time synchronizing your watches. And you make a plan. You measure that you are all a meter apart. And you have 299,792,458 friends. You plan out how to do a wave. You plan it so when you stand up the person next to you stands up a little later and the person next to them a little later and so on until the last person stands up a little over a second later. You made a wave that travelled at the speed of light. And you did it by all standing up when you were supposed to according to the schedule.

What is amazing about that wave is that everyone agrees it went at the speed of light. And everyone agrees the wave started at one end and went to the other.

Now you say it was just a bunch of people looking at their watch and their schedule and standing up when the schedule said to. We could schedule it so they all stand up a little bit faster so the person on the other end stands up a half second instead of a full second after the first one stood up.

Yes you can. And now you made a wave that "travels" at twice the speed of light. So did anything go back in time? Or ... Does the question even make sense.

Remember how I said that when the wave went at the speed of light, everyone agreed. I didn't mean just the people sitting and standing agreed. I mean everyone agreed. People rushing to the right at 99% the speed of light. People rushing to the left at 99% the speed of light. Absolutely everyone agreed that wave went at the speed of light.

This is not true for this second wave. For the second wave, some people will think everyone stood up at the same time. Other people will think the people one one side stood up before people on the other side and yet another person will think the people on the other side stood up before people on the one side.

So some people will think the wave travelled at an infinite speed (everyone standing up together at the same time). Others will think it started at one end and went to the other end. When you are certain it started on the left and went to the right then you think that person standing up on the right happens in the future of the person standing up on the left. And other people disagree.

It isn't an exciting form of time travel, it's just that if you could get from a when-where to another when-where at greater than the speed of light then different people will disagree about which when-where happened first. But it does not matter, they don't actually affect each other, so who cares if you think one happened before the other.

People disagree about what happens when. But when you can get from one place to another at less than the speed of light, then everyone agrees which happened first. And that is great because it means you don't accidentally run into your ancestor just by running fast.

• Fantastic explanation. Two thumbs up. – CuriousOne Sep 15 '15 at 5:59
• @Timaeus. Can you give me a similar scenario using human waves to prove you can't time travel into the future? – Jitter Sep 15 '15 at 6:06
• @Jitter No. Each of the humans in the wave travels to the future. And it is possible to travel to the future. We general build our machines out of parts that themselves travel to the future, so it would be hard to make a machine that can't travel to the future. A black hole might be your best bet if you don't want to see the future. – Timaeus Sep 15 '15 at 6:09
• @Jitter "We are all time travelers moving at the speed of exactly 60 minutes per hour" - Spider Robinson – Steeven Sep 15 '15 at 6:12

At what rate would I be traveling back in time at twice the speed of light?

What does the maths say?

The issue is that you can't travel faster than light. The speed of light $c$ is the maximum speed limit. The faster you move relative to an observer, the slower you age because time passes slower. But only slower, not backwards.

From pretty easy considerations of observing from different frames (see Wikipedia) a time dilation formula is derived:

$$\Delta \tau=\frac{\Delta t}{\sqrt{1-\frac{v^2}{c^2}}}$$

where $\Delta \tau$ is the slower moving observer and $\Delta t$ is the faster moving observer, moving at speed $v$.

For larger $v$, the $v^2/c^2$ part is less negligible, so the denominator is more dominant as it has a value lower than $1$. This extends $\Delta \tau$ to a bigger value than $\Delta t$. Let's try plugging in $v=2c$:

$$\Delta \tau=\frac{\Delta t}{\sqrt{1-\frac{(2c)^2}{c^2}}}=\frac{\Delta t}{\sqrt{1-4\frac{c^2}{c^2}}}=\frac{\Delta t}{\sqrt{-3}}$$

Not working. Imaginary solutions. There is no reel solution to this. Let's try plugging in any value higher than $c$ - that would be $v=kc$ for $k>1$:

$$\Delta \tau=\frac{\Delta t}{\sqrt{1-\frac{(kc)^2}{c^2}}}=\frac{\Delta t}{\sqrt{1-k^2\frac{c^2}{c^2}}}=\frac{\Delta t}{\sqrt{1-k^2}}$$

It should be pretty clear that for any value of $k$ higher than 1, the result is a squareroot of a negative number, meaning imaginary solutions only.

Plugging in exactly $v=c$ gives a denominator of $0$, which could be interpretted as an infinite value of $\Delta \tau$. It that case, time is not running for the observer moving, and this is the extreme case.

Bottomline is that it is not possible to mathematically reach a $\Delta \tau$ which is smaller than $\Delta t$, and it is certainly not possible to reach a negative time. So, backwards time travel is simply not mathematically possible. It is not possible to move faster than light. $c$ is the very maximum limit.

• Using the Goedel universe solution or cosmic strings, is there an answer to my question. – Jitter Sep 15 '15 at 6:32
• @Jitter In a Gödel universe or other spacetime with closed timelike curves there simply isn't always a clear demarcation of a past and a present and a future. For instance you can travel from a when-where to the exact same when-where but when you look around you, classical General Relativity predicts you see nothing weird going on in the near vicinity around you. Each region looks normal with an apparent past and future. Its like if you had a cylinder with time measured around the circle and space along the cylinder length. Locally it looks like a flat normal piece of paper, no weird lengths. – Timaeus Sep 15 '15 at 16:50

If you travel faster than light you wouldn't be going back in time from all reference frames, just some reference frames. If you were going faster than light then from some reference frames you would be traveling faster than light but still going forwards in time and from one reference frame you would be going infinitely fast while from another you would be going back in time. In order to change from moving at FTL forwards in time from a reference frame to backwards in time from that reference frame you would have to first cross through infinite velocity.

Interestingly relativity doesn't technically forbid faster than light travel it just forbids things with real rest mass from traveling faster than light and says that the speed of light is the same in all reference frames, which would still hold even if some of those reference frames were FTL. If there was a particle with imaginary rest mass it would not only travel faster than light but it could never slow down to subluminal velocity. It could go at any speed faster than light though and could change from one FTL speed to another FTL speed. From the reference frame of something with imaginary rest mass the speed of light would still be the same and it would still be moving at subluminal speed from its own reference frame because from its reference frame everything else would be moving at FTL. The speed of an object traveling FTL would not be the same from all reference frames but its direction of motion would be the same from all subluminal reference frames. From the reference frame of something traveling FTL space and time would get swapped so what would be the three spatial dimensions from subluminal reference frames would be time dimensions and what would be the time dimension from subluminal reference frames would be a spatial dimension from the reference frame of something traveling FTL.