# OK I can travel faster than light so where's my damn time machine?

Suppose I have a device that will teleport me from one point to another with a delay equal to (distance/1.5C) so effectively travel at 1.5 times the speed of light. I can teleport to Alpha Centauri and back in a bit under 6 years round trip. I don’t actually accelerate to this rate so I don’t have to worry about infinite energies, but I do violate causality in the sense that there exists a reference frame (with a very high velocity relative to mine) where I arrived at Alpha Centauri before I left earth.

But this isn’t what I really want. I want to go back and kill my grandfather (never did like the old man), or at the very least find the winning lottery number in time to play it. Is there any way to turn my device into something that usefully tells me the future or creates a paradox? Or am I stuck only breaking causality for some comic rays that don’t even care whether or not I exist?

Did you remember to send in the warranty and the coupon along with a copy of the receipt?

Can’t get your time machine without proof, you know.

You’ve got most of it already. As you said, there’s a reference frame wherein you get to alpha Cen before you left. And there’s also a reference frame, for the reverse trip, wherein you get back to Earth from alpha Cen before you left. So switch reference frames before your trip (which can be done with ordinary non-physics-defying rockets), and there you go: You’re now back at Earth before you left.

To be specific, suppose you use your teleporter once to transport yourself 1.5 light years away in the direction of Alpha Centauri, in 1 year. Which we call position p = (x, t) = (1.5, 1).

According to the Lorentz transformation, for an observer at Earth at the moment you left, but moving at velocity v in in the direction of Alpha Centauri, his perception of the time coordinate of p is (in c = 1 coordinates):

t’ = (t - vx)/sqrt(1-v^2) = (1 - 1.5*v)/sqrt(1-v^2)

which is less than zero if v > 2/3. So any objects moving towards Alpha Centauri with a speed greater than 2/3*c will see your jump as going backwards in time.

If you want people on Earth to see you going backwards in time all you have to do is, instead of starting at rest relative to the Earth, start out moving away from Alpha Centauri at a speed greater than 2/3c. Then from your point of view, the Earth is an object moving towards Alpha Centauri at a speed greater than 2/3c – exactly the kind of object that sees you going backwards in time.

(And I expect that I have probably made at least one error in math of direction in the above, but that’s OK, Chronos is around.)

Don’t you remember? We did all this last week. Keep up.

And we went over it AGAIN three weeks from now. That was fun.

According to special relativity there’s something called a ‘plane of simultaneity’ which tilts according to your speed of motion with respect to another location. This plane describes which other locations you are notionally simultaneous with, and (for the purposes of this thought experiment) I presume that these are the locations that an instant teleport machine could reach.

Here’s a nice graphic in .gif form from wikipedia of a plane of simultaneity ‘tilting’ back and forth according to the motion of the observer.
https://en.wikipedia.org/wiki/Relativity_of_simultaneity#/media/File:Relativity_of_Simultaneity_Animation.gif

Now the weird thing (to me) is that the further you teleport, the more pronounced this effect gets; if you teleport as far away as the Andromeda Galaxy then your relative motion need only be as fast as that of a sports car and you are simultaneous with events 24 hours in the future (or past, depending on your direction). So to create a time machine all you need to do is carry your teleport booth in the back of a fast sports car driving round a racetrack in various directions teleporting off to Andromeda and back periodically.

(this is a rather mangled description of the Rietdijk–Putnam argument, which I don’t really understand, so if I’ve got anything wrong please put me right.
https://en.wikipedia.org/wiki/Rietdijk–Putnam_argument )

In general, as long as the geometric mean of your “warp speed” and your conventional speed is greater than c, you can time-travel. So if your rocket engines can get you to .1c, then your warp drive needs an effective speed of 10c, and so on.

So I am correct that I can only really make use of such a teleporter if am able to travel at a significant fraction of light speed conventionally. My teleporter is nice a parlor trick to freak out operators at CERN, but not practical to alter history or tell the future.

But if I upped the ante and developed a teleporter that was truly instantaneous and then got in a rocket moving at 300m/sec teleported to Alpha centari (still moving at 300m/sec away from earth), turn the ship around and travel towards earth at 300m/sec , and then teleport to earth. I should arrive (based on my understanding of leahcim’s equations) about 4 minutes before I left, minus the time it takes to turn around. Even that is a whole lot of work.

Sigh, and they make it look so easy in the comic books.

Ray Romano? Ohhh, you said comic rays. Nevermind.

That’s still plenty of time to ram your rocket into its past self, making it unable to take off in the first place, if that’s your thing.

What’s the point in even trying, when we already know he’ll miss?

Robert Heinlein used this concept in “Time Enough for Love”.

They discovered that the process of transition implied a time component, and thus could be used to move through time. The ship’s AI didn’t like it though; she said it felt like intentionally making a bad landing.
(Or was it “To Sail Beyond the Sunset”? One proceeds out of the other.)

Yeah, but by this time tomorrow, it could be yesterday.

It depends on your frame of reference. If FTL is permitted, in some situations the other proceeds out of one.