# How exactly does FTL travel cause reverse time travel?

So I look at the Minowski diagrams, and I can understand that moving on the y means moving in time, and moving on the x means moving in space, but drawing some diagrams of my own, I find it impossible to see why it would imply FTL travel. All objects travel either directly on the x (instantaneous jump) or forwards. I can understand why you would have some weird visual effects (seeing yourself jump into hyperspace after the jump), but I cannot understand why it would imply time travel, and have given up all attempts to interpret the Minowski diagrams.

Could anyone help by explaining exactly what would happen if a spaceship was equipped with an FTL jump drive?

HallsofIvy
Homework Helper
No one can answer that because no one know exactly what an "FTL jump drive" would do!
It is true that the Lorentz transformations include
$$\frac{t- vx/c^2}{\sqrt{1- v^2/c^2}}$$

If we were to take v> 0, that denominator would be an imaginary number. What does that mean? Another way of looking at is that time appears to move more slowly for a frame as its speed (relative to the reference frame) approaches c. "In the limit", as v approaches c, time intervals go to 0 so one can imagine that, if you take v> c, time intervals would become negative, reversing time.

Of course, this is using one aspect of relativityb (time goes more slowly as speed increases) while denying another (that nothing can move faster than light). The more sensible thing to do is to accept all of the consequences of relativity including tht nothing can move faster than light.

Basically, what I meant by 'FTL jump drive' is that it disappears in one place and instantaneously reappears in another. It isn't affected by relativistic effects. If it were affected by relativistic effects, time runs sideways or something.

The faster you travel, the slower time goes relative to something else. ##t\prime=t\sqrt{1-\frac{v^2}{c^2}}## and so when your velocity squared reaches the speed of light squared ##t\prime=t\sqrt{1-\frac{299,792,458m/s^2}{299,792,458m/s^2}}## you simply end up with ##t\prime=0## meaning that time relative to something else is 0. This means that you effectively enter a changeless state.

The blood in your veins will stop flowing, you will stop breathing, you will not age, your body will not do anything, you enter a stage of eternal hibernation if you will... you could travel at the speed of light for a million years and then stop, the blood would resume flowing, you would start breathing again ect and as far as you're concerned, absolutely 0 time has passed. Nada.

So the fun part, if you go faster than the speed of light, even by 1m/s ##t\prime=t\sqrt{1-\frac{299,792,459m/s^2}{299,792,458m/s^2}}## which equals ##-\frac{1}{299,792,458}## or ##-3.3356......\times10^{-9}##

And that is why if you go faster than light, time will go backwards. Moreover though, with jump drives or more accurately warp drives, you're not going faster than light, you're manipulating space-time, making the distance smaller.

A simple analogy... If two cars at equal speed race round a track, the first car goes round the perimetre, the second car cuts across the track, who will get to the finish line first? The car that took the shortcut of course, but it didn't go faster than the first car, it simply took a shorter route.

I hope that clears things up.

Borek
Mentor
if you go faster than the speed of light, even by 1m/s ##t\prime=t\sqrt{1-\frac{299,792,459m/s^2}{299,792,458m/s^2}}## which equals ##-\frac{1}{299,792,458}## or ##-3.3356......\times10^{-9}##

Nonsense. Square root of a negative number is not another negative number, but an imaginary number. HallsofIvy already wrote that.

Basically, what I meant by 'FTL jump drive' is that it disappears in one place and instantaneously reappears in another. It isn't affected by relativistic effects. If it were affected by relativistic effects, time runs sideways or something.
Note first, that I don't believe time travel is possible.

However, if it was, you would get visual effects that can be modelled using sound. Sound travels at the speed of sound. What happens if something travels faster than sound?
First of all, you would get Cherenkov radiation, which is analogous to the sonic boom.
Secondly, you will loose casuality. Imagine a set of devices that communicate by sound. You can observe that they obey some laws regarding the order of their activation. When you exceed the speed of sound, you invalidate that laws.
Third thing is that you get two "sonic images" for an object that travels faster than sound. Imagine an object that makes a beep, moves faster than sound and makes another beep. You will hear two beeps coming from two separate objects apparently. If the moving objects beeps constantly, you will hear it as two beeping objects, moving in opposite directions and finally meeting at the point where the object's path is closest to you. This was once a proposed interpretation of annihilation of matter and antimatter, but now it's known to be false.

I've just realised that I've made a typographical error in my first post ("imply FTL travel" rather than "reverse time travel"), but all of you got the idea anyway.

So let's say that Alice jumps away from Bob by about one light second, quickly snaps a picture of the moon and returns to him to give him the picture before the moon's light reaches him. I see no reverse time travelling there.

@haael: I don't really get it. I imagine that all you would have would be sonic images that lagged behind the actual object.

stevendaryl
Staff Emeritus
So I look at the Minowski diagrams, and I can understand that moving on the y means moving in time, and moving on the x means moving in space, but drawing some diagrams of my own, I find it impossible to see why it would imply FTL travel. All objects travel either directly on the x (instantaneous jump) or forwards. I can understand why you would have some weird visual effects (seeing yourself jump into hyperspace after the jump), but I cannot understand why it would imply time travel, and have given up all attempts to interpret the Minowski diagrams.

Could anyone help by explaining exactly what would happen if a spaceship was equipped with an FTL jump drive?

First, let's prove a little theorem:

Theorem 1: If FTL travel is possible in every frame, then instantaneous travel is possible.

To see this, let an FTL rocket travel from the point $x=0, t=0$ to the point $x=L, t=L/w$, where $w$ is the speed of the FTL rocket. Now, switch to another frame using the Lorentz transforms.

$(x=0, t=0) \rightarrow (x'=0, t'=0)$
$(x=L, t=L/w) \rightarrow (x'=\gamma L (1-v/w), t' = \gamma L/w (1 - vw/c^2))$

Now, if we choose $v = c^2/w$, then we have the rocket arriving at time
$t'=0$ in the primed frame. So in the primed frame, the rocket arrives instantaneously.

So FTL travel in one frame implies instantaneous travel in a different frame. If instantaneous travel is possible in one frame, then it is possible in every frame, by the relativity principle.

Theorem 2: If instantaneous travel is possible in every frame, then back-in-time travel is possible.

To see this, just pick any two frames, F and F' with a relative (slower-than-light) speed v.

Send a rocket instantaneously according to frame F from
$x=0$ at time $t=0$ to $x=L$ at time $t=0$.

According to frame F', the rocket left at time $t'=0$ and arrived at time $t'=- \gamma vL/c^2$. Now, send the rocket back to the point $x=0$ instantaneously according to frame F'. The rocket arrives at time $t'=- \gamma vL/c^2$. Using the Lorentz transforms again, we see that:

$t' = \gamma (t - vx/c^2) = \gamma t$

So $t = t'/\gamma = - vL/c^2$

So the rocket gets back at a time BEFORE time $t=0$.

First, let's prove a little theorem:

Theorem 1: If FTL travel is possible in every frame, then instantaneous travel is possible.
(snip basically correct reasoning)
So FTL travel in one frame implies instantaneous travel in a different frame. If instantaneous travel is possible in one frame, then it is possible in every frame, by the relativity principle.

Non sequitur - the "relativity principle" itself is not confirmed here.
Consider:
HallsofIvy said:
Of course, this is using one aspect of relativityb (time goes more slowly as speed increases) while denying another (that nothing can move faster than light). The more sensible thing to do is to accept all of the consequences of relativity including tht nothing can move faster than light.
Positive statements (something normally happens) are better founded in experiment than negative ones (something is impossible).

For example parity violation - electromagnetic, gravitational and strong forces are completely symmetric against left and right.

Yet weak interaction exists - and includes complete parity violation.

There is also a CP violation... supposed to entail T violation.

Or does it?

It is claimed that a CPT violation would authomatically involve a Lorentz violation.

Suppose it does. What next?

Imagine that a hidden preferred frame exists. It has no interaction with the common physical processes - so time is dilated as if no preferred frame existed.

Except that the preferred frame does exist - and plays a role in some rare interactions.

Causing FTL.

Pretty clearly, if FTL travel A-B were possible, due to relativity of simultaneity you can always choose a frame where arrival at B is before departure from A.

But how about A-B-A, or A-B-C-A?

Is it possible to have FTL where A-A always takes zero or positive time irrespective of where B or C may be, or what the frame of observer is?

stevendaryl
Staff Emeritus
(snip basically correct reasoning)
Non sequitur - the "relativity principle" itself is not confirmed here.

I don't know what you mean by "not confirmed". If something is possible in one frame, but not in another, then that is a violation of the principle of relativity.

Of course, this is using one aspect of relativity (time goes more slowly as speed increases) while denying another (that nothing can move faster than light).

The proof that FTL implies back-in-time travel is the reason people believe that nothing (or at least nothing that can carry information) can travel faster than light. So basically you're bringing up "nothing can travel faster than light" in the context of a proof that nothing can travel faster than light.

It is claimed that a CPT violation would authomatically involve a Lorentz violation.

Suppose it does. What next?

That means that relativity is wrong. The argument here is that ASSUMING relativity, it follows that FTL implies backward-in-time travel (or the ability to send messages back in time).

Imagine that a hidden preferred frame exists.

Absolutely. There is no problem with FTL if there is a preferred frame.

stevendaryl
Staff Emeritus
Pretty clearly, if FTL travel A-B were possible, due to relativity of simultaneity you can always choose a frame where arrival at B is before departure from A.

But how about A-B-A, or A-B-C-A?

Is it possible to have FTL where A-A always takes zero or positive time irrespective of where B or C may be, or what the frame of observer is?

You can certainly cook up a theory in which
• There is a preferred frame, F.
• Ordinary objects moving at speed v relative to F are contracted by a factor of $\sqrt{1-(v/c)^2}$ in the direction of their motion.
• Ordinary clocks moving at speed v relative to F are slowed by the same factor.
• There is a special rocket (necessarily NOT made out of ordinary matter) that travels at speed $w > c$ relative to frame F.

This theory would be equivalent to SR for experiments involving clocks and rulers made of ordinary matter, but would be different from SR for experiments involving unordinary matter.

As I understand it, when looking at it from the primed perspective, Point A and B shift down the y axis, implying reverse time travel?

As I understand it, when looking at it from the primed perspective, Point A and B shift down the y axis, implying reverse time travel? [/QUOTE]

Don't you mean point A and the point "east" of A?

ghwellsjr
Gold Member
Here is a labeled version of your Minkowski drawing with a line going just from A to B (the markings along the top and right are for the yellow grid lines):

There is nothing special about this worldline. It's just a stationary object located at x=5 with time going from 3 to 4 (according to the frame depicted with the grey grid lines). Here's another spacetime diagram showing just the rest frame of this object:

Now I use the Lorentz Transformation process on points A and B to see what it looks like in a frame moving at 0.5c with respect to the first frame:

Now you can see that the object is moving at -0.5c and its time and space coordinates are the same as the yellow coordinates in the first drawing.

As dauto pointed out, the FTL occurs when the object instantly moves from its location near x=0 to Point A and then later from Point B back to near x=0.

The bottom line is that the Minkowski diagram does not say anything about whether or not an object can actually travel at FTL as you can draw any lines going any way you want. Normally, people use a Minkowski diagram to depict the rest frames of two different objects traveling at different speeds with respect to each other but that speed cannot be equal or greater than c.

By the way, what is the purpose of the blue line in your drawing?

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QuantumPion
Gold Member
An interesting, related paradox involves merely sending messages FTL. If A sends a message to B FTL, and then B sends a response back to A FTL, it is possible for A to receive the response before sending the initial message. This leads to all sorts of logical paradoxes.

QuantumPion
Gold Member
I totally wrote my post first but DrGreg got his in first using his tachyonic antitelephone. :p

@ghwellsjr: The blue line is supposed to be a relativistic object, and the light blue lines its Lorentz-transformed frame of reference.
Looking at your diagram, it seems that the origin of the path to Point A (from the frame of the relativistic object) would be roughly -1.5, 3.5, while point A itself is 4, 0.5. That means that the net movement in time is about -3 units. Still equipped with the jump drive, the ship would be able to jump to its past self and stop it from ever leaving in the first place! Out the window with casualty!

ghwellsjr
Gold Member
@ghwellsjr: The blue line is supposed to be a relativistic object, and the light blue lines its Lorentz-transformed frame of reference.
I don't know how you arrived at that. Usually, when we talk about an object's frame of reference, we mean one in which it is at rest, meaning its worldline is aligned with the time axis or at least parallel with it so that there is no change in the x-axis as a function of time. You didn't draw the dark blue line aligned or parallel with the light blue ct' line so you need to explain what you mean.

Looking at your diagram, it seems that the origin of the path to Point A (from the frame of the relativistic object) would be roughly -1.5, 3.5,
Where did you get these numbers from? I can't figure this out.

...while point A itself is 4, 0.5. That means that the net movement in time is about -3 units. Still equipped with the jump drive, the ship would be able to jump to its past self and stop it from ever leaving in the first place! Out the window with casualty!

Okay, I obviously am misunderstanding these diagrams somewhere with the reference frames. Ignore that.

About the numbers, I assumed that the ship would be jumping from 0, 3 to 5, 3, then from 5, 4 to 0, 4. I then visually estimated the coordinates on the yellow grid, then applied them to the last diagram.

I'm doing something wrong, right?

ghwellsjr
Gold Member
Ok, I see what you mean now by the origin of the path to point A and you determined the coordinates correctly in the yellow frame and your conclusion is that traveling at faster than the speed of light presents all kinds of problems, correct? Does this answer your question?

If something is possible in one frame, but not in another, then that is a violation of the principle of relativity.

Is this only for SR or applicable to GR also?

Consider two observers - one stationary far outside the event horizon of a black hole, and another falling into the black hole.

In the one frame of the outside observer, the in-falling observer never 'quite' reaches the event horizon till the end of eternity, while in the other frame of the in-falling observer he/she crosses the event horizon in finite time. Doesn't this mean that something (an 'event' - of crossing the event horizon) is possible in one frame and not the other? There is no violation of relativity principle either in this situation, I believe.

A.T.
Doesn't this mean that something (an 'event' - of crossing the event horizon) is possible in one frame and not the other?
The principle of relativity applies only to inertial frames, which in curved space-time are only local approximations, and don't extend into infinity. Even in flat space-time you can have an accelerating frame where a horizon forms, and some events are delayed until infinite coordinate time:
http://en.wikipedia.org/wiki/Rindler_coordinates#The_Rindler_horizon

Consider two observers - one stationary far outside the event horizon of a black hole, and another falling into the black hole.
A frame is not a single observer. It's more like continuum of observers.

@ghwellsjr: So what I did with the Minkowski diagram was right? That the path would bend backwards in time?

ghwellsjr
Gold Member
@ghwellsjr: So what I did with the Minkowski diagram was right? That the path would bend backwards in time?
Yes, but that just illustrates that two spatially separated events (the endpoints of your so-called "path") that are simultaneous in one frame are not simultaneous in other frames. In some frames one end of your path happens before the other end and in other frames it's reversed. You can have an object such as a rod that stretches along the path that you describe with its endpoints defined by simultaneous events in its rest frame and the spatial distance between the events is the Proper Length of the rod. In other frames the events have no particular significance.

This is not generally the way that FTL is presented. It is usually just an extrapolation of the thought that in a particular frame as a clock moves faster and faster its rate of time passage goes slower and slower until, it is thought, at the speed of light the clock stops and then when it goes faster than light, it starts running backwards. It's not any more complicated than that but since it is an erroneous thought, it is not supported by any Minkowski diagram nor by the Lorentz Transformation on which the diagrams are based.

But if the Lorentz factor is multiplied by i past lightspeed, that isn't really negative time, is it?

stevendaryl
Staff Emeritus
But if the Lorentz factor is multiplied by i past lightspeed, that isn't really negative time, is it?

The argument that FTL implies back-in-time isn't about applying the Lorentz transforms for a velocity $v > c$. What the argument shows is that if a rocket (or signal) travels faster than light from point A to point B, according to one frame F, then from the point of view of a second frame F' (moving slower than c relative to F), the rocket arrives at B before it left A.

So the "negative time" is not for the rocket that's traveling faster than light. It's for a second, slower-than-light frame.

ghwellsjr
Gold Member
But if the Lorentz factor is multiplied by i past lightspeed, that isn't really negative time, is it?
That's correct, i, is an imaginary number and in fact when you take the square root of a negative number like -4, there are two answers, 2i and -2i, just like when you take the square root of a positive number like 4, there are two answers, 2 and -2.

.Scott
Homework Helper
Basically, what I meant by 'FTL jump drive' is that it disappears in one place and instantaneously reappears in another. It isn't affected by relativistic effects. If it were affected by relativistic effects, time runs sideways or something.
Cool. So there's no "travel" involved. In fact, you won't mind if I simply destroy all information about the traveler at his starting position and recreate him at his destination. And I will use local mass to do it.

That way we are only talking about FTL transportation of information - not of any mass or charge.

To test it, we'll transport a radio transmitter from Earth to a location 1 light-hour away. And we then verify this be receiving its signal exactly 1 hour after the transport.

While doing this, aliens happen to be cruising through our Solar System and observes our experiment. Because they are traveling at a significant fraction of the speed of light, they may not agree that the radio's departure time matches its arrival times. It may see a delay, or, if traveling in the right direction, it may see the arrival occur before the departure - an apparent step back in time.

When the aliens report this to us, we decide to take advantage of it. We make three copies of our transporter placing them at the ends of an equilateral triangle - 1 light-hour on a side. On our first test run, we simply transport an item from point A to B to C and back to A. The round trip time is almost zero.

Then, on our next test run, we accelerate the devices at B and C so that when they transport in their reference frame, it appears to move back in time in our (point A) reference frame. Several minutes before we initiate the transport from point A, our radio arrives from point C!