# FTL and causality

1. Nov 20, 2014

### serinorah

I understand that the questions behind causality and FTL have been brought up numerous times, and I have read several of the old threads. My question isn't with how is causality violated, it is always assumed that any signal sent would appear to travel backwards in time from one point of reference or another. I don't exactly understand why since all examples given to me always seem to make reference between when the events where OBSERVED after having previously stated that when an event is observed does not imply causality violations. Is there an equation which would explicitly clear up how exactly a signal travelling faster than light would move backwards through time. Please don't use an example with the mirrors because if you were to travel faster than light you would leave the photon behind with nothing for it to bounce off of unless you were to stop and wait for it to catch up. This isn't for if FTL is possible since there are plenty of other much more obvious reasons why it isn't. I am just trying to understand why a signal or ship would move backwards in time.

2. Nov 20, 2014

### DrGreg

3. Nov 20, 2014

### stevendaryl

Staff Emeritus
It's really just a fact of relativity that FTL in one frame means back-in-time in another frame. Some terminology: in SR, an "event" means a particular place at a particular time. So "The top of the Empire State Building on July 1, 2004 at 12:00 p.m. Eastern Standard Time" would specify an event. Two events, $A$ and $B$ are said to be "spacelike separated" if they are so far apart that it is impossible to send a light signal (or something slower) from $A$ to $B$. In other words, the time between the events is too short for even light to travel the distance between them.

If two events are spacelike separated, then whether $A$ happens before $B$ or the other way around is relative to the observer.

Now, here's something that is possible in SR: You have two observers, call them Alice and Bob, who are in motion relative to each other. It is possible to have three events, $A$, $B$ and $C$ such that:

• Alice sends an FTL signal from $A$ to $B$, where it is received by Bob.
• Bob sends an FTL response from $B$ to $C$, where it is received by Alice.
• $C$ happens before $A$ (according to both Alice and Bob).
So if this happens, Alice could receive a response to her FTL message (event $C$) BEFORE she sent the message (event $A$).

That can never happen with slower-than-light messages.

4. Nov 21, 2014

### serinorah

Dr. Greg I appreciate the link. It was close to what I was looking for. I had previously tried to use the Lorentz factor to show myself why something would move backwards in time but I couldn't reconcile the fact that anything other than a specific set of numbers produce negative time. Some numbers of c actually turn into positive numbers causing you to move through time faster than the outside world so it didn't make sense to me to say "Ok so a couple of number make time move backwards a couple cause you to to move through time a greater rate but most of them cause you to move through time at a factor of i. This obviously means backwards time movement." Is there no definitive equation for what happens FTL or is it just so accepted that that is what happens that scientists don't care anymore? If it is the later isn't that just like how people were with the world being flat and the sun moving around the earth, the whole you can't prove me wrong because even if most of the data doesn't support what I am saying some of it does so ha.

EDIT: I went over the numerical proof they gave, just to be sure I didn't miss anything. That explains pretty well what's going on and I understood the point they were making but I'm not understanding why they just let c=1.

Last edited: Nov 21, 2014
5. Nov 21, 2014

### PAllen

Letting c=1 is just a choice of units (e.g. use light seconds for distance). Nothing about physics can possibly be affected by choice of units.

6. Nov 22, 2014

### Staff: Mentor

That happens automatically if you measure time in seconds and distance in light-seconds, or time in years and distance in light-years, or the like. The speed of light is one light-year per year, is it not?

You'll see this done a lot, both because it means that we don't have to clutter our equations with a bunch of factors of $c$ and $c^2$ and because it's easier to read. If you choose your units so that the speed of light is 1, you can see at a glance that .5 is half the speed of light; if you use meters and seconds, half the speed of light is the rather less obvious $1.499\times{10}^8$ meters per second.

7. Nov 22, 2014

### stevendaryl

Staff Emeritus
You do know that if Alice is moving at speed $v$ relative to Bob, then the relationship between Alice's time coordinate $t$ and Bob's coordinate time $t'$ is not simply:

$t' = \sqrt{1-\frac{v^2}{c^2}} t$

The full equation is:
$t' = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} (t - \frac{vx}{c^2})$

So the full transformation for $t'$ depends on $x$. If $x$ is very large, then $t'$ can be negative, even if $t$ is positive.

8. Nov 22, 2014

### serinorah

I understand the formula my statement was about an object travelling faster than light then its Lorentz factor would be a mathmatically correct but physically nonsensical answer. Even the formula about for the relationship between their times doesn't work when Alice is traveling faster than light.sure the number is negative but the number itself is some version of i which is useless, so the total answer is negative uselessness. Also does the same causality problems arise when used with something that is moving spacetime itself, like the ergosphere of a rotating black hole. Relative to the rest of the universe anything caught within it, including signals, would be travelling faster than light even if locally they are not. So if someone within that ergosphere were to send a signal out to someone just outside and then they were to send a signal back would there still be a violation?

Also thank you guys for clearing up the c = 1 thing its been a long several days with a sick daughter so I didn't even think of that.

9. Nov 26, 2014

### georgir

Are you familiar with the classification of possible separations between a pair of events?
Two events always have one of three types of relation between them, irrespective of the reference frame used:
- timelike - the normal case, FTL is never required to get from one to the other, the same one always follows the other in all reference frames, and there exists a reference frame where they happen at the same place;
- spacelike - FTL is required to get from one to another in every reference frame, their ordering can change depending on reference frame, and there exists a reference frame where they are simultaneous;
- null - the border case, events that are always separated so that you need exactly lightspeed to go from one to the other.

Allowing unrestricted FTL communication (i.e. communication between any spacelike separated events) is equivalent to allowing backwards-in-time communication between any two timelike separated events, because for any two timelike separated events there exist some other event that is spacelike to both of them and so it can be used as proxy.

Example: a set of two normal timelike events A before B, A="I guess a number and write it down", B="You pick a number and tell it to me". And now a spacelike separated from both M="In a galaxy far far away, a Jedi FTL-force-senses what number you have picked, because B is in his past, then starts to move just a bit so that A is in his future and FTL-mind-tricks me to guess that number".

[EDIT: corrected a pretty bad mistake thanks to ghwellsjr]

Last edited: Nov 26, 2014
10. Nov 26, 2014

### ghwellsjr

You've got your spacelike and timelike interchanged.

11. Nov 26, 2014

### Jonathan Scott

Note that there are no causality problems with signals propagating at up to infinite speed relative to a specific "preferred frame", where the motion relative to any other frame is described by the Lorentz transformation as usual. This is of course in conflict with the principle of relativity and there is no evidence that such a preferred frame exists. However, a preferred frame like this could be considered as a potential deterministic realistic explanation for non-local entanglement effects in quantum mechanics.