<p>Suppose there was a magic gun that fired a bullet at ten times the speed of light relative to the firer. </p>

<p>If I have the only such gun, and I don't move then there is no paradox.</p>

<p>But now suppose these guns are generally available.</p>

<p>At time zero, an observer flies past me at velocity 0.5 c and I give them one of these guns. 7 years later, I fire the gun after them, so in my frame of reference the bullet will catch them at about time  7.36 years when they are about 3.68 ly away. </p>

<p>To get these numbers I just need to solve <span class="math-container">$10*(t-7) = 0.5*t$</span> to calculate when the bullet overtakes the spaceship.</p>

<p>In their frame of reference though, that occurs about 
time 6.38 years after they pass me, at which point, I am about 3.19 ly away from them (we both agree on our relative velocity).</p>

<p>This is calculated using the <a href="https://en.wikipedia.org/wiki/Lorentz_transformation" rel="noreferrer">Lorentz transformations</a>. <span class="math-container">$$t' = \gamma(t - vx)$$</span> taking <span class="math-container">$c = 1$</span>. Here <span class="math-container">$t = 7.38$</span>, <span class="math-container">$\gamma = \frac{1}{\sqrt{1-v^2}} \sim 1.15$</span> <span class="math-container">$v = 0.5$</span> and <span class="math-container">$x = 3.68$</span>.</p>

<p>They now fire back and we can do a similar calculation, but in their frame of reference. So in their frame the bullet will pass me at time <span class="math-container">$t_1$</span> when 
<span class="math-container">$(t_1 - 6.38)*10 = 0.5 t_1$</span> which tells us that <span class="math-container">$t_1 = 6.71$</span>.</p>

<p>Now we use the Lorentz transformation again to determine what time that is in our frame. We find that it is 5.81 years. A bit more than a year before we fired our bullet.</p>

<p>The key thing is that <span class="math-container">$n$</span> times the speed of light in one frame will be backwards in time in another frame.</p>

<p>The diagrams below illustrates this scenario.</p>

<p><a href="https://i.sstatic.net/IjiEk.png" rel="noreferrer"><img src="https://i.sstatic.net/IjiEk.png" alt="enter image description here"></a></p>

<p>Both diagrams show exactly the same events. The one on the left uses the time and space coordinates of the original observer ("you") the one on the right uses the coordinates of the other observer. In bith cases time is on the y axis, space on the x axis. Units are years and light years.</p>

<p>The black diagonal lines are some possible world lines of light rays.
The red lines are lines of constant x (solid) or t (dashed) as determined in "your" frame of reference. The thick red arrow (at x = 0) is your world-line. </p>

<p>The blue lines are lines of constant x (solid) or t (dashed) as determined by the ship. The thick blue arrow is the ship's world-line. As can be seen you meet at the event that you both agree is x = 0, t= 0.</p>

<p>The green line is the bullet that "you" fire. It crosses three solid red lines and goes a bit less than half way from one dashed red line to the next (travels 3+ ly in one 0.36 y). The cyan line is the return bullet. It crosses 3 solid blue lines and goes rather less than half way from one dashed blue line to the next (travels 3+ ly in about 0.3y). </p>

<p>Assuming I've got this right, all the intersections and crossing patterns of red, blue and black lines, and the four world-lines should agree between the two diagrams, since they really do depict the same events.  In particular both diagrams show a causal loop, in which the cyan bullet arrives back at the red observer before the green bullet is fired. They disagree about which bullet travels back in time, but that's relativity for you. </p>

<p>What this tells us practically is that any physics which allows FTL travel but not time travel must <strong>not</strong> be Lorentz invariant, so your gun will work differently in different frames of reference. We know of no such physics at present, and see no clues which suggest it might exist.</p>

CC BY-SA 4.0
2019-04-18T07:21:48.887Z
Thanks... However, I get t'' = 1.005 years with 0.023 for x'. { t'' = (t' - v*x')/(1 - v^2)^(1/2) = (  (1.1-0.9)/sqrt(1-0.9^2)  -  ((1-0.9*1.1)/sqrt(1-0.9^2))*0.9  ) / sqrt(1-0.9^2) ~= 191/190 } I've never actual done lorentz transformations though.  So what did I do wrong, and how does γ appear to become 0.5 in your equation?
Caston
CC BY-SA 4.0
2019-04-18T16:07:36.540Z
You are putting in t' = 1.1 and x' = 1, but those are t and x or the point at which the second bullet is fired. t' and x' for the same event are 0.46 and 0.42.
Steve Linton
CC BY-SA 4.0
2019-04-20T04:03:29.920Z
First of all, when you say " t and x **or** the point... " did you mean " **for** the point at which... ", because I thought t' and x' prime was the time and location of the second bullet being fired.  Secondly, how did you get 1.1 years for t?  1 year plus 1/10th of a year would put the first bullet at the location the other person was when the first bullet was fired (from my PoV).  Wouldn't it need to travel further to get to where the person was then?
Caston
CC BY-SA 4.0
2019-04-20T08:51:13.780Z
When the bullet is fires,"you" think the ship is 0.9 ly away, so at 1.09 y the bullet reaches that point, and the ship has moved on a further 0.081 ly, etc. They meet at about time 1.1.  t and x are the inputs to the Lorentz transformation, which we use twice. So their values are different for the two uses.
Steve Linton
CC BY-SA 4.0
2019-04-23T13:05:29.997Z
Edited answer drastically -- new numbers and diagrams.
Steve Linton
CC BY-SA 4.0
2019-04-24T10:22:46.393Z
OOhh I get it, so the Lorentz transformation causes the other person's spatial x axis to be skewed in the opposite angle to their t axis; therefore multiplying the angle of bullet by upto a factor of 2, pushing anything past 45* from their perspective past 90* from ours.  Thank you.  This is the best answer I've ever received on Stack Exchange!
Caston
CC BY-SA 4.0
2020-10-08T05:41:29.563Z
This is really helpful, thank you!  I'd only add that, in order to demonstrate a violation of causality, the relative velocity of the observers needs to be chosen dependent upon the velocity of the gun.  The slower the (superluminal) gun, the faster the relative velocity needs to be e.g. if the bullet travels only at 1.1c, then v needs to be at least 0.995c in our to demonstrate the causality violation.  But the above does work for any superluminal bullet speed!
popstack
CC BY-SA 4.0
2019-04-17T23:16:28.773Z
2019-04-23T09:31:24.803Z
2019-04-23T09:31:24.803Z
Steve Linton
<p>Sometimes spacetime diagrams can be misleading about paradoxes, though, because they might tend to suggest to you that information traveling from your present here, to what you regard (in your coordinates) as the past <em>somewhere else</em> is 
 paradoxical, but it's actually not obvious if and when such a situation should be regarded as any kind of problem.  That's because what you regard as the past is purely a coordinate issue, it's just the language you have chosen to describe when things happen.  Thus if you use a strange coordinate system, you can send information into what you call the past (in the sense of an earlier time coordinate) without any problems at all.  So there has to be some kind of limitation on the type of coordinate being used before this should be regarded as a paradox.  As such, we want a way to understand what is the paradox in a way that is not married to the coordinates used.</p>

<p>Now, it's true that if we stick to coordinates motivated by the Einstein simultaneity convention, it turns out that we cannot send information anywhere that we regard as the past in those coordinates.  But it's not obvious why that would lead to a paradox if we could, because the only true paradox is if we can send information into our <em>own</em> past <em>here</em> where we are.  If we could do that, it would be a paradox in any coordinate system we are using, because we could tell ourselves not to do something that we've already done.  So your question could be reframed, if it were possible to send information faster than light in Einstein-type coordinates (the coordinates that Lorentz transform into each other), how would that allow us to send information to ourselves in our own past?</p>

<p>To see this, you don't need a spacetime diagram, you only need to understand the relativity of simultaneity.  It's also easiest to use the limit of superfast information, i.e., infinite speed.  If you use Einstein coordinates to match up "today" with somewhere that is stationary with respect to you and is the distance of the Andromeda galaxy, you get the "Andromeda paradox," which says that if you simply break into a sprint toward that place, and reassess "what day is it there", you will find your Einstein coordinates tell you it is now <em>tomorrow</em> there, not today any more.  If you then turn around and sprint away from that place, and reasses, you will conclude it is now <em>yesterday</em> there.  </p>

<p>Thus, by repeating that sprint back and forth, you can make that distant place relive today over and over again.   Of course it would be silly to think beings in that place really would relive that day, but that's what the coordinates give you.  That's not a paradox, that's just exactly what does happen in Einstein coordinates under the Lorentz transformation.  But if you couple this with two-way superluminal signals, taken to the extreme of infinite speed to make it really clear, what could happen is that if when you are standing still, you and someone at this Andromeda distance agree the date is April 17 in both places, when you sprint toward Andromeda you will think it is April 18 there and April 17 here, such that when you send an infinitely fast signal (in that reference frame) you will know they will not receive it until April 18.  But if you send an infinitely fast signal when you are sprinting away (infinitely fast in that reference frame), you will know they will receive it on April 16.</p>

<p>That's only problematic if people at the Andromeda distance can also send an infinitely fast signal, in their reference frame, back to you.  Then if you say "hello" while sprinting away, they will receive that on April 16, and can send back a "same to you" signal that you will receive on April 16!  That's yesterday for you, and you will be puzzled what they were talking about, since you don't even know you will be doing some sprints the next day, let alone that you will be sending messages to Andromeda.  So that's what we said is the real paradox-- the ability to send information back into our <em>own</em> past right where we are.  That requires two-way superluminal communication-- one-way superluminal speed could just be an artifact of weird coordinates chosen.</p>

CC BY-SA 4.0
2019-04-17T20:09:42.257Z
2019-04-17T20:16:24.640Z
2019-04-17T20:16:24.640Z
Ken G
<p>So according to Relativity, time moves at different rates for different observers, and distant observers will actually disagree on the order of events.  Now from what I gather this gets sorted out for objects with relative speeds less than <em>c</em>, but would ultimately cause a paradox for anything moving faster because it would time travel.  However, I don't really see how.  On a spacetime diagram light always moves at a 45° angle, or at least some angle between 0° and 90°, and the only way I see anything effecting the past is if something moves at an angle greater than 90°.</p>

<p>So can someone please give a visual representation that explains the issue?</p>

CC BY-SA 4.0
2019-04-17T07:25:34.083Z
This question would probably be better on Physics.SE, but there are already many questions there on this topic. Here's one brief answer with a clear & simple diagram. https://physics.stackexchange.com/a/13003/123208 BTW, this is special relativity, not general. Also, you were right the 1st time - in Minkowski diagrams light worldlines are at 45° to the space & time axes.
PM 2Ring
CC BY-SA 4.0
2019-04-17T07:48:00.853Z
Also, different observers only disagree on the order of a certain class of events. If event P is in the past lightcone of event Q we say that the line PQ is a timelike spacetime interval, and every observer will agree on that, and that P happened before Q. But if PQ is a spacelike interval, then although all observers will agree on that, only some observers will say that P occured before Q, others will say it's the other way around, and some observers will say that P & Q are simultaneous. [This answer](https://physics.stackexchange.com/a/427796/123208) has an animated spacetime diagram.
PM 2Ring
CC BY-SA 4.0
2019-04-18T11:08:39.163Z
I can't answer your question, Caston. Because there is no motion in spacetime, because it models space at all times. Light moves through _space_, not spacetime. Yes, we can draw diagrams with 45° angles and talk about reference frames and Lorentz transformations, but these things are part of the model, not part of the  reality. The map is not the territory. If I could send some signal faster than light, the signal travels faster than light, and that's all there is to it. It doesn't go back in time. It doesn't affect the past. There is no paradox.
John Duffield
<special-relativity><time-dilation>
Can anyone one show how speeds greater than c cause a paradox on a space time diagram?
Caston
# Can anyone one show how speeds greater than c cause a paradox on a space time diagram?

(Asked by: username_0 on 2019-04-16)

So according to Relativity, time moves at different rates for different observers, and distant observers will actually disagree on the order of events. Now from what I gather this gets sorted out for objects with relative speeds less than *c*, but would ultimately cause a paradox for anything moving faster because it would time travel. However, I don't really see how. On a spacetime diagram light always moves at a 45° angle, or at least some angle between 0° and 90°, and the only way I see anything effecting the past is if something moves at an angle greater than 90°.

So can someone please give a visual representation that explains the issue?

**Tags:** special-relativity, time-dilation
> **Comments:**
> **username_1**: This question would probably be better on Physics.SE, but there are already many questions there on this topic. Here's one brief answer with a clear & simple diagram. https://physics.stackexchange.com/a/13003/123208 BTW, this is special relativity, not general. Also, you were right the 1st time - in Minkowski diagrams light worldlines are at 45° to the space & time axes.  
> **username_1**: Also, different observers only disagree on the order of a certain class of events. If event P is in the past lightcone of event Q we say that the line PQ is a timelike spacetime interval, and every observer will agree on that, and that P happened before Q. But if PQ is a spacelike interval, then although all observers will agree on that, only some observers will say that P occured before Q, others will say it's the other way around, and some observers will say that P & Q are simultaneous. [This answer](https://physics.stackexchange.com/a/427796/123208) has an animated spacetime diagram.  
> **username_2**: I can't answer your question, Caston. Because there is no motion in spacetime, because it models space at all times. Light moves through _space_, not spacetime. Yes, we can draw diagrams with 45° angles and talk about reference frames and Lorentz transformations, but these things are part of the model, not part of the  reality. The map is not the territory. If I could send some signal faster than light, the signal travels faster than light, and that's all there is to it. It doesn't go back in time. It doesn't affect the past. There is no paradox.  

---

## 2 answers

### Answer by username_3 on 2019-04-17. Score: 8

Suppose there was a magic gun that fired a bullet at ten times the speed of light relative to the firer.

If I have the only such gun, and I don't move then there is no paradox.

But now suppose these guns are generally available.

At time zero, an observer flies past me at velocity 0.5 c and I give them one of these guns. 7 years later, I fire the gun after them, so in my frame of reference the bullet will catch them at about time 7.36 years when they are about 3.68 ly away.

To get these numbers I just need to solve $10\*(t-7) = 0.5\*t$ to calculate when the bullet overtakes the spaceship.

In their frame of reference though, that occurs about
time 6.38 years after they pass me, at which point, I am about 3.19 ly away from them (we both agree on our relative velocity).

This is calculated using the [Lorentz transformations](https://en.wikipedia.org/wiki/Lorentz_transformation). $$t' = \gamma(t - vx)$$ taking $c = 1$. Here $t = 7.38$, $\gamma = \frac{1}{\sqrt{1-v^2}} \sim 1.15$ $v = 0.5$ and $x = 3.68$.

They now fire back and we can do a similar calculation, but in their frame of reference. So in their frame the bullet will pass me at time $t\_1$ when
$(t\_1 - 6.38)\*10 = 0.5 t\_1$ which tells us that $t\_1 = 6.71$.

Now we use the Lorentz transformation again to determine what time that is in our frame. We find that it is 5.81 years. A bit more than a year before we fired our bullet.

The key thing is that $n$ times the speed of light in one frame will be backwards in time in another frame.

The diagrams below illustrates this scenario.

[![enter image description here](https://i.sstatic.net/IjiEk.png)](https://i.sstatic.net/IjiEk.png)

Both diagrams show exactly the same events. The one on the left uses the time and space coordinates of the original observer ("you") the one on the right uses the coordinates of the other observer. In bith cases time is on the y axis, space on the x axis. Units are years and light years.

The black diagonal lines are some possible world lines of light rays.
The red lines are lines of constant x (solid) or t (dashed) as determined in "your" frame of reference. The thick red arrow (at x = 0) is your world-line.

The blue lines are lines of constant x (solid) or t (dashed) as determined by the ship. The thick blue arrow is the ship's world-line. As can be seen you meet at the event that you both agree is x = 0, t= 0.

The green line is the bullet that "you" fire. It crosses three solid red lines and goes a bit less than half way from one dashed red line to the next (travels 3+ ly in one 0.36 y). The cyan line is the return bullet. It crosses 3 solid blue lines and goes rather less than half way from one dashed blue line to the next (travels 3+ ly in about 0.3y).

Assuming I've got this right, all the intersections and crossing patterns of red, blue and black lines, and the four world-lines should agree between the two diagrams, since they really do depict the same events. In particular both diagrams show a causal loop, in which the cyan bullet arrives back at the red observer before the green bullet is fired. They disagree about which bullet travels back in time, but that's relativity for you.

What this tells us practically is that any physics which allows FTL travel but not time travel must **not** be Lorentz invariant, so your gun will work differently in different frames of reference. We know of no such physics at present, and see no clues which suggest it might exist.

> **Comments:**
> **username_0**: Thanks... However, I get t'' = 1.005 years with 0.023 for x'. { t'' = (t' - v*x')/(1 - v^2)^(1/2) = (  (1.1-0.9)/sqrt(1-0.9^2)  -  ((1-0.9*1.1)/sqrt(1-0.9^2))*0.9  ) / sqrt(1-0.9^2) ~= 191/190 } I've never actual done lorentz transformations though.  So what did I do wrong, and how does γ appear to become 0.5 in your equation?  
> **username_3**: You are putting in t' = 1.1 and x' = 1, but those are t and x or the point at which the second bullet is fired. t' and x' for the same event are 0.46 and 0.42.  
> **username_0**: First of all, when you say " t and x **or** the point... " did you mean " **for** the point at which... ", because I thought t' and x' prime was the time and location of the second bullet being fired.  Secondly, how did you get 1.1 years for t?  1 year plus 1/10th of a year would put the first bullet at the location the other person was when the first bullet was fired (from my PoV).  Wouldn't it need to travel further to get to where the person was then?  
> **username_3**: When the bullet is fires,"you" think the ship is 0.9 ly away, so at 1.09 y the bullet reaches that point, and the ship has moved on a further 0.081 ly, etc. They meet at about time 1.1.  t and x are the inputs to the Lorentz transformation, which we use twice. So their values are different for the two uses.  
> **username_3**: Edited answer drastically -- new numbers and diagrams.  
> **username_0**: OOhh I get it, so the Lorentz transformation causes the other person's spatial x axis to be skewed in the opposite angle to their t axis; therefore multiplying the angle of bullet by upto a factor of 2, pushing anything past 45* from their perspective past 90* from ours.  Thank you.  This is the best answer I've ever received on Stack Exchange!  
> **username_4**: This is really helpful, thank you!  I'd only add that, in order to demonstrate a violation of causality, the relative velocity of the observers needs to be chosen dependent upon the velocity of the gun.  The slower the (superluminal) gun, the faster the relative velocity needs to be e.g. if the bullet travels only at 1.1c, then v needs to be at least 0.995c in our to demonstrate the causality violation.  But the above does work for any superluminal bullet speed!  

---

### Answer by username_5 on 2019-04-17. Score: 1

Sometimes spacetime diagrams can be misleading about paradoxes, though, because they might tend to suggest to you that information traveling from your present here, to what you regard (in your coordinates) as the past *somewhere else* is
paradoxical, but it's actually not obvious if and when such a situation should be regarded as any kind of problem. That's because what you regard as the past is purely a coordinate issue, it's just the language you have chosen to describe when things happen. Thus if you use a strange coordinate system, you can send information into what you call the past (in the sense of an earlier time coordinate) without any problems at all. So there has to be some kind of limitation on the type of coordinate being used before this should be regarded as a paradox. As such, we want a way to understand what is the paradox in a way that is not married to the coordinates used.

Now, it's true that if we stick to coordinates motivated by the Einstein simultaneity convention, it turns out that we cannot send information anywhere that we regard as the past in those coordinates. But it's not obvious why that would lead to a paradox if we could, because the only true paradox is if we can send information into our *own* past *here* where we are. If we could do that, it would be a paradox in any coordinate system we are using, because we could tell ourselves not to do something that we've already done. So your question could be reframed, if it were possible to send information faster than light in Einstein-type coordinates (the coordinates that Lorentz transform into each other), how would that allow us to send information to ourselves in our own past?

To see this, you don't need a spacetime diagram, you only need to understand the relativity of simultaneity. It's also easiest to use the limit of superfast information, i.e., infinite speed. If you use Einstein coordinates to match up "today" with somewhere that is stationary with respect to you and is the distance of the Andromeda galaxy, you get the "Andromeda paradox," which says that if you simply break into a sprint toward that place, and reassess "what day is it there", you will find your Einstein coordinates tell you it is now *tomorrow* there, not today any more. If you then turn around and sprint away from that place, and reasses, you will conclude it is now *yesterday* there.

Thus, by repeating that sprint back and forth, you can make that distant place relive today over and over again. Of course it would be silly to think beings in that place really would relive that day, but that's what the coordinates give you. That's not a paradox, that's just exactly what does happen in Einstein coordinates under the Lorentz transformation. But if you couple this with two-way superluminal signals, taken to the extreme of infinite speed to make it really clear, what could happen is that if when you are standing still, you and someone at this Andromeda distance agree the date is April 17 in both places, when you sprint toward Andromeda you will think it is April 18 there and April 17 here, such that when you send an infinitely fast signal (in that reference frame) you will know they will not receive it until April 18. But if you send an infinitely fast signal when you are sprinting away (infinitely fast in that reference frame), you will know they will receive it on April 16.

That's only problematic if people at the Andromeda distance can also send an infinitely fast signal, in their reference frame, back to you. Then if you say "hello" while sprinting away, they will receive that on April 16, and can send back a "same to you" signal that you will receive on April 16! That's yesterday for you, and you will be puzzled what they were talking about, since you don't even know you will be doing some sprints the next day, let alone that you will be sending messages to Andromeda. So that's what we said is the real paradox-- the ability to send information back into our *own* past right where we are. That requires two-way superluminal communication-- one-way superluminal speed could just be an artifact of weird coordinates chosen.


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