PHY3101
Pole Vaulter Paradox
x=0
Relativity
x’=0 back front
1
January 29, 2007
*
*
ct’=0
Doors are closed
Twin paradox
*
#3
#2a *
#2 *
ct=0 light
#1
*
The traveling twin leaves the Earth at event #1, arrives at a distant planet and instantaneously turns around at event #2 and finally arrives back on Earth at event #3. While

2 traveling from #1 to #2 the line ct 0 marks the worldline of the traveling twin, and the proper time between events #1 and #2 is ∆ t 0
(2 , 1)
= t 0
2
− t 0
1 interval between #1 and #2 is “time dilated” to be ∆ t
(2 , 1)
= t
2
− t
1
= ∆ t 0
(2 , 1)
/ p
1 − v 2 /c 2 .
This is also the time interval between events #1 and #2a because event #2a is simultaneous with #2 in the Earth’s frame of reference.

3
Faster than light allows time travel.
In the first figure, notice that t = 0 on the x -axis and that x = 0 on the ct -axis. The line labeled v = the x
4
5
0 c ct 0 is the world line of an object moving with a position given by x = vt where
, and this is the ct 0 axis (where x 0
-axis is the line where t 0
= 0) for the “moving” frame of reference. Also
= 0 and also where t = vx/c 2 , which is consistent with the
Lorentz transformations.
* #2
#1
*
*
#−1
Look at the location in spacetime of events #1 and #2. It is clear that x and also that x
2
> 0 and that with this coordinate system.
t
2
1
= 0 and t
1
= 0
> 0. We deduce that event #1 occurs before event #2
#1
*
*
*
#2
A spacetime diagram for the same events but for the primed coordinates is shown in the

4 second figure. Event #1 is at the origin and has coordinates coordinates for event #2 are x 0
2
= ( x
2
− vt
The first figure shows qualitatively that t 0
2
2
) / 1 − v 2 /c 2 and t 0
2 x
= ( t
0
2
= 0 and t
− vx
2
/c 2 ) /
0 p
= 0. The
1 − v 2 /c 2 .
< 0, because event #2 lies below the line t 0 = 0.
Now imagine that it is possible to travel faster than light. In the first diagram, you could then start at event #1 and travel via warp-drive to #2.
Having arrived at #2 you could quickly change to the primed coordinate system of the second figure, and again using warp-drive then return to an event at the same place, in the unprimed coordinates, but just a little before #1 actually occurs. The net effect of this operation is to put you back in your starting place, but a little before you started — you would have gone back in time.
If tachyons actually exist, and we if can control them in some way, then we could send information back into the past.
If we can travel back into the past, or send information back into the past, then we might be able to create a logical paradox, by changing an event that we thought had already occurred. This might be a good reason to claim that “The laws of physics forbid sending information faster than the speed of light.”
All evidence implies that this statement is true. However, it might be the case that information might be able to travel faster than the speed of light, but that we do not have free will (or at least enough free will) to go back and change the past.
Most physicists have thought about such things when they were younger. But, found such ideas not a very fruitful way to increase their understanding.