## Saturday, June 14, 2008

### Why faster than light communication is impossible

Lets consider the following situation - You are 4 light years away from the Earth, on a distant planet X. While you there aliens land on that planet and capture you. They make you tell them from where you came, and tell you that they will go and destroy Earth. You manage to escape, and get to your spaceship. You cannot stop them, and they don't want to look for you - they just go to Earth.
Lets suppose that:
1. Their spaceship can fly at a speed lower than light, but faster than yours spaceship.
2. If the people on Earth are informed in advance about these aliens coming, they will likely be able to defend themselves.
3. You have a device that allows you to send message to Earth that will travel there faster than light.
4. You use the device and send the message in the exact same moment that the aliens leave that planet.

Now, after we have all the information, lets see what will happen.
From the view point of out story hero, he send the message and now goes home.
From the view point of Earth the message is received, and they prepare to fight the aliens.
But what the aliens see?

Lets denote the starting point (the planet X) and the starting time (sending of the signal) by (0,0) respectively, and the end point (Earth) and end time (signal received) by (x,t).
The speed of the alien spaceship is v<1 c="1)" style="text-align: center;">x'=$\gamma (x-tv)$

t'=$\gamma (t-vx)$

For point (0,0) we get (0,0), which is not surprising. For point (x,t) we get ( $\gamma (x-tv)$,$\gamma (t-vx)$).
Firstly lets suppose that t=0. This means that the message went to Earth in zero time. We will get: ($\gamma x, -\gamma xv$). The minus sign means that this event happened before t=o. Thus the signal was received before it was sent. This is clearly impossible - unless a time travel is involved.

Now, what happens if the message is not instant but is still faster than light? In this case t=x/(1+h) - time is distance divided by speed - where h is a positive number (remember I am working with c=1).
Lets look what is the condition for t'>0, for any v (we must find condition for any v because we don't know what is the speed of the alien spaceship):

$\gamma (\frac{x}{1+h}-xv)$>0

We can divide:

$(\frac{x}{1+h}-xv)$>0

And finally we get:

1>v+vh

Rearrange:

1-v>vh

If we will choose v=1-$\epsilon$<1>
$\epsilon$>(1-$\epsilon$)h

If we will now take the limit when epsilon approaches zero, we will get that h=o. This means that it is not possible to send messages faster than light, because otherwise there are always be an observer for whom the order of the events changes - which means time travel.
Note that the way the signal is send is unimportant. The only thing I used is a spaceship which is totally unrelated to the way the device works.

Strictly speaking, this doesn't prove that it is impossible to comminicate faster than light - but the only way to do this involves time travel. Therefore it is logical to assume that it is simply impossible for anything to travel faster than light, including information.

daniel john said...

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