2

Signal can't travel at speed greater than light speed in vacuum which is a assumption of special relativity. But if a signal travels at speed greater than $c$ then it will violate causality. I tried to prove this statement using Lorentz transformation equations. But in the denominator and imaginary number will arise. I need a mathematical proof.

Qmechanic
  • 220,844
Aranyak Ghosh
  • 183

1 Answers1

8

Suppose I have a gun that fires bullets with a superluminal velocity V, and I'm going to use it to shoot you. In our rest frame I'm standing at the origin, $x=0$ and you're standing at some distance $x = D$. I fire the gun at $t=0$ and you die at $t=D/V$. So cause an effect are pretty clear - I fire my gun and as a result of this you die a short time later. (Apologies if this seems unnecessarily bloody :-)

Now suppose my friend Fred is in a spaceship flying past at a (subluminal) velocity $v$. We can use the Lorentz transformations to find out what happens in Fred's rest frame. We'll assume Fred passes me just as I fire, so the gun is fired at the spacetime point $(0,0)$ in both our frames. It just remains to find where in Fred's frame the bullet hits you.

In my frame the bullet hits you at $(t=D/V, x=D)$ so let's use the Lorentz transformations to calculate when the bullet hits you in Fred's frame:

$$\begin{align} t' &= \gamma \left( t - \frac{vx}{c^2} \right) \\ &= \gamma \left( \frac{D}{V} - \frac{vD}{c^2} \right) \\ &= \gamma \frac{D}{V}\left( 1 - \frac{vV}{c^2} \right) \end{align}$$

But if we make the bullet velocity $V \gt c^2/v$ that means $1 - vV/c^2$ is negative, so $t' \lt 0$, and this would mean that in Fred's rest frame you died before I fired the gun.

This is where we have a problem with causality. For any superluminal bullet velocity there is a frame where you died before I fired at you. The only way to avoid this is for the bullet velocity to never exceed $c$.

John Rennie
  • 367,598