Proof for impossibility of FTL Signals Is there is formal proof for the fact that signals must always travel at a speed less than the speed of light in a vacuum?
I understand that special relativity dictates that for massive particles to reach $c$, they would require infinite energy. Thus, allowing only massless particles to travel at the speed of light.
But does this necessarily mean that "no information" can travel faster than light? Or can this be shown from the two postulates of SR?
P.S Jacksons' book introduced "A universal Limiting Speed" as a 3rd postulate.
 A: (Not sure if this is the answer you're looking for)
The "proof" is two-pronged and comes from casuality + experimental verification of SR. The latter simply says that we don't observe violations of Special Relativity, so we should expect that it holds even for speeds that are very close to the speed of light (this is supported by particle physics where particles regularly move at an appreciable fraction of the speed of light).
Given that, then we can use the machinery of Special Relativity to calculate what might happen if we can send signals faster than light. You can set up Lorentz transforms for this. Suppose in the lab frame of reference we have the event "Brutus raises his weapon to kill Caesar" followed by "Brutus kills Caesar" shortly afterwards. This showcases causality, because the first event directly leads to the other.
By defining a new frame of reference, you can use the Lorentz transforms to show that there is a possible frame of reference such that Brutus is killing Caesar today, even though the event in our current frame of reference happened over 2000 years ago. This then leads to the question of whether you can send a signal in this frame of reference to warn Caesar of his upcoming murder. It turns out you can't - you'd need to send a signal that is faster than light.
If you could send a faster than light signal, then this ceases to be the case and you can warn Caesar, thereby altering the past. This would in turn violate causality. What if for example instead of warning Caesar, you had asked Caesar to kill your great-great-great-great grandparents? Then we'd have the so-called grandfather paradox.
If you are willing to give up causality then you can go on to investigate what SR says about particles that move faster than light (see tachyons). For most physicists however, causality is sacred enough that they won't go there.
A: Regarding Caesar murder. Suppose we have a coordinate system at the center of the galaxy, for someone living near Sag A*. They're at rest w.r.t to Earth, this is Frame $S$. So, right now, their point on their world-line is:
$$ E_1 = (t=0,x=0)_S $$
where $t$ is measured in years, and $x$ in light years. The $x$-coordinate points to Earth. Caesar's demise occurred at:
$$ E_0 = (-2065, +26700)_S $$
It's clear that the murder was in their past (as we share the same clocks):
$$ \Delta t = t_1 - t_0 = 2065{\rm y} $$
although the event remains space-like separated:
$$ \Delta s^2 = \Delta t^2 - \Delta x^2 = 
(-2065)^2 - (26700)^2 \approx -(26630)^2 $$
They will not be able to know these Earthly events for another 24-ish ky, nevertheless,
if a message could be sent instantly to Earth, Caesar's fate remains seal.
But, now they hop on a spaceship and accelerate to a mere 8% the speed of the light moving away from Earth. Their coordinate in their new frame $S'$ is:
$$ E_1 = (0,0)_{S'} $$
of course, but now compute Caesar's murder:
$$ E_0 = \big(\gamma(t_0+\beta x_0), \gamma(x_0 + \beta t)\big)_{S'}$$
$$ E_0 \approx (71, 29571)_{S'} $$
Even at mild 24,000 km/s, Caesar's murder is 71 years in the future. If they can send a message to Earth at any speed greater that $29571/71=425c$, Caesar can be warned.
Note that I used $\Delta x'$ to compute the speed, not the standard distance to Sag A* of 26700 light years. There's no preferred frame, so I can use whichever one I want. Once a message moves faster than light, even by a tiny fraction, there is a frame in which it is faster still. There is a frame in which it is instantaneous, and there are frames in which it goes backwards in time.
So: no FTL communication allowed.
