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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.

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(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 says that we don't observe violations of Special Relativity, so we should expect it to hold in general, even in untested domains. In other words, because we've not found violations of SR, we assume (this is admittedly an assumption) we can use SR to investigate what SR says about faster-than-light signals.

Given that assumption, 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. Consider the event "Brutus kills Caesar", a historical event that happened 2000 years ago in our reference frame. 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. You'd need an observer that's moving at an appreciable fraction of light speed situated a long distance away from the Earth, but SR says it can be done.

This then leads to the question of whether this observer can send a signal to warn Caesar of his upcoming murder. It turns out you can't - they'd need to send a signal that is faster than light.

If they could send a faster than light signal, then this ceases to be the case and they can warn Caesar, thereby altering the past. This would in turn violate causality. What if for example instead of warning Caesar, they 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.

NB: One way you might imagine a faster-than-light signal is an actual observer moving faster than light. Such an observer would be able to carry messages, and hence signals. SR says such an observer would see highly unusual things. Let there be two events "Brutus raises his weapon to kill Caesar", and "Brutus kills Caesar". The first event directly causes to the second event, so it seems necessary that every observer should see Brutus raise his weapon before Caesar dies. But you can construct a faster-than-light observer that sees Caesar die before Brutus raises his weapon. How is that possible? It just seems like complete nonsense.

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  • $\begingroup$ So, we assume it to preserve causality? Although could you point me to a resource that explains the whole "caesar killing" part in mathematical detail. $\endgroup$
    – Lost_Soul
    Commented Oct 14, 2021 at 2:33
  • $\begingroup$ @Lost_Soul This is the book I learned it from - page 106: archive.org/details/… $\endgroup$
    – Allure
    Commented Oct 14, 2021 at 2:43
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    $\begingroup$ @Lost_Soul These answer have various diagrams illustrating how FTL signals allow the sending of signals into the past, violating causality. physics.stackexchange.com/a/432251/123208 & physics.stackexchange.com/a/407949/123208 $\endgroup$
    – PM 2Ring
    Commented Oct 14, 2021 at 2:51
  • $\begingroup$ @Lost_Soul causality is ontologically true, so there is no mathematical proof for it. If that bothers you, think of it this way: if causality weren't true, then it wouldn't need a proof to cause it to be true in the first place. $\endgroup$
    – Señor O
    Commented Oct 14, 2021 at 18:01
  • $\begingroup$ @Lost_Soul now that I look at it, the example of Klingons and Federation on the next page in the link above is an even better example of why FTL signals are impossible. $\endgroup$
    – Allure
    Commented Oct 15, 2021 at 2:45
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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.

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  • $\begingroup$ Thanks for working through the math that I didn't do! $\endgroup$
    – Allure
    Commented Oct 15, 2021 at 2:47
  • $\begingroup$ @Allure once I figure out where to put the origins of $S$ and $S'$, it boils down to solving 1 inequality: $t>\beta x$ $\endgroup$
    – JEB
    Commented Oct 15, 2021 at 14:19

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