What is causality (traveling with v>c?)

I’m trying to understand the proof that nothing can move faster than the speed of light. The idea of the proof is that the order of cause and effects would get messed up. And in a way I get the proof - or at least one version I’ve seen - and what they say is the following; take $\Delta t’=\gamma(\Delta t - \frac{\beta}{c^2}\Delta x)$. We see that for the order of events to be reversed, we would need $\frac{\beta}{c^2}\Delta x>\Delta t$, in other words $\frac{\beta}{c}>\frac{c \Delta t}{\Delta x}$.

Right, so I get it algebraically; if light can cross (more than) the distance $\Delta x$ in $\Delta t$, then the fraction on the right would be greater than 1, which implies that no velocity can be greater than c (because we don't allow a reverse in order of events in this case).

In the case that light doesn’t even get to finish $\Delta x$ in $\Delta t$, we are safe to say that the events are not causally connected, so it doesn’t matter in what order they are observed.

But... if you word it like this, aren't we using a circular reasoning here? Events are connected causally if they lie in their light cone, so no object can move faster than the speed of light. But based on what did we decide to define causality in terms of light? What if there was (hypothetically) something that travels with a speed greater than $c$ that in a way would make sense to connect to causality? Then our argument would apply to a different $v_{max}$...

I hope someone understands my problem.

If you review the derivation of the Lorentz transforms again, what you see is that what you're actually using there is the existence of an invariant speed with some value $c$. The assumption that light travels at this speed is then not essential, because you could just as well have replaced light signals by signals traveling at the invariant speed of $c$ regardless if light actually travels at $c$ or not. Also, that $c$ is the maximum speed is not used in the derivations either, yet in the introduction you may encounter such statements. But is the price to pay for having a well readable physics text that doesn't read like a very dry abstract mathematics text where everything is proven as rigorously as possible with the least possible assumptions being made.

So, the conclusion is that there exists a cone defined by the speed $c$ which we call the light cone, within which the order of events are absolute. Then a notion of causality seems to clash with the possibility of the order of events not being absolute, this then precludes signals from traveling faster than $c$. That light actually travels precisely at $c$, and that the phrase "light cone" is an appropriate one, are not necessary assumption in any of these derivations. What you can show is that all massless particles must travel at $c$ and vice versa that particles traveling at $c$ must be massless.

Now, the causality argument can be made in a stronger way as explained here in the section about the two-way example. While saying that the order of causally linked events must be absolute looks reasonable, it would be strange if that were not true, but unless it leads to an outright paradox, it's an assumption that is made that doesn't itself follow from the theory of relativity. But as the argument in the article demonstrates, if we assume that faster than $c$ signals exist, then this leads to a genuine paradox.

• Thank you for your response. I've read both the one-way and two-way example, and I have a question for both: 1) One-way example: Do they assume that $v$ can't exceed $c$? Because that seems what they're doing, so we only have half of the story. 2) Two-way example: I have some difficulty with the idea behind the algebraic conclusion "if $v>\frac{2a}{1+a^2}$, then $T<0$". Doesn't this mean that in general our relative $v$ can't exceed this value? Even if $a\ll c$, I would think the same thing holds for $v$. So how does this result mean that $v\leq c$? – Sha Vuklia Jan 14 '17 at 21:41
• @ShaVuklia In both cases $v$ is the speed between the inertial frames and this is indeed smaller than $c$. One then considers a signal send at a speed $a>c$which leads to a paradox. The only nontrivial assumption is that you can have a signal traveling at a speed $a>c$, there is no problem with having two observers who travel at a speed $v<c$, so if this leads to a paradox then the problem has to be with assuming that $a>c$ is possible. Note that in the two-way example, $c=1$ units were used , so there faster than light means $a>1$ and then Alice can receive her signal before she send it. – Count Iblis Jan 14 '17 at 22:19

What if there was (hypothetically) something that travels with a speed greater than c that in a way would make sense to connect to causality?

If that were the case just replace the c = 299792458 m/sec with your new number and keep the already known equations. But since all massless particles need to travel with the highest possible velocity through the vacuum and photons are massless the propability for that is rather zero.

• Yea, but then your argument is basically "nothing goes faster than the fastest speed". Like, yeah... big deal – Sha Vuklia Jan 14 '17 at 19:21
• Just replace the c with your new fastest speed and keep the equations. – Gendergaga Jan 14 '17 at 19:22
• @ShaVuklia The argument is that there is a fastest speed. This is not trivial; for example, people didn't think it was true before Einstein. – Javier Jan 14 '17 at 19:35
• Einstein postulated that $c$ is the absolute speed of light. But the cause-effect argument shows that $c$ is also the fastest speed. I'm having problems with the argument, because in my eyes it's a circle reasoning. – Sha Vuklia Jan 14 '17 at 19:44
• Well, he also found out that the consistent equations had massless particles go at c. Theyd need to be imaginary to go faster. You are right that if there was a larger max c that applied to the SR equations then light speed would not be invariant. A number of things, they all come together. Not circularity, physics and consistency – Bob Bee Jan 14 '17 at 21:39