# Can someone explain this causality argument about the speed of light from Special Relativity?

I was watching this Yale lecture by R. Shankar.

From 8:29 to 15:05, he gives an argument for why theory of relativity "demands" that it should be impossible for events to influence other events faster than the speed of light and once you accept that there are no logical contradictions in the theory.

For some reason I am not able to follow this argument. Can someone explain clearly what the argument is.

• There are some classic problems that ask you to show that retrocausality is possible if we allow FTL (faster-than-light) signaling. That is to say, I can send a message to my past self telling him what horse to bet on, after the fact. Obviously, things can travel at exactly the speed of light—namely, light. This should convince you, more computationally than philosophically, that light is the universal speed limit—just because retrocausality doesn't make physical sense. – BRSTCohomology Jan 22 '18 at 19:39
• The essence of the argument is that relativity is internally consistent, and if you accept a priori that reality conforms to the theory, then faster-than-light speed is impossible. The problem with many mathematicians is that they don't admit the possibility that anyone may reject the axioms of the theory (or at least may insist that they be proved or supported rather than blindly assumed), so they express themselves in ways that strike ordinary people as logically fallacious - essentially, what he means to say is true, but what he's saying isn't what he means. – Steve Jan 22 '18 at 19:40
• @MarcusAurelius, I would argue that "retrocausality" in the manner you describe is absurd - not because of the limited speed of light, but because what you're describing is not movement in time, it is the resetting of all things in space. That is, to re-run the horse race with a tip from the future, you would have to find a way of putting the horse spatially back in behind the starting gate - not just finding a way to ensure that the horse leaves the gate in a time after you had gained the tip. It's got nothing to do with the speed of light, and everything to do with your confused concepts – Steve Jan 22 '18 at 19:52
• I was just thinking of an example on the fly, unfortunately it didn’t work in this case, sorry! – BRSTCohomology Jan 22 '18 at 19:54
• @MarcusAurelius, don't worry, I find those sorts of examples are common currency in physics. The fact that "retrocausality" has a name shows in how bad shape physical philosophy is. When things exceed the speed of sound, occurrences do not become "retrocausal", they are simply mediated (in a causal fashion) by the thing that goes faster than sound. If anything goes faster than light, the same would be true. – Steve Jan 22 '18 at 20:20

A speed $v$ exists for which $\Delta x =v\Delta t$. Then $\Delta t'=\gamma(\Delta t - c^{-2}u\Delta x)=\gamma\Delta t(1 - c^{-2}uv)$. To preserve the order of events in a Lorentz transformation, we need $\Delta t'$ to have the same sign as $\Delta t$, so $1 - c^{-2}uv>0$ or $uv<c^2$. The special case $v=c$ achievable by a light beam provides the constraint $u<c$ for any relative velocity between reference frames related by Lorentz transformations.