Contradictions caused by moving faster than light There was a Joe Rogan episode with Brian Greene where Joe thinks aliens are watching us because the universe is infinite and there are an infinite number of them. So some of them must be watching.
Brian says "take a 5 billion light year ball and everything is finite then" (5:33). The alien civilization would have to be within 100 thousand light years away to observe modern humans at all. Joe then wonder if there could be some technology that could allow them to observe us unconstrained by the speed of light. Brian shrugs and says "no mechanism we know of" (7:50).
And then there were a lot of comments (on the Youtube video) saying "maybe Brian just lacks imagination".
I know the reason physicists take the speed of light so seriously is that going faster than it would lead to logical contradictions. I'm wondering if there is some simple thought experiment that can be explained to a layman to convince them that faster than light information propagation would lead to logical contradictions (not just physical ones)?
Here is the video: https://www.youtube.com/watch?v=BRo3YXCvgPI
 A: (Assuming you're comfortable with electromagnetic retarded time)A simple thought experiment I thought about a while ago goes like this:
Consider an object A accelerating back and forth creating electromagnetic radiation.
An object B then travels MUCH faster than light.
According to the rest frame of the emmitter of light l. The object B will NEVER feel the electromagnetic fields produced from the accelerating charge . since according to this rest frame the field at a point is only "updated" once distance/c seconds has occurred ( the time it takes for light to reach it) which can easily be seen from the field evaluated at the retarded time. Meaning It is essentially out running the EM wave. Meaning it will feel no force
In the rest frame of object B, B is now at rest  and the emitter is moving at a velocity much greater than C. At T=0, the emmitter produces light and is e.g 1 meter from object B, according to THIS frame. The field produced by the emitter will reach object B after 1m/c seconds as that is the retarded time in this reference frame.
In scenario 1, the object never feels any force. And in frame 1 the object does feel a force. Almost instantly.
This same thought experiment can give you an intuition on time dilation as scenario 1, if the object was moving just under C , it would take ALOT of time for this frame to measure light has reached it. But in the moving frame it would say that it took almost no time. Aka, the moving frame experiences slower time as 1 second for them, is e.g 100 seconds for the stationary frame
A: In order for an idea to lead to a logical contradiction, there needs to be something that it contradicts. Since the idea of faster than light travel is logically conceivable, it has no self-contradiction, and therefore there is no a priori contradiction. Thus, we have the following answer.

Question: Does faster than light travel/communication lead to a logical contradiction?
Answer: No, because it is not a self-contradictory idea.

The only other way for us to reach a contradiction is for faster than light travel/communication to contradict something else. The obvious candidate is special relativity. So now we reduce our question to, does faster than light travel/communication violate special relativity? As I explain below, faster than light travel/communication does not contradict special relativity. Thus, we have the following answer.

Question: Does faster than light travel/communication contradict special relativity?
Answer: No, but there are interesting consequences here that make people think it is unlikely. Faster than light travel/communication implies either (1) travel/communication to the past is possible OR (2) special relativity is incorrect. Since people think special relativity describes the world and since people think travel/communication to the past is unlikely to be possible, it follows that people think faster than light travel/communication is unlikely. (And if you think special relativity is correct and travel/communication to the past is impossible, then you would logically conclude that faster than light travel/communication is impossible.)


Here I explain the conclusion of the above answer.
Two events $(t_{1}, x_{1}, y_{1}, z_{1})$ and $(t_{2}, x_{2}, y_{2}, z_{2})$ are said to be time-like separated if
$$ (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2} - c^{2}(t_{2} - t_{1})^{2} < 0, $$
they are said to be light-like separated if
$$ (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2} - c^{2}(t_{2} - t_{1})^{2} = 0, $$
and they are said to be space-like separated if
$$ (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2} - c^{2}(t_{2} - t_{1})^{2} > 0. $$
Two events that are time-like or light-like separated are those for which you could go from one to the other in less than or equal to the speed of light. Space-like separated events are those that are too distant compared to the time you would need to traverse to get from one to the other.
The main premise of special relativity is that the laws of physics are invariant under Lorentz transformations. In the passive sense, Lorentz transformations are simply a change of coordinates. What this means is that if $K$ is one inertial coordinate system, and $L(K)$ is another coordinate system obtained by a Lorentz transformation $L$, the fundamental equations of motion should be of the same form regardless whether they are written in terms of coordinates $K$ or coordinates $L(K)$.
To give a related example to make this more understandable, consider taking an inertial coordinate system $K$ and consider rotating the coordinates any angle $\theta$ about the $z$-axis to get new coordinates $R(K)$. This is also a change of coordinates, and because the universe doesn't have any "intrinsically preferred direction," it doesn't matter whether we've written the laws of physics with respect to $K$ or $R(K)$. Hence the laws of physics are rotationally invariant.
To understand Lorentz transformations, it might be better to first think about Galilean transformations. A Galilean transformation is simply a change in the velocity of the coordinates (so if you have an inertial coordinate system and another inertial coordinate system passes by at constant velocity, the two coordinate systems differ by a Galilean transformation), and the classical notion that physics is invariant under Galilean transformations is the idea that there is no meaning to absolute speed.
A Lorentz transformation is similar to a Galilean transformation, except it mixes time and space coordinates. Einstein found out that, upon closer examination, physics is not Galilean invariant but Lorentz invariant. The invariance under partial mixing of time and space coordinates leads to time dilation, Lorentz contraction, and the invariance of the speed of light, but most of all, it leads to the idea that different inertial coordinate systems will disagree on the ordering of space-like separated events.
If events $A$ and $B$ are space-like separated, one coordinate system might say $t_{A} > t_{B}$, but another coordinate system might say $t_{A} < t_{B}$. This is not a problem, however, because no communication can occur between space-like separated events. Also, time-like and light-like separated events never change order. Thus, causality is always preserved.
Now the problem with faster than light communication (of any kind) is that it involves sending signals between space-like separated events. Let's say $A$ and $B$ are space-like separated, and I send a signal from $A$ to $B$. In one inertial coordinate system, $t_{A} < t_{B}$ and I sent a signal from location $A$ to location $B$ in a certain time period. But in another inertial coordinate system, $t_{A} > t_{B}$, and I sent a signal from location $A$ to location $B$ to the past.
But if sending signals to the past is possible in one inertial coordinate system, then by Lorentz invariance it must be possible to do this in any inertial coordinate system. Thus, by performing the same thing I did at event $A$, a person at event $B$ can send that signal back to my past self, and essentially a form of time travel is achieved, leading to the grandfather paradox.
Now the grandfather paradox is not a contradiction by itself, because it is possible that we are deterministically fated to avoid creating various paradoxes in our timeline, but the whole idea of sending signals to the past is too much to entertain.
If signaling at faster than the speed of light is possible, there is another possibility, which is that Lorentz invariance is broken, and that there really is an absolute coordinate system after all. If you go back through my reasoning, you would find that Lorentz invariance was the key component to $(1)$ changing the ordering of events $A$ and $B$, and $(2)$ generalizing "sending a signal back in time according to one coordinate system" to "sending a signal back in time in any coordinate system." If laws of physics ultimately depend on an absolute coordinate system, then it is possible to send signals faster than light according to that one coordinate system without any backwards time traveling.

So to summarize, if faster than light communication existed, then either special relativity is incorrect and there is an absolute reference frame, or signaling into the past is possible. Since we have no reason to think that special relativity is incorrect and no reason to think that signaling into the past is possible, we are left with the conclusion that faster than light communication is probably impossible.
Ultimately, it depends on what empirical evidence will show us in the future. Any one of the possibilities I mentioned is possible (with no logical contradictions), but until then, not much more can be said.
A: Maximal Ideal's answer looks at the problem from the standpoint of special relativity, but this wouldn't convince someone who doubts the truth of SR (and, indeed, relativity is just a theoretical model, and despite the strong evidence for it there's no reason to assume it's the complete picture).
However, we do have pretty convincing evidence from experiment that FTL travel is unlikely. In particle accelerators, we've tried pushing various particles faster and faster, and it just so happens that no matter what you do, you can't accelerate them faster than $c$. We can push massive particles to speeds 99.99% of c, but somehow we can never get them equal to $c$, let alone greater. It appears that in nature there's a very real 'wall' at that speed. And this is purely experimental evidence, so there's no need for you to even assume SR.
Further, all particles generated in accelerator collisions are also moving at $c$ (if they're massless), or under $c$. And, further, all particles we've detected from the cosmos, even the most highly energetic ones, coming from quasars or black hole mergers etc., having energies far beyond what we are capable of achieving, are all travelling at equal to or less than $c$. There was some excitement a few years ago about the possibility of having discovered neutrinos that are travelling a bit over $c$, but the 'discovery' happened to be just experimental error.
So if you want to argue that FTL travel is possible, it would require particles that we just haven't discovered yet, obeying laws that violate all the laws that we have strong evidence for. This is what Greene probably means when he says 'no mechanism that we know of.'
A: I think you are overlooking the most fundamental of logical contradictions, which is that your premise is inconsistent with the nature of the explanation you are seeking.
Imagine you stand on a perfect sphere- around you in all directions and at a fixed distance is the horizon. You can never reach the horizon. No matter how fast you move, for how long, or in which direction, the horizon always remains unreachable. It is the same with the speed of light. Asking what would happen if you could exceed the speed of light is like asking what would happen if you could reach the horizon- it is impossible to give a meaningful answer based on the rules of physics, because your premise assumes they no longer apply.
To exceed the speed of light you would have to expend more than an infinite amount of energy, and a quantity more than infinity is a logical contradiction.
If you could exceed the speed of light magically, you could appear to be in several places at once, since you could move between several locations before the image of you at any one of them could reach an observer.
You would be able to see events running backwards, since you would be able to overtake the light from an earlier event after viewing the light from a later one- that effect would ruin the experience of watching a suspense movie, as you would know the ending before seeing the start.
Another contradiction would arise from the law of velocity addition, as a consequence of which two people each moving at greater than the speed of light from a stationary centre but in opposite directions would seem to a third viewer to be moving at less than the speed of light relative to each other. I do not have the time or the will-power to try to figure out the implications of that, but I suspect they would seem odd.
