RUTA sent me a question about the thought experiment I posted in #17 (and had to keep clarifying until #64), and I decided to clean it up a little and post a new version here. This is (my version of) the standard argument for why it can't be possible to send instantaneous messages in a special relativistic universe.
Imagine a computer that's doing inertial motion. We will call it "Alice". I want you to draw a spacetime diagram that represents the coordinates that the inertial coordinate system associated with Alice's motion assigns to the events I'm about to describe. Draw the 0 axis (time) in the "up" direction. Draw the 1 axis (position) to the right. Alice's world line coincides with the 0 axis.
I will write the coordinates (assigned by Alice's coordinate system) of an event as (t,x),
not [strike](x,t)[/strike]. When I talk about the "slope" of a line, I mean dt/dx. This means that a horizontal line has slope 0 and a vertical line has slope ∞. The world line of an object moving with velocity v has slope dt/dx=1/(dx/dt)=1/v. The simultaneity lines of the inertial coordinate system associated with the motion of such an object would be drawn with slope v in this diagram. (This follows from the same synchronization argument that's used to associate an inertial coordinate system with an inertial observer. See any book on SR for an explanation of simultaneity and clock synchronization using light signals).
Now I want you to draw a straight line through the events (-4.5,0) and (8,10). Suppose that this line represents the motion of a second computer. Let's call it "Bob". The slope of this line is (8-(-4.5))/(10-0)=12.5/10=1.25, so Bob's velocity is 1/1.25=0.8. Draw another straight line, through the events (0,0) and (8,10). This line has slope 0.8=v, so it's the set of events that Bob considers simultaneous with (8,10).
Now suppose that both computers are hooked up to tachyon transmitters that can send and receive 1-bit messages, i.e. either "0" or "1". We assume that a tachyon emitted by one of these transmitters will move at infinite speed in the rest frame of the transmitter that emitted them. (It isn't necessary to assume that the speed is infinite, but it makes the diagram a bit easier to draw). Suppose also that Alice is running a program with the following instructions:
IF the message received at t=0 is "1" THEN send the reply "0" at t=8
IF the message received at t=0 is "0" OR if no message is received THEN send the reply "1" at t=8
And suppose that Bob is running a program with these instructions:
IF the message received is "0" THEN immediately send the reply "0"
IF the message received is "1" THEN immediately send the reply "1"
This leads to the following paradox:
If Alice receives 1 at (0,0), she replies by sending 0 at (8,0). When Bob receives that message at (8,10), he replies by sending 0 at (8,10), so Alice receives 0 at (0,0). If Alice receives 0 at (0,0), she replies by sending 1 at (8,0). When Bob receives that message at (8,10), he replies by sending 1 at (8,10), so Alice receives 1 at (0,0). So if Alice receives a message at (0,0), no matter what that message is, we get a contradiction. If she doesn't receive a message at (0,0), she sends 1 at (8,0). Bob receives that message at (8,10), and replies with 1 at (8,10). So Alice receives 1 at (0,0), and we still have a contradiction.
There are of course a few implicit assumptions in this scenario, and the contradiction we found implies that (at least) one of them must be false. These are some of the possibilities:
- Tachyons do not exist.
- Tachyons exist, but reliable tachyon transmitters do not. So they will sometimes send the wrong message or misinterpret the message they received. (I'm not sure if that's sufficient to avoid the paradox).
- Tachyons and reliable transmitters both exist, but the time it takes to detect a tachyon is greater than vL, where v is the speed of the detector in the emitter's rest frame and L is the distance it has traveled since it was emitted. (This is for infinite speed tachyons. The formula would have to be modified for slower ones. We should also be talking about the time it takes to emit the tachyon, but I'm not attempting to give the complete answer here. I'm just trying to explain the basic idea).
(I'm sure there are other possibilities, but these are the ones I know).
Edit: I'm adding a quote from one of my posts that links to this one.
Fredrik said:
Since this leads to nonsense results, the most natural conclusion is that there are no particles that move faster than light.
There are a few other possibilities, for example: 1. The time it takes to emit and/or detect an FTL particle grows at least linearly with the distance it travels. (This would prevent the recipient of the first message to send a reply that reaches the sender of the original message before he sent it). 2. The equations of motion of the matter in the universe doesn't have any solutions that describe someone who both chooses to carry out an experiment like the one I described in the post I linked to, and succeeds. (Yes, this one is even weirder than the first one).