Newbie: “I know very little about this subject, and understand even less, but what follows are my confused ramblings about travelling faster than the speed of light ( C ). Much appreciated if someone could put me straight on any/all the bits I've got wrong :)

If, when travelling faster through space, time slows down for the traveller, does not this mean that we, and everything else in the universe, are actually always moving at the speed of light if we add up all the speeds we are travelling in all the 4 dimensions?

1 - A proton travels through the 3 dimensions of space at C but doesn't move through the dimension of time at all.

1C + 0C = 1C

2 - If I travel at half of C then time moves at half speed for me.

1/2C + 1/2C = 1C

3 - So, I presume, if we don't move physically then we travel through time at the SOL . (My lazy normal state).

0C + 1C = 1C

SO

either nothing can possibly go faster than light because there is no other speed anything can ever move at

OR

to travel through space at more than C you would have to move backwards through time.

The existence of other dimensions might change this, but only if they were big enough for us to fit in them - ie those hypothesized Superstring dimensions are all so small that I can't see how we can possibly be moving in them so we have no speeds to add on to get greater than C.”



Me: “Yes, you're mostly right actually. I'm surprised and quite impressed that you came to that realization on your own quite frankly. You can interpret relativistic dynamics as meaning you always travel at speed c through spacetime. The faster that you move in space, the slower you must then move in time. This is manifest when you talk about four-positions and four-velocities: the four-velocity of object along its own world line is c. But you don't add them like that. You take the squares and add them, just as you would resolving vectors components along orthogonal directions. (That is, you use the Pythagoras' law). So 1: is (1c)^2 + (0c)^2 = (1c)^2 and likewise for 3, but for case 2 you have: (c/sqrt(2))^2 + (c/sqrt(2))^2 = (1c)^2. So for the second case the speed of motion through space and through time is c over sq root 2 and not c over 2.

I'd be careful with how you state your conclusions, though. Moving backwards in time is exactly the manifestation, in this picture, of the breaking of causality caused by objects that travel faster than light. But, none of this proves that that's not possible, we just don't think it is so.”

Newbie: “Thank you for your kind words, corrections, and explanations (even if "just as you would resolving vectors components along orthogonal directions" makes me go "whaaaaat?").

So, when calculating what happens to time if you are travelling faster than light, you need to find the square root of a negative number? And there is no such thing. I would have thought that makes it impossible.

I've also wondered if (if everything is always moving at C in spacetime) it wouldn't make just as much sense to say nothing ever actually moves, it just seems like it does. That's the point where I put on my knotted hankie and grunt ‘My brain hurts’.”

Me: “Crazy as it sounds, it is possible to take the square roots of negative numbers! We define the imaginary number i= sqrt(-1). So, for example, if x=sqrt(-2), then x=+/-sqrt2*i. In general, a complex number will take the form of z=x +iy, where x is the real component, situated along the real number line, and iy is the imaginary part, situated along the imaginary number line, which sits at right angles to the real one.”

Me: “Additionally, let me clarify something first: actually, you don't add the squares you need to subtract them, so don't go quoting the numbers I gave you because they're not right; I just wanted to give you the idea that you take squares and don't just add things up. But actually it's a little more complicated even than that because the squares are in Minkowski space, so you need to subtract components. That probably doesn't make sense to you (hence why I left it easier before), but looking back at it I wanted to clarify that what I wrote before is only a simple picture and not properly correct.

Resolving components of vectors is easier than it sounds and looks; maybe this helps.

So, when calculating what happens to time if you are travelling faster than light, you need to find the square root of a negative number? And there is no such thing. I would have thought that makes it impossible.

The equations you see above (and elsewhere) are derived from a situation where it is implied you can't move faster than light. It's an implicit assumption you make in the construction of the equations. They break down (i.e. give answers that don't make sense, like roots of negative numbers) because they were never constructed in order to work in those situations in the first place. Hence they themselves don't prove that faster than light travel is not possible because you already made that assumption when you constructed them. It's a circular argument. If I derive equations assuming x=0, I can't then claim that the equations tell me that x must be 0, because I myself put that into them in the first place! Thus, if you could travel faster than light, you would need a whole new theory of mechanics and you can't just read it off the current equations.

I've also wondered if (if everything is always moving at C in spacetime) it wouldn't make just as much sense to say nothing ever actually moves, it just seems like it does.

Which "move" do you mean? Do you mean movement in spacetime, or movement in space alone (what we normally call "move"). We all move at the same speed in spacetime, we have no choice about that, but we can move at whatever speed we like in space (as long as it's less than c). That freedom to move in space alone is real -- we really have that choice, and different objects can choose to move differently, so, no, movement isn't an illusion: different objects do move in space differently even though their complete motion in spacetime is still always the same as everything else's. ( )