But I don't see anything in this figure that really makes me do that. And I need it. I neeed my audience to be able to see, very clearly and simply why ftl travel is impossible / absurd. What am I missing? Please someone make me smarter than I am.
Typically we represent the spacetime of special relativity, called Minkowski space, as a plane, just like the normal plane of Euclidean geometry.
This is because they are both flat spaces, so you can carry over intuition from one to the other and the easiest way to encode this intuition is by drawing them the same way, i.e. as a plane.
Now on a 2D plane, if dx and dy are the distances between two points on the x and y axis, then the distance between both of the points is:
for 3D space:
for 4D space:
Of course you could naturally ask the question, what if I put a minus sign in front of one of the terms, I'd get:
So there are two possible four dimensional spaces here. The first one, with the + sign is called 4D Euclidean space and the second one with the - sign is called 4D Minkowski space.
They behave quite differently, but they are both valid spaces mathematically. It just turns out that our universe is the second one, not the first.
You can compare a universe with the first type of distance rule (basically the universe of Aristotles physics) to the real world and it fails to match the behaviour of the real world, unless things are moving very slowly.
So the first thing is, you can't get this minus sign from the triangle. The triangle is just a path through the space or more accurately the composition of three paths. Those paths can be drawn on both Euclidean space and Minkowski space.
In other words, the picture only represents a triangle in a flat space, whether that space is Minkowskian or Euclidean is an extra detail you have to supply.
So, to my mind, you could write down pythagoras' theorem, calculate distances and then ask "why" must that be the rule that triangles obey. The truth is that it isn't the only logical possibility. Then you could suggest the new one, with the minus sign, which is our actual universe.
Good catch, I've corrected it in my original post.
I'm taking this to mean that I can keep working in terms of just two components, distance and time. They are sufficient, in the same way that the distance on the ground from me to the bottom of a flagpole, plus its height from the ground, would be sufficient to calculate the true distance from my feet to the top of the pole.
Yeah, you can do this for simple calculations alright.
Look, in my original dealie, distance is 3, the square is 9; time is 4, the square is 16; 9-16=-7, the root of minus 7 is a broken calculator.
This is telling me that it is slower than light travel that is impossible / absurd !!!! my ftl version comes out a perfectly tolerable 2.64etc.
Please tell I just got the terms backward or something. Please?
For this reason we usually use:
for the distance rule (metric), because it's nicer to have the slower than light distances come out real. It actually makes no difference, since the important thing is that they are imaginary with respect to each other.
The whole point is that displacements which we call "slower than light" are ones with the time displacement being larger than all the spatial ones. These are called time-like displacements.
Faster-than-light displacements have larger space components and are hence called "space-like".
If you use:
then spacelike distances are imaginary, if you use
then timelike is imaginary.
However in calculations of physical quantities the imaginary number drops out and the spacelike displacements always come out with problems (infinite-energy, e.t.c.) regardless of which one you initially give the imaginary value to.
Of course in some cases the time and space displacements are equal and you get 0 as the interval, which is called "light-like" since light moves along this displacements or "null" because of the 0 result.
I should say there is nothing wrong with faster-than-light travel in Special Relativity itself. Some particles can move along time-like displacements (the particles we are made of) and others along space-like displacements (known as tachyons).
All relativity prevents is crossing over between the two domains. No amount of energy can change one type into the other.
It's quantum mechanics however that rules out tachyons completely. Since every particle must be an excitation of a quantum field, there would need to be a tachyon field.
However, first of all, tachyon fields are unstable. Basically two tachyons would have less energy than none, four would have less energy than two and so on, so very quickly the field would produce an infinite number of particles.
However quantum mechanics forbids* an infinite particle state like this and so the only way such a tachyon field can obey quantum mechanics and still exist is if it remains frozen in its "no particle" configuration. However interaction with any other field would kick it out of its ground state, so basically the tachyon field cannot interact with anything and might as well be non-existent.
*This is quite difficult to explain, but basically a field with an infinite particle state like this, simply cannot exist mathematically, attempting to write one down is basically like writing down 1=2. Rather than forbidding it, quantum mechanics says it is logically inconsistent.