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.

*Edited by Son Goku, : No reason given.*

*Edited by Son Goku, : No reason given.*