Is this really true? I can imagine a vector, [1,1,1] in cartesian space. Scale that vector by c and you are now going light speed in all three spatial dimensions.
If you set your velocity to [1,1,1,x], then x (your speed in time) MUST be sqrt(c^2-3). And still, once you've done that, you cannot "scale your velocity", because you cannot change your speed. We can only change direction.
x, y, and z are just shorthand for orthonormal basis vectors. What you’ve described isn’t “traveling in all three dimensions” simultaneously, it’s traveling along on of the dimensions with a different basis.
Undoubtedly there must be something I’m missing here—I’ve taken physics courses but clearly I’m no expert. :)
That is, any vector is going to be inherently one dimensional, regardless of its coordinates. Dimension is a property of a set of vectors, dependent upon how many are linearly independent of the rest of the vectors in the set. What I described doesn’t imply that there are infinite dimensions.
The vector [0,0,1] is traveling through three dimensions just the same as [1,1,1].
> I can imagine a vector, [1,1,1] in cartesian space. Scale that vector by c and you are now going light speed in all three spatial dimensions.
But my point was if look at the magnitude or such a vector it would be >c and therefore exceed the speed limit of c.