4 ms·

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 dimension

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.

admax88qqq 4y ago
What is the length of vector [c,c,c] ?
- random314 4y ago
  Vector [c,c,c] is not physically realizable. First postulate of special relativity is that the magnitude of the velocity vector cannot exceed c
  - admax88qqq 4y ago
    It was a rhetorical question to prompt parent poster to reconsider. > 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.
wbsss4412 4y ago
Said vector doesn’t span all of 3D space though. It is a one dimensional vector. 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.
- gizmo686 4y ago
  But this is true in relativity as well. In an appropriate coordinate system, you are also moving in only 1 of the 4 dimensions. If you are not accelerating, then this coordinate system is an inertial reference frame, and the fact that you appear to be a stationary object moving through time is not mere mathematical curiosity, but the core insight of relativity.
- ickelbawd 4y ago
  That might be true from my own frame of reference but not so to an observer, right? How can we say there are 3 spatial dimensions at all if what you say is true? What you’re describing suggests there are an infinite number of dimensions based on the different frame of reference for all observers. Why do we then believe there to be 3 dimensions? Is this just one of the cases where 3 dimensions is a useful model for calculation but not really true in reality? Undoubtedly there must be something I’m missing here—I’ve taken physics courses but clearly I’m no expert. :)
  - wbsss4412 4y ago
    Unfortunately I am not well versed in relativity, my response was completely rooted in linear algebra. 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].
    - rexer 4y ago
      Not quite, though, right? Because the magnitude of those two vectors is different: the magnitude of the latter one is 1.7. I agree with the rest of your statement.
      - wbsss4412 4y ago
        Yes exactly, I was only focused on the direction, not the magnitude.
hollasch 4y ago
You are assuming that we can alter our speed, which isn't true. Our speed through spacetime is c. If you are at a fixed point in X,Y,Z, then your velocity is c in the direction of time. You can change _direction_ in spacetime, but you can't change your speed. 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.
ickelbawd 4y ago
Whoops. Of course I had to use c for every value so the magnitude exceeds c. FTL travel!

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.

admax88qqq 4y ago

What is the length of vector [c,c,c] ?

random314 4y ago
Vector [c,c,c] is not physically realizable. First postulate of special relativity is that the magnitude of the velocity vector cannot exceed c
- admax88qqq 4y ago
  It was a rhetorical question to prompt parent poster to reconsider. > 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.

wbsss4412 4y ago

Said vector doesn’t span all of 3D space though. It is a one dimensional vector. 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.

gizmo686 4y ago
But this is true in relativity as well. In an appropriate coordinate system, you are also moving in only 1 of the 4 dimensions. If you are not accelerating, then this coordinate system is an inertial reference frame, and the fact that you appear to be a stationary object moving through time is not mere mathematical curiosity, but the core insight of relativity.
ickelbawd 4y ago
That might be true from my own frame of reference but not so to an observer, right? How can we say there are 3 spatial dimensions at all if what you say is true? What you’re describing suggests there are an infinite number of dimensions based on the different frame of reference for all observers. Why do we then believe there to be 3 dimensions? Is this just one of the cases where 3 dimensions is a useful model for calculation but not really true in reality? Undoubtedly there must be something I’m missing here—I’ve taken physics courses but clearly I’m no expert. :)
- wbsss4412 4y ago
  Unfortunately I am not well versed in relativity, my response was completely rooted in linear algebra. 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].
  - rexer 4y ago
    Not quite, though, right? Because the magnitude of those two vectors is different: the magnitude of the latter one is 1.7. I agree with the rest of your statement.
    - wbsss4412 4y ago
      Yes exactly, I was only focused on the direction, not the magnitude.

hollasch 4y ago

You are assuming that we can alter our speed, which isn't true. Our speed through spacetime is c. If you are at a fixed point in X,Y,Z, then your velocity is c in the direction of time. You can change _direction_ in spacetime, but you can't change your speed. 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.

ickelbawd 4y ago

Whoops. Of course I had to use c for every value so the magnitude exceeds c. FTL travel!