Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Need an explanation for Null Vectors

  1. Sep 8, 2014 #1
    So I have an issue understanding how to compute a matrix using the Minkowski metric as a null (read light-like) spacetime vector.
    As best I can understand it, it is a vector which has all velocity in the spacial components and none in time.

    Also, would a vector that transcends the speed of light mean that you are traveling through negative time? I don't really get how I should interpret a vector that is inside the light cone.


    So I have
    [-1 0 0 0]
    [0 1 0 0 ]
    [0 0 1 0 ]
    [0 0 0 1 ]
    for my matrix denoting the Minkowski metric (probably using wrong terminology)
    what would a vector that determines a null vector be?
     
  2. jcsd
  3. Sep 9, 2014 #2
    Definition of Null Vevtor

    $$\Lambda_{\mu\nu}$$ be your matrix, null vector $$x^\mu$$ satisfies the relation

    $$x^\mu\Lambda_{\mu\nu} x^\nu=\Lambda_{\mu\nu} x^\nu x^\mu=0$$

    Best
     
  4. Sep 9, 2014 #3

    pervect

    User Avatar
    Staff Emeritus
    Science Advisor

    I'm not sure why you said velocity?, I hope I haven't misunderstood the question as a consquence of assuming you meant component.

    A vector with only spacelike components and a zero time component would be a space-like vector, not a null vector.

    A vector inside the lightcone would also be a spacelike vector, as in your previous example.


    So I have
    [-1 0 0 0]
    [0 1 0 0 ]
    [0 0 1 0 ]
    [0 0 0 1 ]
    for my matrix denoting the Minkowski metric (probably using wrong terminology)
    what would a vector that determines a null vector be?[/QUOTE]

    An example of a null vector for your metric (which is Minkowskii) would be [1,0,0,1].
    If we let X be your vector, with the components of your vector be ##x^0, x^1, x^2, x^3## , and the nonzero components of your matrix (as above) being written in the following notation ##g_{00} = -1, g_{11}=g_{22}=g_{33}=1##, then the length of your vector is

    [tex]\sum_{\mu=0..3} \sum_{\nu=0..3} g_{\mu\nu} x^\mu x^\nu [/tex]

    which for Minkowskii space is equivalent to

    ##-(x^0)^2 + (x^1)^2 + (x^2)^2 + (x^3)^2##

    and a null vector is just a vector with a length of 0, such as the vector with ##x^0 = x^1 = 1## mentioned previously.
     
  5. Sep 9, 2014 #4

    Meir Achuz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    No. It has a time-like component that equals the magnitude of the space-like component.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Need an explanation for Null Vectors
  1. Null vector help! (Replies: 6)

  2. Null Killing vectors (Replies: 3)

Loading...