1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Proving the dor product of 4-vectors is Lorentz invariant

  1. Feb 13, 2014 #1
    1. The problem statement, all variables and given/known data
    Let A and B be 4-vectors. Show that the dot product of A and B is Lorentz invariant.

    3. The attempt at a solution
    Should I be trying to show that [itex]A.B=\gamma(A.B)[/itex]?

    Thanks
     
  2. jcsd
  3. Feb 13, 2014 #2

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    hi chipotleaway! :smile:

    no, you should be applying the lorentz transformation to A and B, and showing that it doesn't affect A.B :wink:
     
  4. Feb 18, 2014 #3
    Thanks tiny-tim, I've got it now but I had to make one assumption to get the result I wanted, namely if [itex]A=(a_0, a_1, a_2, a_3)[/itex] and [itex]B=(b_0, b_1, b_2, b_3)[/itex], I had to assume [itex]a_0=ct_a[/itex] and [itex]b_0=ct_b[/itex] which is only for 4-position vectors, so I think my result only applies for 4-vectors!

    EDIT: Oh wait, I guess I must've assumed I was dealing with 4-position vectors because I started with
    [itex]A'=\gamma (a_0-\beta a_1, a_1 - ut_a, a_2, a_3)[/itex] and [itex]B'=\gamma (b_0-\beta b_1, b_1 - ut_b, b_2, b_3)[/itex]

    EDIT 2: Added 3rd and 4th components to above expressions
     
    Last edited: Feb 18, 2014
  5. Feb 18, 2014 #4

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    hi chipotleaway! :smile:

    sorry, i'm completely confused …

    what is ta ? :confused:

    (and why do your final expressions only have two components instead of four?)
     
  6. Feb 18, 2014 #5
    My bad, forgot to say what they were! [itex]t_a[/itex] is the time of the event of vector A (so, I think I've only done it for 4-position vectors, [itex]A=(ct_a, a_1, a_2, a_3)[/itex]

    I forgot to put the other two components in as I left it out in my working to save space (fixed now)...but we don't have to apply the transformation to those as well directions do we? Because I think we could just arrange out coordinate system so that the motion takes place in one direction .
     
  7. Feb 18, 2014 #6

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    hi chipotleaway! :smile:
    yes that's ok …

    though you really ought to say that you're choosing the "1" coordinate to be in the direction of u !
    yes, using cta instead ao is probably a good idea, since it makes the dot product easier

    but you can't then mix them both in the same expression …
     
  8. Feb 18, 2014 #7
    Yeah, I should state everything I assume!

    When you say 'mix them both in the same expression'...do you mean the timelike and spacelike components, like
    [itex]a_1-ut_a[[/tex] in [itex]A'[/itex]?
     
  9. Feb 18, 2014 #8

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    Why do you have ##a'_1 = \gamma(a_1-u t_a)## instead of ##a'_1 = \gamma(a_1-\beta a_0)##? Also, you shouldn't have ##a'_i =\gamma a_i## and ##b'_i = \gamma b_i## for i=2 and i=3.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted