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Is the 4-velocity always a space-tipe 4-vector?

  1. May 11, 2009 #1
    Is the 4-velocity always a space-tipe 4-vector?
     
  2. jcsd
  3. May 11, 2009 #2

    dx

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    Re: 4-velocity

    No, 4-velocity is always time-like, unless you're talking about tachyons.
     
  4. May 11, 2009 #3
    Re: 4-velocity

    But |4-velocity|^2 = -1 . So why is it time-lke?
     
  5. May 11, 2009 #4
    Re: 4-velocity

    How can I show that a 4-velocity is always time-like?
     
  6. May 11, 2009 #5

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    Re: 4-velocity

    4-velocities of ordinary particles are timelike because they travel slower than light.
     
  7. May 11, 2009 #6
    Re: 4-velocity

    Ok, so if I have a given 4-velocity how can I show in terms of mathematics that it is time-like ?
     
  8. May 11, 2009 #7

    dx

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    Re: 4-velocity

    Show that it's components satisfy t² - x² - y² - z² = 1.
     
  9. May 11, 2009 #8
    Re: 4-velocity

    why?
     
  10. May 11, 2009 #9

    dx

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    Re: 4-velocity

    What do you mean why? That's the definition of "timelike". A 4-vector is timelike if t² - x² - y² - z² > 0. All 4-velocites of ordinary particles must satisfy t² - x² - y² - z² = 1 (by definition), which is positive, so 4-velocities are timelike.

    BTW, questions like this should be posted in the "Homework & Coursework Questions" forum.
     
    Last edited: May 11, 2009
  11. May 11, 2009 #10
    Re: 4-velocity

    Just calculate its square in the reference frame where the ordinary velocity is equal to zero.
    Then only time-component contributes to the square. A time-like four-vector remains such in the other reference frames.

    Bob.
     
  12. May 11, 2009 #11
    Re: 4-velocity

    Definition of "timelike" : travel slower than light
    WHY a 4-velocity that travel slower than light satisfies t² - x² - y² - z² = 1 ?!!

    BTW, I'm trying to understand the meaning of time-like 4-velocity, I have not written any equation, any text of an exercise
     
  13. May 11, 2009 #12

    dx

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    Re: 4-velocity

    Do you know the definition of 4-velocity?
     
  14. May 11, 2009 #13
    Re: 4-velocity

    I wanted to say :

    Definition of "timelike" 4-velocity :4-velocity of a PARTICLE THAT travels slower than light (ordinary particles). I wrote in a rush.
    (if you're referring to this)
     
  15. May 11, 2009 #14

    dx

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    Re: 4-velocity

    I'm just asking you what the definition of 4-velocity is. What are the components of the 4-velocity of a particle in a reference frame in which it's velocity is (vx, vy, vz)?

    If you can answer this question, then I will be able to answer your question:
     
    Last edited: May 11, 2009
  16. May 11, 2009 #15
    Re: 4-velocity

    ([tex]\frac{v_{x}}{c\sqrt{1-\beta^{2}}}[/tex] , [tex]\frac{v_{y}}{c\sqrt{1-\beta^{2}}}[/tex] , [tex]\frac{v_{z}}{c\sqrt{1-\beta^{2}}}[/tex] , [tex]\frac{i}{\sqrt{1-\beta^{2}}}[/tex] )
     
  17. May 11, 2009 #16

    dx

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    Re: 4-velocity

    What's [itex] i [/itex] in the fourth component?
     
  18. May 11, 2009 #17
    Re: 4-velocity

    complex number : i^2=-1
     
  19. May 11, 2009 #18
    Re: 4-velocity

    Wow. Might you be able to find a textbook that was written after 1936? Because that was about the last time anyone ever used the ict convention. In that convention, the norm of the 4-velocity will always be -1, and is still timelike, because that is what timelike means in that convention. Get a newer book. Seriously. We've learned a lot since World War II.
     
  20. May 11, 2009 #19

    dx

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    Re: 4-velocity

    No, 4-vectors don't have imaginary components.

    The components of the 4-velocity uµ are

    [tex] u_0 = \frac{1}{\sqrt{1-\beta^{2}}} [/tex]

    [tex] u_1 = \frac{v_{x}}{c\sqrt{1-\beta^{2}}} [/tex]

    [tex] u_2 = \frac{v_{y}}{c\sqrt{1-\beta^{2}}} [/tex]

    [tex] u_3 = \frac{v_{z}}{c\sqrt{1-\beta^{2}}} [/tex]

    Now, calculate [tex] u_0^2 - u_1^2 - u_2^2 - u_3^2 [/tex]. You should find that it is equal to 1.
     
  21. May 11, 2009 #20
    Re: 4-velocity

    :) very funny, however my prof is able to find a textbook that was written after 1936. Infact I don't find a textbook that uses the ict convention!But I had to study on the prof's notes. The professors should be renovate their lessons, I should learn from them!
     
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