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In 4-momentum, why is E the 4th component?

  1. Oct 31, 2012 #1
    Assuming that c is a "conversion factor" to convert between space and time,

    Then, in 4-vector, we have x_1 through x_3, and t, where, x/c = t

    x/c = t, (where t = time, c= lightspeed, x = spacial dimension)

    If we do what we did to space to get time, to momentum,

    p/c = m*v/c = m (x/t) / c = m(x/c)/t = mt/t = m

    we actually end up with mass... not energy...

    So, there is an inconsistency, and I must have made a mistake somewhere. Can you please describe the method and mathematical proof and context for calling energy the component of momentum that is in the time dimension?

    Thank you very much. -- j
     
  2. jcsd
  3. Oct 31, 2012 #2

    tiny-tim

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    hi jaguar7! :smile:

    you're assuming that a change of velocity should affect different dimensions equally

    but a change of velocity is a rotation between two dimensions (t and x, say), and its matrix is
    Code (Text):
    cosh  sinh
    sinh  cosh
    just as a rotation between two space dimensions is
    Code (Text):
    cos   sin
    -sin  cos
    … a rotation does not affect dimensions equally! :wink:
     
  4. Oct 31, 2012 #3
    Hi, tiny-tim. Thank you for your response.

    I'm afraid I'm not very good with matrices. I suppose I'll have to review that. I've been looking for my old books. They've been mysteriously difficult to find after I moved...

    I understand that a change in velocity is a rotation between dimensions.

    Does that mean I can't use the term velocity in the mathematics? Or that I must use a "4-velocity"? I'm not sure how I would go about doing that... though I would very much like to learn how, somehow... :p

    Thank you, again, tim. -- j
     
    Last edited: Oct 31, 2012
  5. Oct 31, 2012 #4

    tiny-tim

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    hi jaguar7! :smile:
    i don't think i've ever seen the term "4-velocity"

    (4-momentum and 4-force, yes)
     
  6. Oct 31, 2012 #5
    Thanks. =)

    Still, though, how would one go about showing mathematically that energy is the 4th component of momentum, I wonder...?
     
  7. Oct 31, 2012 #6

    pervect

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    You'd just want to show that (E,P) transformed as a 4-vector using the lorentz transform for an isolated point particle.

    To do this, you might first prove that the four-velocity is a 4-vector, then consider the prodect mass * 4-velocity, where mass is the invariant mass.
     
  8. Oct 31, 2012 #7

    jtbell

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    http://en.wikipedia.org/wiki/Four-velocity
     
  9. Nov 1, 2012 #8
    I think it is a fourth component because of the simple relation
    E2-P2=m02,c=1 I have put.
     
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