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Lorentz transformation & relativistic momentum

  1. Jan 27, 2016 #1
    1. The problem statement, all variables and given/known data
    We now specify the velocity v to be along the positive x1-direction in S and of magnitude v. We also consider a frame [itex] \overline{S} [/itex] which moves at speed u with respect to S in the positive x1-direction.

    question 1 : Write down the transformation law for [itex] p^\mu [/itex].
    question 2: Write [itex] \overline{p} ^\mu [/itex] also in terms of the speed [itex] \overline{v} [/itex] and its corresponding gamma factor.

    For the first question I got the following answer which I believe is right:
    [itex] \begin{pmatrix}
    \overline{p}^0 \\ \overline{p}^1 \\ \overline{p}^2 \\ \overline{p}^3 \end{pmatrix}
    \begin{pmatrix} \gamma & -\gamma \beta & 0 & 0 \\
    - \gamma \beta & \gamma & 0 & 0 \\
    0 & 0 & 1 & 0\\
    0 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} p^0 \\ p^1 \\ p^2 \\ p^3 \end{pmatrix}[/itex]
    [itex] \overline{p}^0 = \gamma (p^0 - \beta p^1) [/itex] (1)
    [itex] \overline{p}^1 = \gamma (p^1 - \beta p^0) [/itex] (2)
    [itex] \overline{p}^2 = p^2 [/itex] (3)
    [itex] \overline{p}^3 = p^3 [/itex] (4)
    [itex] \overline{p}^\mu = \Lambda ^\mu _\nu p^\nu [/itex] (5)
    2. Relevant equations
    [itex] \textbf{p} = m\eta = \frac{ m \eta}{\sqrt{1 - u^2/c^2}} [/itex] (6)
    [itex] p^0 = m \eta ^0 = \frac{ mc}{\sqrt{ 1 - u^2/c^2}} [/itex] (7)
    [itex] p^\mu p_\mu = -(p^0)^2 + ( \textbf{p} \bullet \textbf{p}) = -m^2c^2 [/itex] (8)

    3. The attempt at a solution

    Now I believe I have to substitute equation 7 for p0 in equation 1/2
    and equation 6 for p1/2/3 in equation 1-4
    The only problem now is that equations 6 and 7 assume a certain u2 but I need to get something expressed in [itex] \overline{v}[/itex]. According to my textbook u in equation 6/7 is :the velocity of a travelling object of mass m .
    To me it sounds like this means that I can replace all the u's by
    [itex] \overline{v}[/itex]'s.
    But I already know that in the next question I need to get something expressed in [itex] \overline{v}[/itex], [itex] v[/itex] and [itex] u [/itex]. So maybe that probably means that my theory of how to substitute the u's is wrong.

    Of course we can also use Einstein's velocity addition law: [itex] \overline{u} = \frac{u - v}{ 1 - uv/c^2} [/itex] where [itex] \overline{u} [/itex] is the velocity between the two reference frames.

    Thanks in advance for any help!
     
  2. jcsd
  3. Jan 27, 2016 #2

    ChrisVer

    User Avatar
    Gold Member

    In the equations (1) and (2), can you write the expression of [itex]\gamma[/itex] in terms of velocity?
     
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