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Homework Help: Acceleration 4 vector question

  1. Apr 15, 2010 #1
    This is question 5.8 from Rindler's Relativity: Special, General, Cosmological book, with A, U the 4 acceleration and velocity; a, u the 3 acceleration and velocity. It has 3 parts which I think I've answered correctly below, but need help with the last part.

    So could you have a look and comment on it?

    a) Prove that, in any inertial frame where a is orthogonal to u, [tex]\textbf{A}= \gamma^{2} (\textbf{a},0)[/tex], and conversely.

    my answer:

    [tex]\dot{\gamma}=\gamma^{3}/c^{2}\textbf{a}\cdot\textbf{u}[/tex]

    [tex]\textbf{a}\cdot\textbf{u} = 0 [/tex]

    [tex]\textbf{A}= \gamma d\textbf{U}/dt = \gamma d/dt(\gamma\textbf{u},\gamma c) = \gamma(\dot{\gamma}\textbf{u} + \gamma\textbf{a}, \dot{\gamma}c ) [/tex]

    and so

    [tex]\textbf{A}= \gamma^{2} (\textbf{a},0)[/tex] --- (1)

    which proves the first part.

    Given (1), then

    [tex] \dot{\gamma}c = 0 [/tex] and so

    [tex]\dot{\gamma}=\gamma^{3}/c^{2}\textbf{a}\cdot\textbf{u} = 0 [/tex]

    which is only possible for [tex]\textbf{a}\cdot\textbf{u} = 0 [/tex]

    Hence the converse is also true.

    b) Deduce that for any instantaneous motion it is possible to find an inertial frame S_|_ in which a, u are orthogonal to one another.

    my answer:

    In s_|_ , [tex]\textbf{A}= \gamma^{2} (\textbf{a},0)[/tex]

    I need to show [tex]\dot{\gamma}c[/tex] in A can transform to 0 in S_|_:

    [tex]\gamma_{v}(\gamma_{u}\dot{\gamma_{u}} - v(\gamma_{u}\mathbf{a_{x}} + \dot{\gamma_{u}}\mathtbf{u_{x}})/c^{2}) = 0 [/tex]

    [tex] v = \gamma_{u}\dot{\gamma_{u}}c^{2} / (\gamma_{u}\mathbf{a_{x}} + \dot{\gamma_{u}}\mathbf{u_{x}})[/tex]

    [tex] v = \gamma_{u}^{4}\mathbf{a}\cdot\mathbf{u} / (\gamma_{u}\mathbf{a_{x}} + \dot{\gamma_{u}}\mathbf{u_{x}})[/tex]--- (2)

    Hence there exists a v and therefore s_|_.

    c) Moreover, prove that there is a whole class of such frames, all moving relative to S_|_ in directions orthogonal to a.

    My answer:

    The stationary frame can be rotated giving a new a_x, u_x in (2) and hence another v and s_|_. But I can't show they're moving relative to S_|_ in directions orthogonal to a.

    Thanks for your help.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
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