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Homework Help: Modern Physics - Extention of the Lorentz Transformation?

  1. Mar 10, 2010 #1
    1. The problem statement, all variables and given/known data
    Conventionally, the Lorentz Transformation relates two reference frames that begin at the same location and time with one reference frame moving at a constant velocity [tex]{\vec{v}}[/tex] along a positive [itex]{x}[/itex]-axis (which is common to both reference frames) with respect to the other reference frame. It follows that the transformation relating the two reference frames: [tex]{K(x,y,z,t)}[/tex] and [tex]{K^{\prime}({x^{\prime}},{y^{\prime}},{z^{\prime}},{t^{\prime}})}[/tex] is the following,
    [tex]{x^{\prime}} = {{\gamma}{\left({x-vt}\right)}}[/tex]
    [tex]{y^{\prime}} = {y}[/tex]
    [tex]{z^{\prime}} = {z}[/tex]
    [tex]{t^{\prime}} = {t}[/tex]

    Consider the following, what would the Lorentz Transformation equations be if one reference frame was moving with a constant velocity [tex]{\vec{v}}[/tex] along a radial direction [itex]{\vec{r}}[/itex] (which is common to both reference frames) with respect to the other reference frame? Given reference frames: [tex]{K(x,y,z,t)}[/tex] and [tex]{K^{\prime}({x^{\prime}},{y^{\prime}},{z^{\prime}},{t^{\prime}})}[/tex]; find this Lorentz Transformation.

    2. Relevant equations
    Knowledge of Transformations.
    Einstein's Two Postulates on Relativity (The Principle of Relativity and The Constancy of the Speed of Light).
    [tex]
    {{{x}^{2}}+{{y}^{2}}+{{z}^{2}}} = {{{c}^{2}}{{t}^{2}}}
    [/tex]
    [tex]
    {{{{x}^{\prime}}^{2}}+{{{y}^{\prime}}^{2}}+{{{z}^{\prime}}^{2}}} = {{{c}^{2}}{{{t}^{\prime}}^{2}}}
    [/tex]

    3. The attempt at a solution
    Conventionally, in a Lorentz Transformation we are only concerned with the constant velocity [tex]{\vec{v}}[/tex] of one reference frame moving along a common [itex]{x}[/itex]-axis between both reference frames with respect to the other reference frame. Consequently, the vector components of [tex]{\vec{v}}[/tex] are:
    [tex]{{\vec{v}} = {{v}_{x}}{\hat{i}}[/tex]

    Taking reference frame: [tex]{K^{\prime}({x^{\prime}},{y^{\prime}},{z^{\prime}},{t^{\prime}})}[/tex]; as the reference frame moving at constant velocity [tex]{\vec{v}}[/tex] with respect to reference frame [tex]{K(x,y,z,t)}[/tex] along a common [itex]{\vec{r}}[/itex] direction we note that velocity [tex]{\vec{v}}[/tex] now has vector components: [tex]{{\vec{v}} = {{{{v}_{x}}{\hat{i}}}+{{{v}_{y}}{\hat{j}}}+{{v}_{z}}{\hat{k}}}}}[/tex]. It follows then that the Lorentz Transformation equations must also reflect the displacements along the axes: [itex]{x}[/itex], [itex]{y}[/itex], and [itex]{z}[/itex]; but mathematically how would I show this?

    Thanks,

    -PFStudent
     
    Last edited: Mar 10, 2010
  2. jcsd
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