(adsbygoogle = window.adsbygoogle || []).push({}); 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

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

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