# Modern Physics - Extention of the Lorentz Transformation?

1. Mar 10, 2010

### PFStudent

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 $${\vec{v}}$$ along a positive ${x}$-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: $${K(x,y,z,t)}$$ and $${K^{\prime}({x^{\prime}},{y^{\prime}},{z^{\prime}},{t^{\prime}})}$$ is the following,
$${x^{\prime}} = {{\gamma}{\left({x-vt}\right)}}$$
$${y^{\prime}} = {y}$$
$${z^{\prime}} = {z}$$
$${t^{\prime}} = {t}$$

Consider the following, what would the Lorentz Transformation equations be if one reference frame was moving with a constant velocity $${\vec{v}}$$ along a radial direction ${\vec{r}}$ (which is common to both reference frames) with respect to the other reference frame? Given reference frames: $${K(x,y,z,t)}$$ and $${K^{\prime}({x^{\prime}},{y^{\prime}},{z^{\prime}},{t^{\prime}})}$$; 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).
$${{{x}^{2}}+{{y}^{2}}+{{z}^{2}}} = {{{c}^{2}}{{t}^{2}}}$$
$${{{{x}^{\prime}}^{2}}+{{{y}^{\prime}}^{2}}+{{{z}^{\prime}}^{2}}} = {{{c}^{2}}{{{t}^{\prime}}^{2}}}$$

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

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

Thanks,

-PFStudent

Last edited: Mar 10, 2010