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%TCIDATA{<META NAME="Title" CONTENT="Reference/Physics/The Lorentz Transformation">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Thu Feb 13 16:20:51 1997}
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\section{The Lorentz Transformation\label{The Lorentz Transformation}}

The Lorentz transformation 
\[
(\vec{x}^{\prime },t^{\prime })=(\vec{x}^{\prime }(\vec{x},t),t^{\prime }(%
\vec{x},t)) 
\]
leaves the wave equation invariant if $c$ is invariant: 
\[
\frac{\partial ^{2}}{\partial x^{2}}+\frac{\partial ^{2}}{\partial y^{2}}+%
\frac{\partial ^{2}}{\partial z^{2}}-\frac{1}{c^{2}}\frac{\partial ^{2}}{%
\partial t^{2}}=\frac{\partial ^{2}}{\partial x^{\prime }{}^{2}}+\frac{%
\partial ^{2}}{\partial y^{\prime }{}^{2}}+\frac{\partial ^{2}}{\partial
z^{\prime }{}^{2}}-\frac{1}{c^{2}}\frac{\partial ^{2}}{\partial t^{\prime
}{}^{2}} 
\]
The general form of the Lorentz transformation is given by: 
\[
\vec{x}^{\prime }=\vec{x}+\frac{(\gamma -1)(\vec{x}\cdot \vec{v})\vec{v}}{%
|v|^{2}}-\gamma \vec{v}t~~,~~t^{\prime }=\frac{\gamma (t-\vec{x}\cdot \vec{v}%
)}{c^{2}} 
\]
where 
\[
\gamma =\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}} 
\]
The velocity difference $\vec{v}~^{\prime }$ between two observers
transforms according to: 
\[
\vec{v}~^{\prime }=\left( \gamma \left( 1-\frac{\vec{v}_{1}\cdot \vec{v}_{2}%
}{c^{2}}\right) \right) ^{-1}\left( \vec{v}_{2}+(\gamma -1)\frac{\vec{v}%
_{1}\cdot \vec{v}_{2}}{v_{1}^{2}}\vec{v}_{1}-\gamma \vec{v}_{1}\right) 
\]
If the velocity is parallel to the $x$-axis, this becomes $y^{\prime }=y$, $%
z^{\prime }=z$ and: 
\begin{eqnarray*}
&&x^{\prime }=\gamma (x-vt)~,~~~x=\gamma (x^{\prime }+vt^{\prime }) \\
&&t^{\prime }=\gamma \left( t-\frac{xv}{c^{2}}\right) ~,~~~t=\gamma \left(
t^{\prime }+\frac{x^{\prime }v}{c^{2}}\right) ~,~~~v^{\prime }=\frac{%
v_{2}-v_{1}}{1-\frac{v_{1}v_{2}}{c^{2}}}
\end{eqnarray*}
If $\vec{v}=v\vec{e}_{x}$ holds: 
\[
p_{x}^{\prime }=\gamma \left( p_{x}-\frac{\beta W}{c}\right) ~,~~~W^{\prime
}=\gamma (W-vp_{x}) 
\]
With $\beta =\dfrac{v}{c}$, the electric field of a moving charge is given
by: 
\[
\vec{E}=\frac{Q}{4\pi \varepsilon _{0}r^{2}}\frac{(1-\beta ^{2})\vec{e}_{r}}{%
(1-\beta ^{2}\sin ^{2}(\theta ))^{3/2}} 
\]
The electromagnetic field transforms according to: 
\begin{eqnarray*}
\vec{E}^{\prime } &=&\gamma (\vec{E}+\vec{v}\times \vec{B}) \\
\vec{B}^{\prime } &=&\gamma \left( \vec{B}-\frac{\vec{v}\times \vec{E}}{c^{2}%
}\right)
\end{eqnarray*}
Length, mass and time transform according to: 
\[
\Delta t_{\mathrm{r}}=\gamma \Delta t_{\mathrm{0}}\text{,\quad }m_{\mathrm{r}%
}=\gamma m_{0}\text{,\quad }l_{\mathrm{r}}=l_{0}/\gamma 
\]
with $_{\mathrm{0}}$ the quantities in a co-moving reference frame and $_{%
\mathrm{r}}$ the quantities in a frame moving with velocity $v$ with respect
to\ it. The proper time $\tau $ is defined as: 
\[
d\tau ^{2}=ds^{2}/c^{2} 
\]
so 
\[
\Delta \tau =\Delta t/\gamma 
\]
For energy and momentum the following holds: 
\begin{eqnarray*}
W &=&m_{\mathrm{r}}c^{2}=\gamma W_{0} \\
W^{2} &=&m_{0}^{2}c^{4}+p^{2}c^{2} \\
p &=&m_{\mathrm{r}}v=\gamma m_{0}v=Wv/c^{2} \\
pc &=&W\beta
\end{eqnarray*}
where $\beta =v/c$. The \textit{force} is \textit{defined} by 
\[
\vec{F}=d\vec{p}/dt 
\]

4-vectors have the property that their modulus is independent of the
observer: their components \textit{can} change after a coordinate
transformation but not their modulus. The difference of two 4-vectors
transforms also as a 4-vector. The 4-vector for the velocity is given by 
\[
U^{\alpha }=\frac{dx^{\alpha }}{d\tau } 
\]
The relation with the ``common'' velocity $u^{i}:=dx^{i}/dt$ is: 
\[
U^{\alpha }=(\gamma u^{i},ic\gamma ) 
\]
For particles with nonzero rest mass the following holds: 
\[
U^{\alpha }U_{\alpha }=-c^{2} 
\]
For particles with zero rest mass (so with $v=c$) the following holds: 
\[
U^{\alpha }U_{\alpha }=0 
\]
The 4-vector for energy and momentum is given by: 
\[
p^{\alpha }=m_{0}U^{\alpha }=(\gamma p^{i},iW/c) 
\]
So we have 
\[
p_{\alpha }p^{\alpha }=-m_{0}^{2}c^{2}=p^{2}-W\,^{2}/c^{2} 
\]

\subsection{\ Related Reference Topics}

{\small \hyperref{Calculus}{}{}{../Mathematics/cal_contents.tex}}

{\small \hyperref{Differential Equations}{}{}{../Mathematics/ODE_contents.tex%
}}

{\small \hyperref{Table of Integrals}{}{}{../Mathematics/TAB_integrals.tex}}

{\small \hyperref{Table of Derivatives}{}{}{%
../Mathematics/TAB_derivatives.tex}}

{\small \hyperref{Table of Physical Constants}{}{}{PhysicalConstants.tex}}

{\small \hyperref{Table of SI Units}{}{}{SIUnits.tex}}

\subsection{\ Related Scientific Notebook Topics}

{\small \hyperref{Numbers, Functions, and Units}{}{}{../../Help/DM2.tex}}

{\small \hyperref{Calculus}{}{}{../../Help/DM7.tex}}

{\small \hyperref{Vector Calculus}{}{}{../../Help/DM9.tex}}

{\small \hyperref{Differential Equations}{}{}{../../Help/DM10.tex}}

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