Ethan's Notes

    Lorentz Transformation

    #Physics
    Blog explaining hyperbolic trig functions

    $x'=\gamma(x-ut)$

    • The Lorentz-transformed $x$-coordinate $x'$ in $S'$ of $x$ in $S$
    • $u$ is the velocity of $S'$ relative to $S$

    $t'=\gamma(t-\frac{ux}{c^2})$

    • The Lorentz-transformed time $t'$ in $S'$ of $t$ in $S$
    • $t$ and $x$ are native to $S$

    $v'_x=\frac{v_x-u}{1-\frac{uv_x}{c^2}}$

    $\displaystyle v_{AC}=\frac{v_{AB}+v_{BC}}{1+\left( \frac{v_{AB}v_{BC}}{c^{2}} \right)}$

    • Einstein's velocity addition rule
    • $\displaystyle v_{AB}$ is the speed of object $\displaystyle A$ relative object $\displaystyle B$

    $$\begin{bmatrix}

    t'\\x'\\y'\\z'
    \end{bmatrix} =
    \begin{bmatrix}
    \gamma & -\gamma\beta & 0 & 0\\-\gamma\beta & \gamma & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1
    \end{bmatrix}
    \begin{bmatrix}
    t\\x\\y\\z
    \end{bmatrix} =
    \begin{bmatrix}
    \cosh \theta & -\sinh \theta & 0 & 0\\-\sinh \theta & \cosh \theta & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1
    \end{bmatrix}
    \begin{bmatrix}
    t\\x\\y\\z
    \end{bmatrix}$$

    • $\beta = \frac{v}{c}$
    • $\gamma = \frac{1}{\sqrt{1 - \beta^2}} = \cosh \theta$
    • $\gamma\beta = \sinh \theta$
    • The matrix may be called $\displaystyle \Lambda$

    $\displaystyle \bar{x}=\Lambda^{\mu}_{\nu}x^{\nu}$

    • Concise Einstein notation form of the above equation
    • The superscript represents row, subscript represents column

    $s^2 = (ct)^2 - \vec r^2$

    • $s$ is the preserved interval for Lorentz Transformations

    $\vec r = (x, y, z)$

    $$g_\text{Euclidean} = \begin{bmatrix}

    	1 & 0 & 0\\\\	0 & 1 & 0\\\\	0 & 0 & 1
	\end{bmatrix}$$

    $$g_\text{hyperbolic} = \begin{bmatrix}

    	1 & 0 & 0 & 0\\\\	0 & -1 & 0 & 0\\\\	0 & 0 & -1 & 0\\\\	0 & 0 & 0 & -1
	\end{bmatrix}$$

    $$g_\text{minkowski} = \begin{bmatrix}

    	-1 & 0 & 0 & 0\\\\	0 & 1 & 0 & 0\\\\	0 & 0 & 1 & 0\\\\	0 & 0 & 0 & 1
	\end{bmatrix}$$

    $$s^2 = \begin{bmatrix}

    	ct & x & y & z
	\end{bmatrix} g_\text{hyperbolic}
	\begin{bmatrix}
	ct\\\\	x\\\\	y\\\\	z
	\end{bmatrix} = (ct)^2 - (x^2 + y^2 + z^2)$$

    $$\phi = \begin{bmatrix}

    	\omega / c & k_x & k_y & k_z
	\end{bmatrix} g_\text{hyperbolic}
	\begin{bmatrix}
	ct\\\\	x\\\\	y\\\\	z
	\end{bmatrix} = \omega t - \vec k \cdot \vec r$$

    $$k = \begin{bmatrix}

    		\omega / c, k_x, k_y, k_z
		\end{bmatrix}$$