Blog explaining hyperbolic trig functions
x′=γ(x−ut)
- The Lorentz-transformed x-coordinate x′ in S′ of x in S
- u is the velocity of S′ relative to S
t′=γ(t−c2ux)
- The Lorentz-transformed time t′ in S′ of t in S
- t and x are native to S
vx′=1−c2uvxvx−u
vAC=1+(c2vABvBC)vAB+vBC
- Einstein’s velocity addition rule
- vAB is the speed of object A relative object B
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}$$
*