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4-vectors
Using index notation, the time and space coordinates can be combined into a single "4-vector" [math]x^{\mu},\ \mu=0,\ 1,\ 2,\ 3[/math], that has units of length, i.e. ct is a distance.
[math]\begin{bmatrix}
x_0 \\
x_1 \\
x_2 \\
x_3
\end{bmatrix}=
\begin{bmatrix}
ct \\
x \\
y \\
z
\end{bmatrix}[/math]
Using the Lorentz transformations and the index notation,
[math]
\begin{cases}
t'=\gamma (t-vz/c^2) \\
x'=x' \\
y'=y' \\
z'=\gamma (z-vt)
\end{cases}
[/math]
[math]\begin{bmatrix}
x_0' \\
x_1' \\
x_2 '\\
x_3'
\end{bmatrix}=
\begin{bmatrix}
\gamma (x_0-vx_3/c) \\
x_1 \\
x_2 \\
\gamma (x_3-vx_0)
\end{bmatrix}
=
\begin{bmatrix}
\gamma (x_0-\beta x_3) \\
x_1 \\
x_2 \\
\gamma (x_3-vx_0)
\end{bmatrix}[/math]
Where [math]\beta \equiv \frac{v}{c}[/math]
We can express the space time interval using the index notation
[math]ds^2\equiv c^2 dt^{'2}-dx^{'2}-dy^{'2}-dz^{'2}= c^2 dt^{2}-dx^2-dy^2-dz^2[/math]
[math]ds^2\equiv dx_0^{'2}-dx_1^{'2}-dx_2^{'2}-dx_3^{'2}= dx_0^{2}-dx_1^2-dx_2^2-dx_3^2[/math]
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