4-vectors

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4-vectors

Using index notation, the time and space coordinates can be combined into a single "4-vector" $x^{\mu},\ \mu=0,\ 1,\ 2,\ 3$, that has units of length, i.e. ct is a distance.

$\begin{bmatrix} x_0 \\ x_1 \\ x_2 \\ x_3 \end{bmatrix}= \begin{bmatrix} ct \\ x \\ y \\ z \end{bmatrix}$

Using the Lorentz transformations and the index notation,

$\begin{cases} t'=\gamma (t-vz/c^2) \\ x'=x' \\ y'=y' \\ z'=\gamma (z-vt) \end{cases}$

$\begin{bmatrix} x_0' \\ x_1' \\ x_2 '\\ x_3' \end{bmatrix}= \begin{bmatrix} \gamma (x_0-vx_3/c) \\ x_1 \\ x_2 \\ \gamma (x_3-vx_0) \end{bmatrix} = \begin{bmatrix} \gamma (x_0-\beta x_3) \\ x_1 \\ x_2 \\ \gamma (x_3-vx_0) \end{bmatrix}$

Where $\beta \equiv \frac{v}{c}$

We can express the space time interval using the index notation

$ds^2\equiv c^2 dt^{'2}-dx^{'2}-dy^{'2}-dz^{'2}= c^2 dt^{2}-dx^2-dy^2-dz^2$

$ds^2\equiv dx_0^{'2}-dx_1^{'2}-dx_2^{'2}-dx_3^{'2}= dx_0^{2}-dx_1^2-dx_2^2-dx_3^2$

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