4-vectors

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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]\Rightarrow s^2\equiv c^2 t^{'2}-x^{'2}-y^{'2}-z^{'2}= c^2 t^{2}-x^2-y^2-z^2[/math]

[math]s^2\equiv x_0{'2}-x_1^{'2}-x_2^{'2}-x_3^{'2}= x_0^{2}-x_1^2-x_2^2-x_3^2[/math]

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