Four-vector notation

Remembering that the 4-vector product is written as

$$ \tilde a\cdot \tilde b = a^0 b^0-a^1b^1-a^2 b^2 -a^3 b^3 $$

to make the minus signs inevitable we will use the Minkowski notation

Contravariant vector

$$ a^\mu\equiv (a^0,a^1,a^2,a^3)^T $$

💃 Example: $x^\mu =(ct,x,y,z)$

Covariant vector

$$ a_\mu\equiv (a_0,a_1,a_2,a_3)\equiv(a^0,-a^1,-a^2,-a^3) $$

💃 Example: $x_\mu=(ct,-x,-y,-z)$

Scalar product

$$ \tilde a \cdot \tilde a = a^\mu b_\mu = a^\mu b_\mu $$

🗒️ Notes:

• Repeated indices are summed
• Only if there is one contravariant and covariant is the result Lorentz invariant (expand for proof)

Metric tensor

To convert from one to another we use

$$ g^{\mu \nu}=g_{\mu \nu} =\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{bmatrix} $$

we can see this works:

$$ \begin{aligned} g^{\mu \nu} a_{\nu}&=\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{bmatrix} \begin{pmatrix} a^0 \\a^1 \\ a^2 \\a^3\end{pmatrix}=\begin{pmatrix} a^0 \\-a^1 \\ -a^2 \\-a^3\end{pmatrix}=\begin{pmatrix} a_0 \\a_1 \\ a_2 \\a_3\end{pmatrix}=a^\mu \\ g_{\mu \nu} a^{\nu}&=\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{bmatrix} \begin{pmatrix} a_0 \\a_1 \\ a_2 \\a_3\end{pmatrix}=\begin{pmatrix} a_0 \\-a_1 \\ -a_2 \\-a_3\end{pmatrix}=\begin{pmatrix} a^0 \\a^1 \\ a^2 \\a^3\end{pmatrix}=a_\mu \end{aligned} $$