# Pre-Quantum Electrodynamics
Author: Jean-Sébastien Caux

#### Covariant and Contravariant Four-Vectors
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\paragraph{Four-vectors.} Let's introduce the standard notations \[ x^0 \equiv ct, \hspace{10mm} \beta \equiv \frac{v}{c}, \hspace{10mm} x^1 = x, ~~x^2 = y, ~~x^3 = z. \] The Lorentz transformation then reads

{\bf Lorentz transformation (motion along \(x\) at velocity \(v\))} \[ \bar{x}^0 = \gamma \left( x^0 - \beta x^1 \right), ~~~~\bar{x}^1 = \gamma \left( x^1 - \beta x^0 \right), ~~~~\bar{x}^2 = x^2, ~~~~\bar{x}^3 = x^3 \] or in matrix form \[ \left( \begin{array}{c} \bar{x}^0 \\ \bar{x}^1 \\ \bar{x}^2 \\ \bar{x}^3 \end{array} \right) = \left( \begin{array}{cccc} \gamma & -\gamma \beta & 0 & 0 \\ -\gamma \beta & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right) \left( \begin{array}{c} x^0 \\ x^1 \\ x^2 \\ x^3 \end{array} \right) \]

This can be compactly written as \[ \bar{x}^\mu = \sum_{\nu = 0}^3 \Lambda^\mu_\nu x^\nu. \]

\paragraph{Covariant and contravariant vectors.} Four-vectors with upper index are called {\it contravariant}. Their lower-index counterparts are called {\it covariant} vectors and are obtained by using the Minkowski metric \(g_{\mu \nu}\) according to

\[ a_\mu = \sum_{\nu = 0}^3 g_{\mu \nu} a^\nu, \hspace{10mm} g_{\mu \nu} = \left( \begin{array}{cccc} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right) \]

\paragraph{Scalar products} are defined as the in-product of covariant/contravariant four-vectors,

\[ \sum_{\mu = 0}^3 a^\mu b_\mu \equiv a^\mu b_\mu \]

where in the right-hand side we have introduced the {\bf Einstein summation convention}, namely that any repeated index is implicitly summed over. As you can trivially check, it doesn't matter which vector is co/contravariant: \(a^\mu b_\mu = a_\mu b^\mu\). Scalar products are Lorentz-invariant and thus take the same value in all inertial systems.

\paragraph{Invariant intervals.} Generalizing the notion of the norm of a vector, the scalar product of a four-vector with itself is known as the invariant interval. Because of the geometry of spacetime, the invariant can take positive or negative values. The nomenclature goes as follows:

\begin{center} \begin{tabular}{cc} $a^\mu a_\mu > 0$ & $a^\mu$ is {\it spacelike} \\ $a^\mu a_\mu < 0$ & $a^\mu$ is {\it timelike} \\ $a^\mu a_\mu = 0$ & $a^\mu$ is {\it lightlike} \end{tabular} \end{center}

For two events \(A\) and \(B\), the difference \[ \Delta x^\mu \equiv x_A^\mu - x_B^\mu \] is called the {\bf displacement four-vector} and its self-scalar product is the {\bf invariant interval} between the two events: \[ I \equiv \Delta x^\mu \Delta x_\mu = -c^2 \Delta t^2 + |{\boldsymbol x}|^2 \] where \(t\) is the time difference between the events and \({\boldsymbol x}\) is their spatial separation vector.

\paragraph{Spacetime diagrams.} These are also know as {\it Minkowski diagrams}. Time is on the vertical axis, space on the horizontal one. The trajectory of a particle is known as its {\bf world line}. Light is represented as propagating at lines at 45 degrees, defining the {\bf forward} and {\bf backward light cones}. Lorentz transformations, which preserve all invariant intervals, move spacetime points around but leave them on the same hyperboloid.

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Author: Jean-Sébastien Caux

Created: 2022-03-07 Mon 20:38