#### Covariant and Contravariant Four-Vectors
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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

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 \[ \bar{x}^\mu = \sum_{\nu = 0}^3 \Lambda^\mu{}_\nu x^\nu. \]

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

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

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 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.

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:

\(a^\mu a_\mu > 0\) | \(a^\mu\) is spacelike
\(a^\mu a_\mu < 0\) | \(a^\mu\) is timelike
\(a^\mu a_\mu = 0\) | \(a^\mu\) is lightlike

For two events \(A\) and \(B\), the difference \[ \Delta x^\mu \equiv x_A^\mu - x_B^\mu \] is called the displacement four-vector and its self-scalar product is the 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.

Spacetime diagrams: these are also know as Minkowski diagrams. Time is on the vertical axis, space on the horizontal one. The trajectory of a particle is known as its world line. Light is represented as propagating at lines at 45 degrees, defining the forward and 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: 2022-03-24 Thu 08:42