Minkowski space

The basic objects we will be manipulating in the theory of special relativity are tensors. Basically, a tensor is an object with contravariant indices (superscripts) and/or covariant indices (subscripts): \[\begin{align} \require{physics} & \text{contravariant:} \thinspace x^\mu % & \text{covariant:} \thinspace x_\mu \end{align}\]

\(x^\mu\), called the 4-position, or an event, has one time component and three space components: \[\begin{equation} (x^\mu) % = % \begin{pmatrix} % x^0 \\ x^1 \\ x^2 \\ x^3 % \end{pmatrix} % % = % \begin{pmatrix} % x^0 \\ (x^i) % \end{pmatrix} % % = % \begin{pmatrix} % ct \\ \vb{r} % \end{pmatrix} % \thinspace , \end{equation}\] in which we will use the notation \(( \cdot )\) to signify `the matrix representation of’. In this definition, we have already introduced that we will use Greek indices \[\begin{equation} \mu, \nu, \kappa, \lambda, \alpha, \beta, \ldots = 0, 1, 2, 3 \end{equation}\] to describe space-time coordinates, and we will use Roman indices \[\begin{equation} i, j, k, l, \ldots = 1, 2, 3 \end{equation}\] to designate space coordinates only.

Events \(x^\mu\) are the vectors, i.e. the elements, of the so-called Minkowski (vector) space. Minkowski space is additionally equipped the Minkowski metric tensor \(\eta\): \[\begin{equation} \eta \equiv (\eta_{\mu \nu}) = \mqty(\dmat[0]{1,-1,-1,-1}) \thinspace , \end{equation}\] which is symmetric \[\begin{equation} \eta_{\nu \mu} = \eta_{\mu \nu} \end{equation}\] and equal to its own inverse \[\begin{equation} \eta^{\mu \nu} \eta_{\nu \sigma} = % \delta^\mu_\sigma % \thinspace . \end{equation}\] In these previous equations, we have already introduced Einstein’s summation convention: indices that appear once both contravariantly and covariantly are dummy indices and are implicitly summed over.

Using the Minkowski metric, we can change contravariant indices to covariant indices: \[\begin{equation} x^\mu = \eta^{\mu \nu} x_{\nu} % \thinspace , \end{equation}\] and vice versa: \[\begin{equation} x_\mu = \eta_{\mu \nu} x^{\nu} % \thinspace . \end{equation}\] and we define the inner product between two tensors \(a\) and \(b\) as: \[\begin{equation} a_\mu b^\mu = \eta_{\mu \nu} a^\mu b^\nu % \thinspace , \end{equation}\]

We can then define the space-time interval \(\dd{s}^2\) as \[\begin{align} \label{eq:space-time_interval} \dd{s}^2 % &= \eta_{\mu \nu} \dd{x}^\mu \dd{x}^\nu \\ % &= c^2 \dd{t}^2 - \norm{\dd{\vb{r}}}^2 \\ % &= c^2 \dd{t}^2 - (\dd{x}^2 + \dd{y}^2 + \dd{z}^2) % \thinspace , \end{align}\] such that the space-time distance \(\dd{s}\) is given by \[\begin{equation} \dd{s} = \sqrt{ c^2 \dd{t}^2 - \norm{\dd{\vb{r}}}^2 } % \thinspace . \end{equation}\]

The \(4\)-gradient is defined as the naturally covariant vector \[\begin{equation} \label{eq:4-gradient} \partial_\mu \equiv \pdv{x^\mu} % = \begin{pmatrix} % \dfrac{1}{c} \dfrac{\partial}{\partial t} % & \dfrac{\partial}{\partial x} % & \dfrac{\partial}{\partial y} % & \dfrac{\partial}{\partial z} % \end{pmatrix} \thinspace , \end{equation}\] and the d’Alembertian is defined as \[\begin{equation} \square % \equiv \partial_\mu \partial^\mu % = \frac{1}{c^2} \pdv[2]{t} - \laplacian % \thinspace , \end{equation}\] and it acts as the generalization of the Laplacian to Minkowski space.