Minkowski space

Minkowski space is the vector space $R^{d+1}$ endowed with the Minkowski metric

$ds^{2}=dx_{}^{2}-c^{2}dt^{2}$.

(In many papers, the opposite sign of the metric is used, but the difference is purely notational). We use the usual summation, raising, and lowering conventions. The quantity c is the speed of light and shall be normalized to equal 1, except when referring to the non-relativistic limit of a relativistic equation.

The d'Lambertian operator

$\Box :=\partial _{a}\partial ^{a}=\Delta -\partial _{t}^{2}$

is naturally associated to this metric, the same way that the Laplace-Beltrami operator is associated with a Riemannian metric.

Minkowski space is the natural domain for various physical equations associated with relativity, and in particular with semilinear wave equations. It is the simplest example of a Lorentzian manifold.

In addition to the obvious symmetries of space translation, time translation, spatial rotation and reflection, and time reversal symmetry, Minkowski space also has the important symmetry of Lorentz boosts.