Minkowski spacetime - Knowlpedia

Definition

Four-dimensional Minkowski spacetime is the affine space underlying $\mathbb R^4$ , equipped with the translation-invariant Lorentzian metric

\eta=-dt^2+dx^2+dy^2+dz^2.

After choosing an origin it is identified with Minkowski vector space, whose quadratic form is the same displayed $(-+++)$ form: $q(v)=\eta(v,v)$ . The affine formulation distinguishes events from displacement vectors. The coordinate field $\partial_t$ supplies the standard time orientation.

Causal structure

For a displacement $v$ ,

\eta(v,v)<0,\quad \eta(v,v)=0,\quad \eta(v,v)>0

mean respectively timelike, null, and spacelike. The null cone separates the two timelike cones. Translations preserve this structure, as do linear transformations in the Lorentz group.

Symmetry

The full affine isometry group is the Poincaré group

\operatorname{ISO}(1,3)=\mathbb R^{1,3}\rtimes O(1,3).

Its identity component uses the proper orthochronous Lorentz group and preserves both spatial orientation and time orientation. Minkowski spacetime is flat: its Levi–Civita connection has zero curvature in inertial coordinates.

Differential operators

The scalar d’Alembert operator is

\Box_\eta=\partial_t^2-\partial_x^2-\partial_y^2-\partial_z^2

in the convention used in this collection. A spin structure and constant spin frame give the Minkowski Dirac operator.

Dimension and notation

The notation $\mathbb R^{1,3}$ is also used for the underlying quadratic vector space. More generally, $n$ -dimensional Minkowski spacetime has signature $(1,n-1)$ , with negative directions listed first in this collection. Some physics texts instead use $\eta=\operatorname{diag}(1,-1,-1,-1)$ ; all signs in wave and Clifford formulas must then be translated together.

References

Barrett O'Neill, Semi-Riemannian Geometry With Applications to Relativity, Academic Press, 1983. Publisher record. Relevant: Chapters 1 and 14.
Robert M. Wald, General Relativity, University of Chicago Press, 1984. Publisher record. Relevant: Appendix C.