nLab Minkowski space

Contents

Context

Riemannian geometry

Riemannian geometry

Contents

Definition

For $d-1 \in \mathbb{N}$, $d$-dimensional Minkowski space is the Lorentzian manifold whose underlying smooth manifold is the Cartesian space $\mathbb{R}^d$ and whose pseudo-Riemannian metric is at each point the Minkowski metric.

This is naturally a spacetime.

Properties

Isometries

The isometry group of Minkowski space is the Poincaré group. The study of Minkowski spacetime with its isometries is also called Lorentzian geometry. This is the context of the theory of special relativity.

Gravitational stability

Theorem

Minkowski spacetimes is a stable? solution of the vacuum Einstein equations.

This is due to (ChristodoulouKlainerman 1993).

References

Due to

• Hermann Minkowski, Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern, Math. Ann. (1910) 68: 472, reprinted from: Nachrichten der Kgl. Ges. d. Wiss. zu Göttingen, Math.-phys. Kl., Sitzung vom 21. Dezember 1907 (doi:10.1007/BF01455871)

The views of space and time that I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of both will retain an independent reality.

(Address to the 80th Assembly of German Natural Scientists and Physicians, (Sep 21, 1908), see WikiQuote)