Skip to main content

Mathematics for Physics

182 Minkowski Space

A set of events equipped with the Minkowski metric that is used to represent flat spacetime.

Unified spacetime. Minkowski space is one flat four-dimensional geometry of three space dimensions and one time dimension. A flat spacetime is a four-dimensional arena with zero curvature. This principle is used to represent every occurrence as a point of one geometric structure.

Invariance of the line element. The line element has the same value for every inertial observer. A line element is the infinitesimal squared separation of two nearby events. This principle is used to compute observer-independent separations of nearby events.

The line element is

\[ ds^{2} = -c^{2}\, dt^{2} + dx^{2} + dy^{2} + dz^{2} \]

where

  • \(ds\) is the infinitesimal spacetime interval.
  • \(c\) is the speed of light.
  • \(t\) is time.
  • \(x, y, z\) are Cartesian spatial coordinates.

The opposite signs of time and space. The metric gives time the opposite sign from space. A metric is the rule that computes squared lengths of spacetime separations. This principle is used to tell timelike, spacelike, and lightlike separations apart.

Global flatness. The components of the metric are constant at every coordinate, so the geometry has no local warping. This principle is used to describe physics when gravity can be neglected.

Lorentz and Poincaré symmetry. The geometry is unchanged under rotations, translations in space and time, and boosts. A boost is a coordinate change between observers in uniform relative motion. This principle is used to obtain the transformations that keep the form of the laws.

Null causal boundaries. Paths with \(ds^{2} = 0\) are the paths of light. A null path is a trajectory of zero interval. This principle is used to draw the light cone that bounds cause and effect.

Note: Also called Minkowski spacetime. Also called flat spacetime.

182.1 References

  1. Carroll, S. Spacetime and Geometry: An Introduction to General Relativity. Cambridge University Press, 2021. — flat spacetime; curvature zero; invariant line element.
  2. Emam, M. H. Covariant Physics. Oxford University Press, 2021. — opposite signs of time and space in the metric; timelike, spacelike, and lightlike classification.
  3. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — null paths and the light cone.
  4. Susskind, L., & Friedman, A. Special Relativity and Classical Field Theory: The Theoretical Minimum. Basic Books, 2017. — invariance of \(ds^{2}\); Lorentz and Poincaré symmetries.