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Mathematics for Physics

181 Minkowski Metric

A constant metric matrix with one negative and three positive diagonal entries that is used to compute interval distance in flat spacetime.

The metric tensor of flat spacetime. The Minkowski metric is a symmetric array of numbers that assigns a scalar to any pair of spacetime vectors. A metric is a rule for computing that scalar. This principle is used to define lengths and angles in flat spacetime.

The Minkowski metric is

\[ \eta_{\mu\nu} = \operatorname{diag}(-1,\, 1,\, 1,\, 1) \]

where

  • \(\eta_{\mu\nu}\) are the components of the Minkowski metric.

The line element. The metric determines the line element, whose sign distinguishes timelike, spacelike, and lightlike separations. A line element is the infinitesimal squared distance between two neighboring events. This principle is used to classify the separation of nearby events.

The line element is

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

where

  • \(x^{\mu}\) are spacetime coordinates with \(x^{0} = ct\).
  • \(ds\) is the infinitesimal spacetime interval.
  • \(c\) is the speed of light.
  • \(t\) is time.
  • \(x, y, z\) are Cartesian spatial coordinates.

Constant metric components. The metric components are constant in inertial Cartesian coordinates. This principle is used to describe spacetime when gravity can be neglected.

Invariant scalar products. The scalar product formed with the Minkowski metric has the same value for every inertial observer. A four-vector is a spacetime vector whose components mix under a Lorentz transformation. This principle is used to write physical laws that keep the same form in every inertial frame.

Note: Also called the Minkowski metric tensor. Also called the flat metric.

181.1 References

  1. Carroll, S. Spacetime and Geometry: An Introduction to General Relativity. Cambridge University Press, 2021. — source for the heading explanation.
  2. Carroll, S. Spacetime and Geometry: An Introduction to General Relativity. Cambridge University Press, 2021. — Minkowski metric \(\eta_{\mu\nu}=\operatorname{diag}(-1,1,1,1)\); invariant scalar products.
  3. Emam, M. H. Covariant Physics. Oxford University Press, 2021. — Minkowski metric signature conventions; classification of separations.
  4. Susskind, L., & Friedman, A. Special Relativity and Classical Field Theory. Basic Books, 2017. — line element and scalar products.
  5. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — invariance of spacetime products in inertial frames.