115 Minkowski Space

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

definition [d] (Minkowski Space = Minkowski Spacetime = Flat Spacetime) The four-dimensional flat Lorentzian manifold of special relativity, with Minkowski metric \(\eta_{\mu\nu} = \operatorname{diag}(-1,1,1,1)\) and line element

\(ds^{2} = -c^{2}\, dt^{2} + dx^{2} + dy^{2} + dz^{2}\) .

where

\(\eta_{\mu\nu}\) is the Minkowski metric tensor.
\(ds\) is the infinitesimal spacetime interval.
\(c\) is the speed of light.
\(t\) is time.
\(x, y, z\) are Cartesian spatial coordinates.
\(x^{\mu} = (ct,\, x,\, y,\, z)\) are spacetime coordinates of an event.

Note:

events are points of the manifold.
\(c\) is often set to \(1\).
the Riemann curvature vanishes.

definition [d] (Minkowski Space = Minkowski Spacetime) The same flat spacetime with the opposite metric signature \(\eta_{\mu\nu} = \operatorname{diag}(1,-1,-1,-1)\) and line element

\(ds^{2} = (c\, dt)^{2} - dx^{2} - dy^{2} - dz^{2}\) .

where

\(x^{\mu} = (ct,\, x,\, y,\, z)\) are spacetime coordinates.
\(\eta_{\mu\nu}\) is the Minkowski metric.
\(c\) is the speed of light.
\(ds\) is the infinitesimal spacetime interval.

Note:

the two signatures are a convention, not different geometries.

115.1 References

Carroll, S. Spacetime and Geometry: An Introduction to General Relativity. Cambridge University Press, 2021. — source for the heading explanation.
Carroll, S. Spacetime and Geometry: An Introduction to General Relativity. Cambridge University Press, 2021. — Minkowski spacetime; signature \((-+++)\).
Griffiths, D. J. — Minkowski metric and flat spacetime as cited in notebook.
Emam, M. H. Covariant Physics. Oxford University Press, 2021. — Minkowski metric conventions.

114 Minkowski Metric

116 Moving Clocks