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Mathematics for Physics

114 Minkowski Metric

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

definition [d] (Minkowski Metric = Minkowski Metric Tensor = Flat Metric) The constant metric tensor of Minkowski spacetime,

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

with line element

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

where

  • \(\eta_{\mu\nu}\) are the components of the Minkowski metric.
  • \(\mu, \nu\) are spacetime indices.
  • \(x^{\mu}\) 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.

Note:

  • \(\mu, \nu\) run over \(0,1,2,3\).
  • \(\eta_{\mu\nu}\) is diagonal and constant in inertial Cartesian coordinates.
  • \(c\) is often set to \(1\).

definition [d] (Minkowski Metric = Minkowski Metric Tensor) The constant metric tensor

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

with line element

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

where

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

Note:

  • the choice of overall sign is a convention, opposite to \((-+++)\).