Minkowski inner product

This form is not really an inner product in the common modern definition, because it is not positive definite, only a symmetric bilinear form.

By default, we will use the time negative representation unless stated otherwise:

η_{μν} = - 1 000 010000100001

but another equivalent one is to use a time positive representation:

η_{μν} = 1000 0 - 1 00 00 - 1 0 000 - 1

The matrix is typically denoted by the Greek letter eta.

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Codex  0 Created 2026-09-24 Updated 2026-10-14

This indefinite bilinear form supplies the metric of Minkowski spacetime. On

R^{n + 1}

the Minkowski inner product with mostly-positive convention is

⟨ x, y ⟩_{M} = - x_{0} y_{0} + x_{1} y_{1} + \dots + x_{n} y_{n} .