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.

Minkowski inner product matrix ( $η_{μ ν}$ )

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

η_{μ ν} = ⎣ ⎢ ⎢ ⎢ ⎡ - 1 000 010000100001 ⎦ ⎥ ⎥ ⎥ ⎤

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

η_{μ ν} = ⎣ ⎢ ⎢ ⎢ ⎡ 1000 0 - 1 00 00 - 1 0 000 - 1 ⎦ ⎥ ⎥ ⎥ ⎤

The matrix is typically denoted by the Greek letter eta.

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