Minkowski space

${\mathbb{R}}^4$ with a weird dot product-like operation called the Minkowski inner product.

Because the Minkowski inner product product is not positive definite, the norm induced by an inner product is a norm, and the space is not a metric space strictly speaking.

The name given to this type of space is a pseudometric space.

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

The matrix representation of the symmetric bilinear form that is the Minkowski inner product.

Since that is a symmetric bilinear form, the associated matrix is a symmetric matrix.

By default, we will use the time negative representation unless stated otherwise: but another equivalent one is to use a time positive representation: The matrix is typically denoted by the Greek letter eta.