Semi-indefinite inner product and generalized Minkowski spaces

Abstract

a b s t r a c tIn this paper we develop the theories of normed linear spaces and of linear spaces with indefinite metric, for finite dimensions both of which are also called Minkowski spaces in the literature.In the first part of this paper we collect the common properties of the semi-and indefinite inner products and define the semi-indefinite inner product as well as the corresponding semi-indefinite inner product space. We give a generalized concept of the Minkowski space embedded in a semi-indefinite inner product space using the concept of a new product, which contains the classical cases as special ones.In the second part we investigate the real, finite-dimensional generalized Minkowski space and its sphere of radius i. We prove that it can be regarded as a so-called Minkowski-Finsler space, and if it is homogeneous with respect to linear isometries, then the Minkowski-Finsler distance of its points can be determined by the Minkowski product.

Isometries of Minkowski geometries

“…He said that the semi-inner product [·, ·] has the Lipschitz property if for every x from the unit ball there is e real number κ such that for every y and z from the unit ball holds |[x, y] − [x, z]| ≤ κ y − z . We note that the differentiability property for the semi-inner product (defined in [10]) implies the Lipschitz property of the product, too. Let A be a diagonalizable linear operator of V , and let λ 1 > λ 2 > .…”

“…Nevertheless, the phrase "Minkowski space" is applied for two different theories: the theory of normed linear spaces and the theory of linear spaces with indefinite metric. It is interesting (see [10], [11], [12]) that these essentially distinct theories have similar axiomatic foundations. The axiomatic build-up of the theory of linear spaces with indefinite metric comes from H. Minkowski [25] and the similar system of axioms of normed linear spaces was introduced by Lumer much later in [19].…”

A formula for the g-angle between two subspaces of a normed space

“…One may observe that the angle θ in X satisfies the following basic properties (see [3]). In a normed space, the concept of angles between two vectors has been studied intensively (see, for instance, [1,2,5,8,11,12,13]). Here we shall be interested in the notion of angles between two subspaces of a normed space using a semi-inner product.…”

A Physical Space-modeled Approach to Lagrangian Equations with Bundle Structure for Minkowski 4-space

“…To follow these research in physics as a first step, we prefer to solve Lagrangian energy equations on the Minkowski 4-space which is based on a jet bundle structure.Works on generalized Minkowski spaces in the references [2,3,12], the time-space manifolds and studies for time-dependent Lagrangians can be seen in the references [4,17].…”