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The invariants of a pair of directions in geometry [En1] and their interpretation

Misiak A., Stasiak E., Szmuksta-Zawadzka M.

Demonstratio Mathematica

2013

Vol. 46, nr 2

361--371

Solving a certain functional equation, we find all invariants of a pair of directions in n-dimensional pseudo-Euclidean geometry of index one [En1]. In (n-1)-dimensional space we construct a model for these directions by means of concepts characteristic of Euclidean geometry. Because it is a pseudo-orthogonal group, not orthogonal, that operates in this model, the distance between two points and the measure of an angle are not invariants. Using these changeable quantities we construct invariant quantities.

Geometria a fizyka

Jancewicz B.

Postępy Fizyki

2006

T. 57, z. 1

2-9

General relativity has taught us that the scalar product and the metric determined by it are dynamical quantities. Therefore, it is desirable to introduce physical quantities independently of the scalar product. This article describes 16 types of directed quantities introduced in 3-dimensional space. A system of mechano-electrical units is proposed such that the fundamental units correspond to as many scalar quantities as possible, allowing to recognize from the unit of a given physical quantity its type as a directed quantity.