Pseudovector

A pseudo- vector, and rotation vector, axial vector or axial vector, called in physics a vector quantity, which if you look at the mirror image of the investigated physical system at a point reflection, their direction maintains, while polar or thrust vectors reverse direction in such a case.

For example, the angular velocity of a body on a circular path, because of rotation does not change at a point reflection is described by an axial vector, while the web speed is in such a case in the opposite direction and is therefore described by a polar vector.

The direction of an axial vector is with respect to an orientation of the room, usually the right- defined. Axialvektoren typically occur when a physical relationship by the cross product (which in right-handed coordinate systems, the right-hand rule used ) is expressed.

Definition

Transformation behavior under a motion of the system

At a given physical system, consider a second, which at any time from the first seen by the same spatial movement ( ie, by a length- and angle- preserving mapping, no movement in the kinematic sense). Here are unequally intimate ( orientation -reversing ) movements allowed. Thus, at a fixed time T is a particle in the first system, which is located at the point P (t), on a particle at the position P '(t) shown in the moving system. Mass and charge of the particle remain unchanged. For continuous distributions this means that a density is mapped to a density. It is said that a physical quantity has a certain transformation properties under the movement when this transformation maps the physical size to the appropriate size in the moving system. For example, ' the transformed velocity v ' is the moving particle at the position P, which is determined by the speed V of the original particle at the point P.

Axial and polar vectors

Sets up the motion only of translations and rotations together, the transformation behavior is the same for all vectorial quantities. Considering, however, the case of a point reflection in the space at the center, ie, where and are the position vectors of a particle and its mirror image are, so there are two cases to distinguish. A polar vector, such as the velocity of the particle is characterized in that it as the position vectors transformed. An axial vector, such as the angular velocity of the particle is, on the other hand displayed under the point reflection on themselves. The property of a vector quantity to be axial or polar, already the transformation behavior sets under any movement. Because every movement can be represented by a sequential execution of translations, rotations and reflections point.

Active and passive transformation

These observations on the transformation behavior of a vector quantity with an active movement of the system has nothing to do with the transformation behavior of the components of the vector under an ordinary coordinate transformation. The latter is the same for axial and polar vectors, namely the components of a tensor of rank one. So there are real vectors in terms of the tensor calculus, which is why the term pseudo- vector is misleading in this context. In fact, there are authors who do not clearly separate these different terms. Many authors describe a non- intimate motion of the system as a coordinate transformation with a simultaneous change in the orientation with respect to which the cross product is to be calculated. This corresponds to a passive transformation, the observer undergoes the same transformation as the coordinate system. Clearly this means that the right hand at a point reflection of the coordinate system to a left hand. Mathematically, this is realized by introducing a Pseudotensors whose components are given regardless of the orientation of an orthonormal coordinate system by the Levi- Civita symbol. (Also called tensor density of weight -1) This completely antisymmetric pseudotensor therefore is not a tensor. In this sense, the term pseudo- vector is to be understood that in this analysis at a point reflection of the coordinate system changes its direction ( its components, however, remain unchanged). This passive view of the same results regarding the differentiation of axial and polar vectors, such as the active one.

Examples

• For the relation of the position vector, velocity and angular velocity of a particle. Under a point reflection is easily checked after that. Position and velocity vector are therefore polar vectors. Thus applies to the angular velocity of the mirror particle. So you have to apply, ie, the angular velocity is axial vector.
• From the formula for the Lorentz force it follows that the magnetic field must have an axial vector, since the force is proportional to acceleration, and thus a polar vector.
• The angular momentum is defined as. It follows, that the angular momentum is an axial vector.
• The vorticity with the nabla operator is an axial vector.

Calculation rules

• The cross product of two polar or two axial vectors is an axial vector.
• The cross product of a polar and an axial vector is a polar vector.
• The scalar product of two polar or two axial vectors is a scalar (ie, retains its sign under any movement).
• The scalar product of a polar and an axial vector is a pseudo-scalar (ie, changes its sign under a point reflection ).

Connection with antisymmetric tensors

An axial vector can be assigned to an antisymmetric tensor of rank two over Hodge duality. In terms of coordinates belongs to a vector, the 2- form of the Levi- Civita symbol and using the summation convention. Written as a matrix is obtained:

For the magnetic field B as obtained in this way, the spatial components of the electromagnetic Feldstärketensors. This relationship can be used to generalize equal to the likes of the angular momentum for spaces of dimension three. Only in R3 has an anti-symmetric 2- form as many independent components as a vector. In R4, for example, there are not four, but six independent components.

Example: Reflection of a rotating disk on a plane

Consider the example of a rotating horizontal disk. The disc had a red top and a yellow underside. The rotation is described by the angular velocity vector. Suppose that the direction of rotation is such that the angular velocity vector of the red top facing away upwards. Now we consider the horizontal mirror image of this rotating disk. The vertical proportions of position and velocity vectors have been reversed, the top is now yellow and the bottom red. The direction of rotation, however, is preserved. So the angular velocity vector has not reversed by the reflection.