Frame-indifference of cross products, rotations, and the permutation tensor

Abstract

ABSTRACT: Under improper transformations, the traditional transformation laws for cross products, the permutation tensor, and rotations are incorrect. For a cross product, using a counter-example the left-hand rule is proved wrong. The unique rule for a cross product is the right-hand rule. Coordinate systems have handedness, while a cross product is frame-indifference. Since the permutation tensor is defined as a triple product including a cross product, the law for the permutation tensor is false. For a rotation, its pseudovector representation is incorrect, because the mirror is an auxiliary device to produce the virtual image rather than a new coordinate system or reference frame. Keywords: Pseudovector, Cross product, Improper transformation, Handed coefficient, Permutation tensor