Subtleties of invariance, covariance and observer independence

Abstract

The role of the observers is frequently obscured in the literature, either by writing equations in a coordinate system implicitly pertaining to some specific observer or by entangling the invariance and the observer dependence of physical quantities. Using examples in relativistic kinematics and classical electrodynamics, we clarify the confusion underlying these misconceptions.

Nature of Electric and Magnetic Fields; How the Fields Transform

“…The recent example is in [17]; it is only the electric part (the magnetic part is zero there). Similarly, in the component form these relations are presented, e.g., in [18] and in the basis-free form with the abstract 4D quantities in [7,8,15] and in, e.g., [19]. But, it has to be noted that from all of them only Oziewicz, see [11] and references to his papers in it, exclusively deals with the abstract, basis-free 4D quantities.…”

“…KS [8] use the tensor formalism with the abstract index notation, but in section 3 in [8] they made almost the same mistakes as in BT [7]. In section 7.2 similar shortcomings in the treatment of the angular momentums and spin that are made in section 4 in [8] are discussed and objected. In section 8, the mathematically correct definitions with the 4D GQs of the orbital angular momentums and spins are discussed.…”

“…They have not noticed that under the LT the electric field 4D vector must transform as any other 4D vector transforms. In section 7.1, it is discussed the derivation of the AT from the paper by Klajn and Smolić (KS) [8]. KS [8] use the tensor formalism with the abstract index notation, but in section 3 in [8] they made almost the same mistakes as in BT [7].…”

“…In section 7.1, it is discussed the derivation of the AT from the paper by Klajn and Smolić (KS) [8]. KS [8] use the tensor formalism with the abstract index notation, but in section 3 in [8] they made almost the same mistakes as in BT [7]. In section 7.2 similar shortcomings in the treatment of the angular momentums and spin that are made in section 4 in [8] are discussed and objected.…”

Does three-dimensional electromagnetic field inherit the spacetime symmetries?

“…Since by assumption p / ∈ W it follows from here that α = 0 and thus £ ξ E a = 0. Note, however, that symmetry inheritance £ ξ F ab = 0 contradicts the equation (23) under the assumption F ab = 0, unless µ = 0. Again, boundary points of the set where B = 0 are covered by the argument of continuity.…”