We discuss transormation laws of electric and magnetic fields under Lorentz transformations, deduced from the Classical Field Theory. It is found that we can connect the resulting expression for a bivector formed with those fields, with the expression deduced from the Wigner transformation rules for spin-1 functions of massive particles. This mass parameter should be interpreted because the constancy of speed of light forbids the existence of the photon mass.
The Lorentz transformation is derived without the postulate of the universal limiting speed, and the general Edwards transformation is obtained by using the principle of permutation invariance (covariance). It is shown that the existences of the one-way universal limiting speed (in the Lorentz transformation) and the constancy of the two-way average speed of light (in the Edwards transformation) are the necessary consequences of the principle of permutation invariance that is consistent with the postulate of relativity. The connection between the Edward transformation and the general coordinat…
In this paper it is exactly proved by using the Clifford algebra formalism that the standard transformations of the three-dimensional (3D) vectors of the electric and magnetic fields E and B are not the Lorentz transformations of well-defined quantities from the 4D spacetime but the 'apparent' transformations of the 3D quantities. Thence the usual Maxwell equations with the 3D E and B are not in agreement with special relativity. The 1-vectors E and B, as well-defined 4D quantities, are introduced instead of ill-defined 3D E and B.
The Lorentz Transformation is derived from only three simple postulates: (i) a weak kinematical form of the Special Relativity Principle that requires the equivalence of reciprocal space-time measurements by two different inertial observers; (ii) Uniqueness, that is the condition that the Lorentz Transformation should be a single valued function of its arguments; (iii) Spatial Isotropy. It is also shown that to derive the Lorentz Transformation for space-time points lying along a common axis, parallel to the relative velocity direction, of two inertial frames, postulates (i) and (ii) are suffi…
The article shows how the factorization of an arbitrary Lorentz transformation is performed. That is, representation of an arbitrary Lorentz transformation as a sequence of spatial rotation and boost or boost and spatial rotation. Relations are obtained that determine the required boosts and turns.
Electric and magnetic Hertz potentials are a formalism for obtaining solutions of Maxwell's equations from solutions of the inhomogeneous wave equation, with polarisation and magnetisation as the sources. We provide an overview of their covariant transformation properties, and examine an application of Hertz potentials to the Lorentz transformation of localised pulses by Lekner, who obtained a result seeming to contradict Von Laue's theorem. We show that Lekner's result of total energy-momentum not transforming as a four-vector was due to an erroneous transformation of Hertz potentials, that d…
It is demonstrated how the right hand sides of the Lorentz Transformation equations may be written, in a Lorentz invariant manner, as 4--vector scalar products. The formalism is shown to provide a short derivation, in which the 4--vector electromagnetic potential plays a crucial role, of the Lorentz force law of classical electrodynamics, and the conventional definition of the magnetic field in terms spatial derivatives of the 4--vector potential. The time component of the relativistic generalisation of the Lorentz force law is discussed. An important physical distinction between the space-tim…
The concept of an observer and their associated rest space is defined in a pre-metric (i.e., projective-geometric) context that relates to time+space decompositions of the tangent bundle to space-time. The transformation from one observer to another when the two are in a state of relative motion is then defined, and its relationship to the Lorentz transformation is discussed. The group of all linear transformations that preserve the observer quadric, which generalizes the proper-time hyperboloid in Minkowski space, is defined and the reductions to some of its subgroups are described, as well a…
Using the standard formalism of Lorentz transformation of the special theory of relativity, we derive the exact expression of the Thomas precession rate for an electron in a classical circular orbit around the nucleus of a hydrogen-like atom.
We derive rotation free Lorentz Transformation (LT) between two inertial reference frames without using the second postulate of Einstein, i.e., we do not assume the invariant speed of light (in vacuum) under LT. We find a general transformation rule between two inertial frames where a speed, invariant under that transformation, arises naturally. This idea first came into light by Mathematician Ignatowski [1] around the year, 1910. Without loss of novelty, here we present our derivation while reviewing more recent literature in this subject.
In the present work, we will derive the expression for the change in direction of the electric field using the Lorentz transformation for inertial frame or un-accelerated frame. The acceleration also influences the observed rate of a moving clock in addition to the influence due to its velocity. So we will also derive the expression for the change in direction of the electric field using the Lorentz-type transformation (as given by Friedman) for non-inertial frame or uniformly accelerated frame.
In this paper we show how to get the Lorentz transformations from E=mc^2, the laws of conservation of energy and momentum, and the special relativity principle. To this end we first deduce the law of addition of relativistic velocities
Besides the defining space-time symmetries (homogeneity and isotropy) of inertial frames, the derivation of Lorentz transformation requires postulating the principle of relativity and the existence of a finite speed limit. In this article, we point out that the existence of a finite speed limit can be readily inferred from the nature of allowed inertial frames. We also show that the principle of relativity can be obtained from the defining space-time symmetries of every inertial frame. Therefore, if the conventional definition of inertial frames is augmented properly, the special theory of rel…
The standard classic special relativistic transformation of the electromagnetic (EM) field under proper Lorentz transformations is revisited. As to the pure Lorentz-boosts, popular treatments on EM transformation contemplate ideal geometries generating special static charge and steady current distributions and in conjunction, invoke parallel and perpendicular (to the boost-velocity) components of the fields so engendered; the outcomes subsequently being suitably generalized. We demonstrate ab initio, the EM field transformation under arbitrary oblique uniform relative boost velocities from a b…
In special relativity, spacetime algebra (STA) provides a powerful and insightful approach to an invariant formulation of physics. However, in this geometric algebra of spacetime, relativistic physics is usually considered a misnomer: STA provides us with an invariant (coordinate-free) formulation and hence it seems that (passive) Lorentz transformations are misplaced. However, in order to interpret experimental results, it is important to relate invariant physical quantities to some reference frame. In this paper, passive Lorentz transformations are derived using STA but without the second po…
A new relativistic transformation in the velocity space (here named the differential Lorentz transformation) is formulated solely from the principle of relativity and the invariance of the speed of light. The differential Lorentz transformation is via transforming physical quantities, instead of space-time coordinates, to make laws of nature form-invariant. The differential Lorentz transformation may provide a way to resolve the incompatibility of the theory of special relativity and the quantum theory.
It is demonstrated how the right hand sides of the Lorentz Transformation equations may be written, in a Lorentz invariant manner, as 4--vector scalar products. This implies the existence of invariant length intervals analogous to invariant proper time intervals. This formalism, making essential use of the 4-vector electromagnetic potential concept, provides a short derivation of the Lorentz force law of classical electrodynamics, the conventional definition of the magnetic field, in terms of spatial derivatives of the 4--vector potential and the Faraday-Lenz Law. An important distinction betw…
There are many papers devoted to derivation of Lorentz Transformations (LT). Many people have pro posed alternative derivations. Their analysis allows looking at LT and their consequences from different standpoints. At the same time it is important to choose the simplest and general way of derivation of LT for textbooks, to make them more accessible to a wide circle of laymen interested in the elativistic physics and to leave the W.Voight's, J.Larmor's, G.Lorentz's, A.Einstein's and other derivations for the history. A short, clear and simple derivation of generalized LT based on the Principle…
Homogeneity of space and time, spatial isotropy, principle of relativity and the existence of a finite speed limit (or its variants) are commonly believed to be the only axioms required for developing the special theory of relativity (Lorentz transformations). In this paper it is shown, however, that Lorentz transformation cannot actually be derived without the explicit assumption of time isotropy (time-reversal symmetry) which is logically independent of the other postulates of relativity. Postulating time isotropy also restores the symmetry between space and time in the postulates of relativ…
In 1887 Voigt published a paper dedicated to the Doppler effect in which he demanded form invariance to the wave equation in inertial frames and obtained a set of spacetime transformations now known as the Voigt transformations. In 1905 Poincaré showed that the wave equation was also invariant under the Lorentz transformations. Voigt and Lorentz transformations are then closely related, but this relation is not widely known in the standard literature. In this paper we derive the Lorentz transformations from the invariance of the D'Alembert operator $\big(\Box^{2}=\Box'^{2}\big)$ and the Voigt …
We use the method of field decomposition, a technique widely used in relativistic magnetohydrodynamics, to study the small velocity approximation (SVA) of the Lorentz transformation in Maxwell equations for slowly moving media. The "deformed" Maxwell equations derived under the SVA in the lab frame can be put into the conventional form of Maxwell equations in the medium's comoving frame. Our results show that the Lorentz transformation in the SVA up to $O(v/c)$ ($v$ is the speed of the medium and $c$ is the speed of light in vacuum) is essential to derive these equations: the time and charge d…
