Classical Electrodynamics, 3rd ed.

Doppler Shift from a Composition of Boosts with Thomas Rotation: a Spacetime Algebra Approach

“…and hence (a ∧ b) 2 ∈ R. Now we should note that Minkowski spacetime has a nondefinite metric. Otherwise we would get strange results that are strongly unphysical [22].…”

“…There are three different observers: (i) an observer in frame S (the laboratory) with proper velocity s = ce 0 ; (ii) an observer in frame S 1 (where antenna E 1 is located) with proper velocity u 1 = cf 0 ; (iii) an observer in frame S 2 (where antennas E 2 and R 2 are located) with proper velocity u 2 = cg 0 . Observers u 1 and u 2 are moving away from observer s with known relative velocities v 1 and v 2 , according to (2). This is a problem of relativistic composition of non-parallel relative velocities and was already solved in [21]: one has to determine v = βcv in terms of v 1 = β 1 cv 1 and v 2 = β 2 cv 2 .…”

“…Any good textbook on classical electrodynamics treats the general Doppler shift from the relativistic point of view as the transverse Doppler shift is a purely relativistic effect [1][2][3]. From the physical perspective, the Doppler shift can be geometrically grasped using a simple formalism introduced by Herman Bondi, called the K-calculus [4].…”

Retardation in Special Relativity and the Design of a Relativistic Motor

“…This law has numerous experimental verifications and seems to be one of the cornerstones of physics. However, by the middle of the nineteenth century, Maxwell had formulated the laws of electromagnetism in his famous four partial differential equations [3][4][5] which were formulated in their current form by Oliver Heaviside [6]. One of the consequences of these equations is that an electromagnetic signal cannot travel at speeds exceeding that of light.…”

“…One of the consequences of these equations is that an electromagnetic signal cannot travel at speeds exceeding that of light. This was later used by Albert Einstein [4,5,7] (among other things) to formulate his special theory of relativity which postulates that the speed of light is the maximal allowed velocity in nature. According to the principles of relativity, no signal (even if not electromagnetic) can propagate at superluminal velocities.…”