Search

We reveal relations between superluminal and subluminal spatiotemporally localized optical pulses and the phenomena of the relativistic aberration of a wavevector and the Doppler shift. It turns out that such waves are Lorentz-transformed versions of simple optical waves generated in another reference frame. The requirement of a null Doppler shift is shown to give rise to a speed associated with the relativistic velocity addition law of a double (two-step) Lorentz transformation. The effects of such a transformation are examined both in terms of four-coordinates and in the spectral domain. It is established that a subluminal pulse reverses its direction. In addition to a change in direction, the propagation term of a superluminal pulse becomes negative. The aberration due to a double Lorentz transformation is examined in detail for propagation invariant superluminal waves (X wave, Bessel X wave), as well as intensity-invariant superluminal and subluminal waves. Detailed symmetry considerations are provided for the superluminal focus X wave and the subluminal MacKinnon wavepacket.

A generalization of the Hubble law in the framework of the special theory of relativity received. The relativistic Doppler effect allows you to find a nonlinear relationship between redshift and distance.

We report on complementary tests and measurements regarding our recent determination of the Boltzmann constant, kB, by means of Doppler broadening thermometry, also providing additional information as compared to previous articles. A revised uncertainty budget is illustrated, including some new components that were ignored in previous spectroscopic experiments, and better quantifying other components that were estimated to be negligible. In particular, we consider the relativistic Doppler effect, the perturbation caused by the finite bandwidth of the detection system and the influence of the spontaneous emission content of the probe laser. These new components do not increase the global uncertainty which still amounts to 24 ppm. Our value for the Boltzmann constant is 1.380 631 (33) × 10−23 J K−1, which is the best determination reported so far by using an optical method.

In the past few years, the perturbation due to the finite speed of light was among the most inconsistent in corner-cube absolute gravimeters. For the relativistic treatment of the perturbation based on Lorentz transformation, the relation between Doppler shift in special relativity and time delay in classical domain is easily misunderstood, leading to spurious conclusions. To avoid these issues, we apply a 'relativistic geometrical method' based on the motion of photons to calculate the frequency shift in corner-cube absolute gravimeters, where the misunderstood relation has been corrected. Additionally, we find that the modern corner-cube absolute gravimeters cannot sense effects of the special relativity.

The electromagnetic field is determined for a time-varying electric dipole moving with a constant velocity that is parallel to its moment. Graphics are used to visualize this field in the rest frame of the dipole and in the laboratory frame when the dipole is moving at relativistic speed. Various phenomena from special relativity are clearly illustrated by these graphics and explained with simple calculations; these include the constancy of the speed of light in inertial frames, the Doppler effect, the headlight effect, and the concentration of field lines. In addition, the energy and linear momentum of the radiated field are determined and shown to satisfy the transformation and invariance required by special relativity.

It has been almost 100 years since Einstein formulated his special theory of relativity in 1905. He showed that the basic space–time symmetry is dictated by the Lorentz group. It is shown that this group of Lorentz transformations is not only applicable to special relativity, but also constitutes the scientific language for optical sciences. It is noted that coherent and squeezed states of light are representations of the Lorentz group. The Lorentz group is also the basic underlying language for classical ray optics, including polarization optics, interferometers, the Poincar´ es phere, one-lens optics, multi-lens optics, laser cavities, as well multilayer optics.

For the relativistic plasma, how to fix the Lorentz factors of the particles is an important but difficult problem. We resolve this problem by demonstrating the exact relation between the average Lorentz factor and temperature in relativistic plasmas. A rather simple relation is also obtained for the ultra-relativistic case.

Since the Lorentz boost is symmetric under exchange of x and ct, special relativistic phenomena will also manifest this symmetry. Firstly, simultaneity becomes paired with 'colocality' ('at the same place'), and the 'Relativity of Colocality' becomes the dual to the well-known 'Relativity of Simultaneity'. Further, Lorentz time contraction arises from reversal of the observation conditions pertaining to time dilation, expressible figuratively as 'Moving clocks run slow, but moving time runs fast'. Symmetry also dictates that the most fundamental observational modes are: (1) the simultaneous observation of length, a process involving both the relativity of simultaneity and length contraction; and (2) the colocal measurement of duration, involving both the relativity of colocality and time contraction. Only the first of these modes is well known. The adoption of this symmetrical lexicon provides a necessary logical basis for interpretational studies of observation and measurement in special relativity.

The Lorentz transformation equations for emission rates are calculated from the equations for the time dilation, the Doppler effect and aberration. Radiation from accelerated charges and from diffuse surfaces in motion is considered. The theory is applied to a model of an expanding star, and the modification of its apparent magnitude due to the expanding motion is calculated.

In a paper (Zanchini E 2012 Phys. Scr. 86 015004), the two experiments on the transverse Doppler shift in the rotating accelerating systems performed by Hay et al and Kundig are reinterpreted. Contrary to the widely accepted view, the author concluded that this transverse Doppler effect is not purely a relativistic effect. We argue that his interpretation is based on a fundamental misunderstanding of the valid scope for the expressions of the relativistic Doppler effect. Here, we also clarify that the expressions of the relativistic Doppler effect are strictly valid only when dealing with a plane light wave propagating along an arbitrary direction, or with a spherical light wave produced by a point light source in motion along the direction of the line connecting the source and the observer.

The conceptually simple problem of plane electromagnetic (EM) waves in moving media is used to illustrate differences between the conventional Lorentz transformation of electrodynamics and the new relativistic transformation of electrodynamics, which has been recently proposed. The problem is first examined in accordance with the conventional Lorentz transformation. Peculiarities are pointed out for the case where plane EM waves propagate in a medium which moves with ‘superluminal’ speeds against the waves. By contrast, those found peculiarities are resolved by the new relativistic transformation. Reasons why the usual Lorentz transformation of electrodynamics does not fulfill the principle of relativity are given. The conventional interpretation of the principle of relativity should be examined critically.

The general relativistic dispersion relations of transverse waves parallel propagating in magnetized Vlasov plasmas, are reconsidered. In a previous paper (Lazar and Schlickeiser 2003 Can. J. Phys. 81 1377) we have derived these equations by a rigorous relativistic kinetic theory and by a correct analytical continuation for all complex frequencies. Detailed expressions are more explicitly presented here for residues to completely describe the damped waves ? forward or backward moving waves, and left-handed or right-handed polarized waves. The cyclotron damping occurs when the circularly-polarized wave frequency nearly coincides with the gyrofrequency of plasma particle, modified by the relativistic Doppler effect in its own referential. The rigorous analysis of relativistically correct dispersion equations leads to an important correction of nonrelativistic theory: the superluminal EM waves undergo no cyclotron damping in magnetized equilibrium plasmas.

It is shown that the Lorentz transformation for the time coordinate is the sum of two times, the first being associated with the time dilation effect, the second with the relativity of the 'time of flight' of the photon, from the source to the receiver. The Lorentz transformation for the x coordinate is the sum of two terms, the first expressing a shift in the positive direction of the common axes, whereas the second represents the 'photographed' x coordinate of the source of light as it appears on a snapshot taken by an observer O1, at a moment when observer O' is in front of him.

Red-shifted or blue-shifted frequencies of lights are shown in geometric proportions to the speed of light and of the light source.

A unified treatment of the Doppler effect both classically and relativistically in four-dimensional space is presented. The treatment is based on treating the displacement of a one-dimensional wave as a function of position and time, and then taking the wave's phase line as its world line.

The well known non-relativistic Doppler effect is discussed from a more general point of view where the positions of both the wave source and the receiver are arbitrary continuous functions of time. The basic relation between the frequency of the emitted wave (also an arbitrary continuous function of time) and the frequency of the received wave is found in an unexpectedly simple way. Two examples are examined.

A geometrical description of the composition law for pure inertial transformations in relativity is developed, in close analogy with the description of rotations by means of turns. This affords a very intuitive and visual image of all problems involving a composition of velocities, and clearly shows the appearance of hyperbolic geometry in relativistic kinematics. The pertinent geometrical constructions as well as some applications are discussed. For the case of light the preceding constructions are surprisingly simple, and from them emerges a connection between the geometrical idea of parallelism angle and the physical Doppler effect. A slide rule which can be used to carry out all needed computations is also described.

It is illustrated how relativistic mechanics can be developed from experiments without introducing the Lorentz transformations. From the experimental result that the limiting speed of accelerated electrons is the speed of light, it is postulated that the numerical value of the limiting speed of accelerated particles has the same numerical value in all inertial reference frames. Together with the principle of relativity this postulate can be used to develop the Lorentz transformations.

Suggests an effect of the one-way velocity of light which can be detected by observing the angle of stellar aberration carefully.

The author attempts to show in an explicit form the dynamical origin of the change in energy of Doppler-shifted photons. That is, it shows, from a consideration of the dynamical motion of a source atom, that an emitted photon suffers an increase or decrease of energy, relative to that of a photon from a stationary source, exactly equal to the change given by the Doppler effect. The calculations are fully relativistic but are conveniently limited to observation of the photons in the line of motion of the source. Demonstration of the dynamical origin of the change will probably make it more real for some students.

Einstein's first deduction of the inertia-energy relationship is modified so that an isotropic light signal is considered instead of two rays of light. In this modified Gedanken experiment aberration must explicitly be taken account of. If aberration is neglected, one does not obtain the correct Lorentz transformation for the energy of the light signal.