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Introductory textbooks for the theory of special relativity discuss the time dilation effect after introducing the Lorentz transformations. Reading the dictum 'moving clocks run slow' a student is likely to have the impression that this is a measured effect. On the other hand, after considering the relativistic Doppler effect the same student might be puzzled that this effect depends on the relative emitter-observer direction, contrary to Einstein's time dilation. Simple thought experiments are presented here where the inferred time dilation and the observed relativistic Doppler effect give quite opposite answers. Before solving the puzzle, we show how easily one can get confused and obtain apparently contradictory results. [ABSTRACT FROM AUTHOR]

It is commonly accepted that a rod after having been accelerated and then left free contracts by the Lorentz factor , and also that a clock after having been accelerated and then returned to rest exhibits time dilation in accordance with the proper time formula . Although both issues have been debated numerous times, no models have been proposed to trace the dynamical nature of these phenomena. The purpose of the present paper is to propose a simple relativistic model of an elastic rod, which allows exact analytical solutions for a number of problems. The solutions of two problems are presented: (i) acceleration of the rod by a constant force applied at one of its ends during a finite time interval, resulting in the Lorentz contraction, (ii) acceleration and deceleration of a light clock, resulting in time dilation. [ABSTRACT FROM AUTHOR]

Well-developed tools from cartography are adapted and used to visualize curved and flat spacetime to further understanding of relativistic effects, while interpretation of these tools is aided through their application to the familiar world map. First spacetime is cut into two-dimensional subspaces that are mapped to the Minkowski or Euclidean plane using map projections. Then, exploiting the cartographic technique of map interruptions, the two-dimensional subspaces are further cut into 'wedges' to minimize the distortion. The mental exercise of sewing the wedges back together aids in visualizing and interpreting the geometry of the spacetime slice. Map distortion indicators alert the reader to the inevitable effects of distortion. This wedge diagram provides a simple yet powerful tool to visualize both flat and curved spacetime geometry. It is more general than the familiar embedding diagram and avoids the pitfall of confusing the space in question with the embedding space. The wedge diagram is applied first to Minkowski spacetime with the cartographers' azimuthal equidistant map projection and, when combined with the wedge technique, provides a simple visualization of the twin paradox. The wedge diagram is then applied to the exterior Schwarzschild spacetime to visualize the physical significance of spacetime curvature. In particular it is shown how gravitational time dilation results from intrinsic curvature of a radial-time slice of spacetime. Then this diagram is applied to the Friedmann–Robertson–Walker (FRW) spacetime revealing how expansion of the Universe results from a qualitatively similar curvature of a radial-time slice, but with the roles of space and time reversed. The analogy between the two spacetimes suggests novel interpretations of the two spacetimes. We interpret the Schwarzschild spacetime as time expanding with increasing distance from the horizon. And we interpret the FRW spacetime as the Universe expanding as the influence of the big bang peters out. [ABSTRACT FROM AUTHOR]

We consider gravitational time dilation between stationary observers and present examples which contradict the statement that ‘time slows down as gravity increases’. We furthermore show that this statement cannot be true in general, if gravity strength is defined in an observer independent manner. We also provide a pedagogical introduction to gravitational time dilation, and discuss aspects of this phenomenon, which are often omitted in textbooks on general relativity. [ABSTRACT FROM AUTHOR]

In 1907, Einstein suggested an experiment with flying atoms for corroborating time dilation. In that paper, the flying atom was conceived as a flying clock: the reference to the Doppler effect was only indirect (the experiments by Stark to the first order of ν/c). In 1922, Schrödinger showed that the emission of a light quantum by a (flying) atom is regulated by the conservation laws of energy and linear momentum. Therefore, the Doppler effect for photons is the consequence of the energy and momentum exchange between the atom and the photon: a central role is played by the quantum energy jump ΔE of the transition (a relativistic invariant). The first realization of the experiment devised by Einstein was by Ives and Stilwell (1938). Since then and until now, experiments of this kind have been repeated in search of better precision and/or a deviation from the predictions of special relativity. The striking feature is that all the papers dealing with these experiments completely neglect Schrödinger's dynamical treatment. There are different origins of this omission: pragmatic (agreement between formulae, wherever coming from, and experiments), historical (deep rooting of the wave theory of light), and epistemological (neglect of basic epistemological rules). [ABSTRACT FROM AUTHOR]

The design of the FMEL, a floor-mirrored Einstein-Langevin 'light clock', is introduced. The clock provides a physically intuitive manner to calculate and visualize the time dilation effects for a spatially extended set of observers (an accelerated 'frame') undergoing unidirectional acceleration or observers on a rotating cylinder of constant radius. In both cases, the rate of the clock agrees with the 'clock hypothesis' and approximates an 'ideal' clock. Inertial and accelerated observers will agree on the length of standard metre sticks that are parallel to the primary mirrors of the clock and they will agree on the simultaneity of events that occur along that line. It is found that the path of the light in this clock is not a least-time path between the mirrors, but it is the least-time path for the motion of objects restricted to the accelerated frame. It is possible to send signals between observers in the accelerated frame at speeds greater than the speed of light in their frame, but this does not imply a violation of causality. [ABSTRACT FROM AUTHOR]

The aim of this study is to investigate pre-service teachers' understanding of and difficulties with some core concepts in the special theory of relativity. The pre-service teachers (n = 185) from the Departments of Physics Education and Elementary Science Education at Dokuz Eylul University (in Turkey) participated. Both quantitative and qualitative research methods were used in this study. Students' understanding of and difficulties with core elements (time, length, mass and density) were tested using a paper-and-pencil questionnaire (including four questions) and in-depth interviews after the instruction of related modern physics topics. The analyses of the collected data were based on quantitative and qualitative techniques. The results indicate that pre-service teachers at different academic levels have specific and considerable difficulties with proper time, time dilation, proper length, mass and relativistic density concepts. In this paper, the conclusions of the study and implications for physics teaching are discussed. [ABSTRACT FROM AUTHOR]

The apparent length of a relativistically moving rod and visualization of the Einstein time dilation are studied. The description of events in space-time is given geometrically as viewed from two different inertial frames that are in relative motion with one another. These relativistic phenomena are graphically illustrated using one-dimensional Loedel diagrams. The equations that describe these effects are then deduced using elementary geometry. The aim of this paper is to outline a procedure that complements the standard algebraic presentation of these effects, and show that the kinematic concepts of special relativity can be deduced from the geometry. [ABSTRACT FROM AUTHOR]

The issue of the condition for simultaneous arrival of light signals emitted simultaneously from the opposite ends of a rigidly accelerating spaceship, raised in Ben-Ya’acov (2016 Eur. J. Phys.37 055601), is resolved. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Relativistic rigid motion suggests a new version for the so-called ‘twin paradox’, comparing the ages of two astronauts on a very long spaceship. Although there is always an instantaneous inertial frame in which the whole spaceship, being rigid, is simultaneously at rest, the twins’ ages, measured as the proper-times along their individual world lines, are different when they are located at remote parts of the spaceship. The age, or proper-time, difference depends on the distance at rest between the astronauts and the rapidity difference between start to end. The relation of the age difference with the relative Doppler shift of light signals transmitted between the astronauts and implications for the possibility to assign a common age (proper-time) to complex, spatially extended, relativistic systems are also discussed. [ABSTRACT FROM AUTHOR]

The gravitational redshift forms the central part of the majority of the classical tests for the general theory of relativity. It could be successfully checked even in laboratory experiments on the earth's surface. The standard derivation of this effect is based on the distortion of the local structure of spacetime induced by large masses. The resulting gravitational time dilation near these masses gives rise to a frequency change of any periodic process, including electromagnetic oscillations as the wave propagates across the gravitational field. This phenomenon can be tackled with classical electrodynamics assuming a curved spacetime background and Maxwell's equations in a generally covariant form. In this paper, we show that in a classical field theoretical context the gravitational redshift can be interpreted as the propagation of electromagnetic waves in a medium with corresponding conductivity σ = g/(μ0c³), where g is the gravitational acceleration and μ0 is the vacuum magnetic permeability. Moreover, the energy density of the wave remains proportional to its frequency in agreement with Planck's postulate. [ABSTRACT FROM AUTHOR]

We treat, as an illustrative example of gravitational time dilation in relativity, the observation that the centre of the Earth is younger than the surface by an appreciable amount. Richard Feynman first made this insightful point and presented an estimate of the size of the effect in a talk; a transcription was later published in which the time difference is quoted as ‘one or two days’. However, a back-of-the-envelope calculation shows that the result is in fact a few years. In this paper we present this estimate alongside a more elaborate analysis yielding a difference of two and a half years. The aim is to provide a fairly complete solution to the relativity of the ‘aging’ of an object due to differences in the gravitational potential. This solution—accessible at the undergraduate level—can be used for educational purposes, as an example in the classroom. Finally, we also briefly discuss why exchanging ‘years’ for ‘days’—which in retrospect is a quite simple, but significant, mistake—has been repeated seemingly uncritically, albeit in a few cases only. The pedagogical value of this discussion is to show students that any number or observation, no matter who brought it forward, must be critically examined. [ABSTRACT FROM AUTHOR]

Dynamics and radiation of a relativistic charged particle moving in a linear restoring force field is studied. Solutions to the equations of motion are presented in a compact form in terms of Jacobi elliptic functions. The intensity of radiation, its angular distribution and radiation spectrum of a relativistic oscillator are investigated. It is shown that, unlike a non-relativistic oscillator, the maximum intensity is emitted not at the turning points, but at some intermediate points of the particle’s trajectory. The emission spectrum consists of lines at frequencies that are multiples of the oscillation frequency of the oscillator. As the oscillator energy increases, the number of harmonics in the spectrum increases. The maximum in the spectrum of highly relativistic oscillator occurs on high numbers of harmonics. [ABSTRACT FROM AUTHOR]

In this paper we review the derivation of the light bending equation, obtained before the discovery of general relativity (GR). This is intended for students learning about GR, and for specialists who will shed new light on, and make new connections between, these historic derivations. Since 1915 it has been well known that the observed bending of light has two contributory factors: the first is directly deduced from the equivalence principle alone and was obtained by Einstein in 1911; the second comes from the spatial curvature of spacetime. In GR, these two components are equal, but other relativistic theories of gravitation can give different values to those contributions. In this paper we give a simple explanation, based on the wave-particle picture, of why the first term, which relies on the equivalence principle, is identical to the one obtained by a purely Newtonian analysis. In this context of wave analysis, we emphasize that the dependency of the velocity of light on the gravitational potential, as deduced by Einstein, concerns the phase velocity. Then we wonder whether Einstein could have envisaged already, in 1911, the second contribution, and therefore the correct result. We argue that considering a length contraction in the radial direction, along with the time dilation implied by the equivalence principle, could have led Einstein to the correct result. [ABSTRACT FROM AUTHOR]

Interactive visualization is a valuable tool for introductory or advanced courses in general relativity as well as for public outreach to provide a deeper understanding of the visual implications due to curved spacetime. In particular, the extreme case of a black hole where the curvature becomes so strong that even light cannot escape, benefits from an interactive visualization where students can investigate the distortion effects by moving objects around. However, the most commonly used technique of four-dimensional general-relativistic ray tracing is still too slow for interactive frame rates. Therefore, we propose an efficient and adaptive polygon rendering method that takes light deflection and light travel time into account. An additional advantage of this method is that it provides a natural demonstration of how multiple images occur and how light travel time affects them. Finally, we present our method using three example scenes: a triangle passing behind a black hole, a sphere orbiting a black hole and an accretion disk with different inclination angles. [ABSTRACT FROM AUTHOR]

Thought experiments involving a light clock are common in introductory treatments of special relativity, because they provide a simple way of demonstrating the non-intuitive phenomenon of time dilation. The properties of the ray or pulse of light that is continuously reflected between the parallel mirrors of the clock are often stated vaguely and sometimes involve implicitly other relativistic effects, such as aberration. While this approach is adequate for an introduction, it should be supplemented by a more accurate analysis of the light clock once the formulae for the Lorentz transformation and the transformation of the electromagnetic field have been developed. A simple yet accurate electromagnetic model for the light clock is presented for this purpose. In this model, the ray of light in the qualitative treatment is replaced by a guided wave in a parallel-plate waveguide. Expressions for the electromagnetic field and energy density within the waveguide are determined in the inertial frame in which the clock is at rest and the laboratory frame in which the clock is moving with constant velocity. The analytical expressions and graphical results obtained clearly demonstrate the operation of the clock and time dilation, as well as other interesting relativistic effects. (Contains 5 figures.)

The null result of the Michelson-Morley experiment and the constancy of the one-way speed of light in the 'rest system' are used to formulate a simple problem, to be solved by elementary geometry techniques using a pair of compasses and non-graduated rulers. The solution consists of a drawing allowing a direct visualization of all the fundamental effects of standard relativistic kinematics, namely time dilation, length contraction and relativity of simultaneity. Moreover, it also provides an immediate image of other important and more subtle aspects, often passed by in relativity courses, such as the conventionality of simultaneity thesis, possible non-invariance of the one-way speed of light and compatibility between the Lorentz-Poincaré and Einstein-Minkowski philosophies. The geometric scheme so constructed constitutes a powerful tool to clearly illustrate both traditional and not-so-traditional aspects of special relativity teaching. [ABSTRACT FROM AUTHOR]

This work ends a trilogy devoted to a journey into the foundations of special relativity. The first paper debated the meaning of the constancy of the two-way speed of light and its close relation to the conceptualization of time. The second one addressed the question of the possible constancy of the one-way speed of light and the trivial--but, unfortunately, even now somewhat controversial--question of the compatibility between the assumption of a special system of reference and Einstein's special relativity. The present study deals with the principle of relativity. Its historical evolution is reviewed and a 'weak' formulation is defended. It is emphasized that many assertions usually associated with special relativity, such as the 'relativity of time dilation' and 'relativity of space contraction' are indeed philosophical statements, as it has been established already by several authors in the past. Nonetheless, most teachers and scientists still believe nowadays they are implied by the theory and by the group property of the Lorentz transformation. This is by no means so, as it is reviewed and elucidated with the simple example on space contraction. It is argued that the lack of knowledge of the true value of the one-way speed of light in empty space leaves the theory undetermined. Einstein's special relativity corresponds to a simple and very elegant solution to this problem, allowing the study of relative motion without any concern with the study of absolute motion, which is considered to be superfluous. However, its standard interpretation is minimalist and even misleading. A large number of researchers have discussed this question, mostly within the conventionality of simultaneity thesis. The typical formulation of this thesis provides some new physical insight and points out the problem, but does not solve it. In contrast, it often leads to a labyrinth of difficult language which is herein clarified. [ABSTRACT FROM AUTHOR]

We present an in depth analysis and a new derivation of the Doppler factor in the Liénard–Wiechert (LW) potentials, based on geometrical considerations in Minkowski space. We argue that, contrary to a common assumption, the methods used for deriving the Doppler factor in the case of an electrically charged extended particle are not applicable in the case of a point particle. A geometrical interpretation of the Doppler factor is nonetheless found in the later case, based on the electromagnetic interaction model of Fokker. In this model the interaction takes place between infinitesimal worldline segments with end points connected by light signals, just like the points where light pulses are emitted and detected in general. This analogy reveals that the relativistic Doppler effect is the missing link between the classical Doppler effect and the Doppler factor in the LW potentials. [ABSTRACT FROM AUTHOR]

The classic experiment on muon decay is proposed in the undergraduate or master physics curriculum, making use of a single scintillation detector and observing in coincidence the delayed signal due to the electron following muon decay. Two experimental setups, with different geometries and scintillation materials, were considered, making use of standard equipment available in most undergraduate labs. The results obtained in the two cases were compared and discussed. For a better understanding of the results, and to also evaluate the possible effect of material surrounding the detector, detailed GEANT simulations were carried out for both experimental setups. Such simulations are also intended as an additional student activity to virtually exploit in a short time the effect of modifying detector and material configuration, which cannot be achieved during a single student session. The results of this experiment and its interpretation also proved to be a useful way to discuss with students many topics of interest in modern physics, such as—among others—the time dilation, the dependence of muon lifetime in matter, the Fermi weak interaction coupling constant and the parity violation in weak interactions. [ABSTRACT FROM AUTHOR]

Using geometric algebra and calculus to express the laws of electromagnetism we are able to present magnitudes and relations in a gradual way, escalating the number of dimensions. In the one-dimensional case, charge and current densities, the electric field E and the scalar and vector potentials get a geometric interpretation in spacetime diagrams. The geometric vector derivative applied to these magnitudes yields simple expressions leading to concepts like divergence, displacement current, continuity and gauge or retarded time, with a clear geometric meaning. As the geometric vector derivative is invertible, we introduce simple Green's functions and, with this, it is possible to obtain retarded Liénard–Wiechert potentials propagating naturally at the speed of light. In two dimensions, these magnitudes become more complex, and a magnetic field B appears as a pseudoscalar which was absent in the one-dimensional world. The laws of induction reflect the relations between E and B, and it is possible to arrive to the concepts of capacitor, electric circuit and Poynting vector, explaining the flow of energy. The solutions to the wave equations in this two-dimensional scenario uncover now the propagation of physical effects at the speed of light. This anticipates the same results in the real three-dimensional world, but endowed in this case with a nature which is totally absent in one or three dimensions. Electromagnetic waves propagating exclusively at the speed of light in vacuum can thus be viewed as a consequence of living in a world with an odd number of spatial dimensions. Finally, in the real three-dimensional world the same set of simple multivector differential expressions encode the fundamental laws and concepts of electromagnetism. [ABSTRACT FROM AUTHOR]

The article 'Comment on "A note on Purcell's basic explanation of magnetic forces"' (2020 Eur. J. Phys. 41 068001) disputes the validity of the results presented in D'Abramo (2020 Eur. J. Phys. 41 025203). In this reply, we provide several arguments against that criticism and confirm the original conclusions. [ABSTRACT FROM AUTHOR]

We derive here, from first principles, the energy–momentum density of a perfect fluid in the form of an ideal molecular gas, in an inertial frame where the fluid possesses a bulk motion. We begin from the simple expressions for the energy density and pressure of a perfect fluid in the rest frame of the fluid, where the fluid constituents (gas molecules) may possess a random motion, but no bulk motion. From a Lorentz transformation of the velocity vectors of the molecules moving in different directions in the rest frame of the fluid, we compute their energy–momentum vectors and number densities in an inertial frame moving with respect to the rest frame of the liquid. From that, we arrive at the energy–momentum density of the fluid in a frame where it has a bulk motion. This way, we explicitly demonstrate how a couple of curious pressure-dependent terms make an appearance in the energy–momentum density of a perfect fluid having a bulk motion. In addition to an ideal molecular gas, we compute the energy–momentum density for a photon gas also, which of course matches the energy–momentum density expression obtained for a molecular gas having an ultra-relativistic random motion. [ABSTRACT FROM AUTHOR]

The launch in 2019 of the new international system of units is an opportunity to highlight the key role that the fundamental laws of physics and chemistry play in our lives and in all the processes of basic research, industry and commerce. The main objective of these notes is to present the new SI in an accessible way for a wide audience. After reviewing the fundamental constants of nature and its universal laws, the new definitions of SI units are presented using, as a unifying principle, the discrete nature of energy, matter and information in these universal laws. The new SI system is here to stay: although the experimental realizations may change due to technological improvements, the definitions will remain unaffected. Quantum metrology is expected to be one of the driving forces to achieve new quantum technologies of the second generation. [ABSTRACT FROM AUTHOR]

Our goal is to present a derivation of Lorentz's transformations from gedanken experiments with electromagnetic forces without having to assume Einstein's postulate of the constancy of the speed of light. Instead, we propose a different second postulate, namely that electric charge is independent of the state of motion of the system, and we restrict the first postulate in order to state the relativity of electromagnetic forces only. We argue that assuming that charge is invariant is much more intuitive than assuming that light speed is constant. In this derivation, furthermore, the constancy of a limit speed c is obtained as a result of the velocity addition rule. Finally, we discuss the epistemological and didactic implications of this formulation. [ABSTRACT FROM AUTHOR]