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We demonstrate that a de Broglie wave of a particle in a gravitational field turns towards the region of a smaller gravitational potential, causing the particle to fall. This turning is caused by clocks running slower in the smaller potential. We use the analogy of ocean waves that are slower in shallower water and turn towards beaches. This approach explains the free fall qualitatively and quantitatively without postulating motion along geodesics and with only elementary algebra., Comment: Added Reference 20, W. K. Wootters, Why Things Fall, Foundations of Physics 33, 1549 (2003)

According to Einstein’s general theory of relativity, a clock runs more slowly if it is close to a large gravitating object. This principle was highlighted in the movie “Interstellar,” in which the main character spends several hours on a planet orbiting a massive black hole, and returns to find that his young daughter has become an elderly woman. In the movie, one hour on the planet equates to seven years on Earth. The same effect can be measured on a much smaller scale by changing locations within Earth’s gravitational field. Using atomic clocks, we have compared the ticking of time at Colorado College (elevation 1845 m above sea level), at the United States Air Force Academy (elevation 2165 m), and near the summit of Pikes Peak (elevation 4288 m). Our results confirm that clocks do indeed tick more quickly at higher altitudes, in accordance with Einstein’s theory. This work was carried out as a collaboration of physics majors and faculty at Colorado College (CC) and the United States Air Force Academy (USAFA). The experimental technique is straightforward and our results can be reproduced by other students and teachers, given access to an atomic clock and reasonably large changes in elevation. Indeed, a similar endeavor has been reported by amateur clock enthusiasts.

Smartphones and tablets are packed with sensors that allow us to take experimental data, essentially making them mobile physics labs. Apps exist that make it easy to capture and analyze data from these sensors, allowing users to study diverse phenomena such as free fall acceleration, the speed of sound, radioactivity, and many others. Commonly, the use of apps in the physics classroom focuses on existing apps rather than on the creation of apps themselves. This is for a good reason-writing an app usually requires advanced knowledge of programming languages and experience with app design. These skills are generally regarded as beyond the typical physics student. However, there are app development environments, such as MIT App Inventor 2 for Android, which make the process of creating an app and utilizing the sensors much more accessible to students. Putting the ability to create and configure apps within the reach of students can also help integrate computation into the curriculum. In this article, we discuss our creation of an Android app, Time Dilation Calculator, done as part of a senior undergraduate capstone project, which uses App Inventor and a mobile device's global positioning system (GPS) receiver to calculate the time dilation effect of special relativity.

Undergraduate physics majors typically begin their study of modern physics with special relativity. It is here that physics students first encounter the counterintuitive concepts of time dilation and length contraction. Unfortunately, the derivations of these results are often cloaked in several layers of analysis that render them rather mysterious and otherworldly for novice students. Here we show how a very simple thought experiment leads directly to a student-friendly derivation of time dilation and length contraction. The only assumption needed is the constancy of the speed of light.

An experiment to study the effect of relativistic time dilation on secondary muon fluxes observed at different altitudes is described in this article. Muons, produced as secondary particles from the interaction of primary cosmic rays with the upper atmosphere, form a natural and abundant source of subatomic ‘‘clocks’’ moving at very high speeds. The measured muon flux on a mountain relative to that measured at sea level can be compared to predictions from calculations that take into account the relativistic time dilation in the muon frame. Situations under which such an experiment can be successfully performed are explored with a day‐long field trip to a nearby mountain. This experiment has been developed at Smith College as a module in the Five College cooperative undergraduate advanced laboratory course (other participating institutions are Amherst College, Mount Holyoke College, and the University of Massachusetts).

The possibility of explaining time dilation as a dynamic cause‐and‐effect phenomenon is explored by calculating the rates of three elementary electromagnetic clocks in a stationary and in a moving reference frame. The operation of the clocks is based on the interaction between a field‐experiencing electric point charge and different field‐producing electric charge configurations. The calculations show that, when the clocks move, they run as predicted by the special relativity theory. The slowing down of the three moving clocks is due to a different electromagnetic field produced by the moving field‐producing charges (and hence due to a different force acting on the field‐experiencing charges) as well as due to a change of the effective mass of the moving field‐experiencing charges. Thus, for the clocks under consideration, time dilation can be considered a dynamic cause‐and‐effect phenomenon and not merely a kinematic effect, as time dilation is usually explained in conventional presentations of the speci...

We give a variation of the twin paradox of special relativity in which the relationship of acceleration of the rocket twin and time dilation is clarified.

An experiment has been performed to demonstrate the relativistic time dilation as a large effect, using only comparatively simple equipment. μ-mesons incident on top of Mt. Washington, New Hampshire, were selected to have speeds in the range between 0.9950 c and 0.9954 c. The number of these which survived to reach sea level was measured in Cambridge, Massachusetts. The number expected without time dilation was calculated from the distribution of decay times of these μ-mesons (i.e., the mean life as measured in both this experiment and others) and from the known distance of descent. The observed time dilation factor is 8.8±0.8 to be compared with the effective time dilation factor calculated for mesons of these speeds in our detection geometry 1/(1 − v2/c2)12 = 8.4 ± 2.

It is shown that, contrary claims notwithstanding, apparently no one before Einstein in 1905 ventured the idea that all clocks in motion might go slow. And thus, however logical it would have been, there never existed a theory based on the axioms of length contraction and time dilation relative to the ether.

We present a comprehensive discussion of the formulation of the kinematics of special relativity, i.e., the Lorentz transformation. We begin with a concise new proof that the principle of relativity implies that the transformation of event coordinates between inertial reference frames is linear. We then give a clear derivation of the pre-Lorentz transformation, which follows from the principle of relativity. We then show that the pre-Lorentz transformation and the inertial invariance of the speed of light together result in the Lorentz transformation. This, of course, is essentially the traditional formulation. We next present two additional formulations, one using Lorentz–Fitzgerald contraction and one using time dilation, instead of inertial invariance. This is reasonable since Lorentz–Fitzgerald contraction and time dilation are about as well established as and are arguably less abstract than inertial invariance, and thus may profitably be used instead of inertial invariance to complete the formulation. We then present a complete proof that the pre-Lorentz transformation and the requirement of closure upon composition together imply that the transformation is either a Galilean transformation or a generalized Lorentz transformation. This is noteworthy in that it gets ever so close to the Lorentz transformation without invoking light. In the course of this, we obtain a generalized velocity addition rule, which reduces to the velocity addition rule of special relativity. We next show that the generalized Lorentz transformation, together with inertial invariance, Lorentz–Fitzgerald contraction, and time dilation, used one at a time, yields three more formulations. We then show that the unspecified, nonzero, constant speed in the generalized Lorentz transformation can be determined without any reference to light, thereby obtaining a seventh formulation. Light plays no explicit role in the four formulations employing Lorentz–Fitzgerald contraction and time dilation and plays no role whatsoever in the seventh formulation. Thus, and this is a fact which should be strongly emphasized, the formulation of special relativity in no way depends upon the nature of electromagnetic radiation. We conclude by briefly discussing these seven formulations of the kinematics of special relativity and some associated implications.

Captain Einstein is a virtual reality (VR) movie that takes you on a boat trip in a world with a slow speed of light. This allows for a direct experience of the theory of special relativity, much in the same spirit as in the Mr. Tompkins adventure by George Gamow (1939). In this paper, we go through the different relativistic effects (e.g., length contraction, time dilation, Doppler shift, light aberration) that show up during the boat trip, and we explain how these effects were implemented in the 360 ° video production process. We also provide exercise questions that can be used-in combination with the VR movie-to gain insight and sharpen the intuition on the basic concepts of special relativity.

Einstein’s special theory of relativity includes many non-intuitive and apparently paradoxical conclusions about space and time. One of these is time dilation, the fact that a clock moving relative to an observer runs slower than an identical clock at rest in the observer’s reference frame. This is clearly exhibited in the extended mean lifetime of elementary particles moving at high speed, compared to their mean lifetime when at rest, and must be taken into account when processing signals from atomic clocks carried by satellites in the global positioning system (GPS). My intent in this note is not to derive this or other relativistic effects; such derivations may be found in many textbooks. Rather, I would like to share two figures I prepared while teaching elementary physics quite a few years ago, to help understand the apparent paradox that two observers, in relative motion, both measure the other’s clock to run slower than their own. It is my hope that instructors can reproduce these two figures and find them useful in discussions to illustrate these basic concepts. The sketches are only approximately drawn to scale.