Minkowski Diagrammatics: A Computable Geometry for Lorentz-Equivariant Modeling

Tam Hunt, UC Santa Barbara, tam.hunt@psych.ucsb.eduThe Lorentz transformations form the mathematical core of the 1905 theory of Special Relativity as well as the earlier version of relativity created by Lorentz himself, originally in 1895 but developed further in the ensuing years. These two theories interpret the physical significance of the transformations quite differently, but in ways that are generally not considered to be empirically distinguishable. It is widely believed today that Einstein’s Special Relativity presents the superior interpretation. A number of lines of evidence, however, from cosmology, quantum theory and nuclear physics present substantial evidence against the Special Relativity interpretation of the Lorentz transformations, challenging this traditional view. I review this evidence and suggest that we are now at a point where the sum of the evidence weighs against the Special Relativity interpretation of the transformations and in favor of a Lorentzian or neo-Lorentzian approach instead.1. IntroductionI’m sitting in a public square in Athens, Greece, biding my time as I write these words. The battery on my phone ran out as I was trying to navigate to my lodgings on my first night in this historic city, forcing me to stop and charge my phone for a little while. I’m waiting for the passage of time.The nature of time has been debated vigorously since at least the age of Heraclitus and Parmenides in ancient Greece. “All things flow,” said Heraclitus. “Nothing flows,” said Parmenides as a counter-intuitive rejoinder, suggesting that all appearances of change are an illusion. How could Parmenides make the case that nothing flows, nothing changes? It would seem, from easy inspection of the world around us that indeed all things do flow, all things are always changing. So what was Parmenides talking about?Parmenides’ arguments illustrate well the rationalist approach that Plato was later to more famously advocate, against the empiricist or “sensationist” approach that Heraclitus and Aristotle too would champion as a contrary approach. Parmenides and Plato saw reason as the path toward truth and they were not afraid to allow reason to contradict what seemed to be obvious sensory-based features of the world. Apparent empirical/sensory facts can deceive and, for these men, Parmenides, Plato and their followers, reason alone was the arbiter of truth. Wisdom entailed using reason to see through the world’s illusions to the deeper reality.Heraclitus and Aristotle, to the contrary, stressed the need to be empirical in our science and philosophy (science and philosophy were the same endeavor in the era of classical Greece). Reason was of course a major tool in the philosopher’s toolbox for these men too, but it seems that reason unmoored from evidence should not be used to trump the obvious facts of the world. The Aristotelian approach is to find a pragmatic balance between empirical facts and reason in attempting to discern the true contours of reality.Einstein was firmly in the camp of Parmenides and Plato (Popper, et al. 1998). He famously considered the passage of time, the distinction between past, present and future, to be a “stubbornly persistent illusion.” This view of time, as an illusory construct hiding a deeper timeless world, was based on his theories of relativity. Einstein and his co-thinkers held this view, of time as illusory, despite the obvious passage of time in the world around us, no matter where we look. The widely-held view today is that Einstein finally won the long war, decisively, between Heraclitus and Parmenides. Despite appearances, nothing flows and the passage of time is just that: only appearance.I suggest in this paper, however, that this conclusion is premature. Einstein’s thinking is indeed an example of rationalism trumping empiricism and it is time for us to take a more empirical approach to these foundational questions of physics and philosophy. Today’s physics lauds empiricism rhetorically, but in practice a rationalist approach often holds sway, particularly with respect to the nature of time.2. An overview of Special Relativity and Lorentzian RelativityIn discussing the nature of time with respect to modern physics, I will focus on the Special Theory of Relativity (SR) and avoid discussion of the general theory. Einstein’s 1905 theory of relativity adopted the Lorentz transformations directly, unchanged from Lorentz’s own version of these equations (Einstein 1905, Lorentz 1895 and 1904, in Lorentz 1937). Einstein’s key difference from Lorentz’s version of relativity (first put forth in 1895, but developed further in later work) was to reinterpret Lorentz’s equations, based on a radically different assumption about the nature of physical reality. Lorentz interpreted the relativistic effects of length contraction and time dilation—which follow straightforwardly from the Lorentz transformations—as resulting from interaction with an ether that constituted simply the properties of space (Lorentz’s ether was not some additional substance that pervades space, as was the case in some earlier ideas of the ether). Einstein, to the contrary, interpreted these effects as resulting from the dynamics of spacetime, a union of space and time into a single notion, and dismissed the ether as “superfluous.”Because Lorentz’s and Einstein’s versions of relativity both use the Lorentz transformations, they will yield in many cases the same empirical predictions. The prevailing view today, then, is that while these two theories are empirically indistinguishable there are other considerations, relating to parsimony primarily, that render special relativity the preferred approach. I discuss below, however, why we now have good empirical reasons to distinguish between these two interpretations—in favor of the Lorentzian approach.Length contraction and time dilation occur as a result of the assumed absolute speed of light because either space or time, or both, must distort if we consider the speed of light to be invariant. This is because speed is measured simply by dividing distance traveled by the time elapsed; and if the speed of light remains the same in all circumstances then space and/or time must distort in order to maintain this invariance. As an object travels closer and closer to the speed of light, its length must decrease (length contraction) and/or the time elapsed must increase (time dilation) – but only from the perspective of an observer in a different inertial frame. In the original inertial frame there is no length contraction or time dilation.“Moving clocks run slow” is a good shorthand for relativistic time dilation, but again only from the perspective of a different inertial frame. Time moves at the same rate for an observer in the moving frame of reference, no matter what one’s speed in relation to other frames. Relativistic effects only occur when considering the relationship between two different frames of reference, not in the same frame.

The Lorentz transformations and special relativity are unable to provide a realistic physical explanation of the behavior of matter and light. We will show that all these phenomena can be explained using Newton's physics and mass-energy conservation, without space contraction or time dilation. We have seen previously that the principle of mass-energy conservation requires that clocks run at a slower rate in a moving frame, and physical bodies become longer because of the increase of the Bohr radius. These results allow us to answer the question: With respect to what does light travel? For example, when we move away at velocity v from a source emitting light at velocity c, the relative motion of the radiation is observed from the Doppler shift. How can we explain logically that these photons appear to reach us at velocity c and not (c-v)? The conventional explanation relies on special relativity, but it implies an esoteric space-time distortion, which is not compatible with logic. This paper gives a physical explanation how the velocity of light is really (c-v) with respect to the observer, even if the observer's tools always measure a velocity represented by the number c. We explain how this problem is crucial in the Global Positioning System (GPS) and in clock synchronization. The Lorentz' transformations become quite useless. This apparent constant velocity of light with respect to a moving frame is the most fascinating illusion in science.

It is known that neutrinos propagate faster than light. For that reason the Einstein’s special theory of relativity cannot be applied to these phenomena. On the other hand the Matscie’s special theory of relativity based on the Matscie’s transformation is valid for any velocity including the velocities greater than the velocity of light in free space. The relativistic phenomena consisting of the length contraction and the time dilation can be verified successfully by the Matscie’s special theory of relativity. In this case, the velocity of light in free space acts as the critical velocity. Only about 81.6% of the rest mass of a body can be converted into energy. At very low velocities, the kinetic energy of a moving body is practically the same as that in the classical mechanics. And also at velocities of much higher than the critical velocity, it almost reduces to the classical expression. For moderate velocities, the Matscie’s special theory of relativity reduces to the Einstein’s special theory of relativity. For velocities close to (below or above) the velocity of light in free space, the kinetic energy of a moving body differs from that predicted by the classical mechanics.

In the year 1905, Albert Einstein presented his theory of Special Relativity and revolutionized our understanding about matter and energy by telling that they were the same. He derived the famous E = mc 2 formula from his realization that space and time have the same status. This realization, in turn, was complimented by his assumption that the speed at which light travels in vacuum is constant in any reference frame. Now what is this reference frame? It is not difficult to visualize the concept of reference frame. Imagine that you are traveling in a train and you throw a ball to a co-passenger in the same direction the train was running. The average speed of the ball should be the distance between you and your co-passenger divided by the time taken by the ball to reach your co-passenger. However, if someone outside the train watches it, one would find that the distance traveled by the ball is the sum of the distance traveled by the train during the time you throw the ball and your co-passenger receives it and the distance traveled by the ball from you to your co-passenger. Consequently, the outside observer would find a higher speed of the ball. The most important thing to notice here is that the time taken by the ball to move from one point to the other will appear to be the same to you and to the outside observer although the distance traveled by the ball would differ. You and your co-passenger measure the speed of the ball inside the train which is one reference frame attached with the train. The outside observer measures the speed of the ball in another reference frame attached to the Earth. One reference frame is moving at a constant speed with respect to another reference frame. If another train passes you with a different speed as compared to the speed of your train and a passenger inside that train measures the speed of the ball, he or she will derive a different speed of the ball. So, the speed of the ball is “relative” to the reference frame wherein it is measured. Einstein considered that light would not follow this rule; the speed of light would remain the same in all reference frames. In order to describe any cosmic event, one has to consider both space and time together. Einstein’s Special Theory of Relativity becomes significant only when the speed of an object is comparable to the speed of light. But nothing can exceed the speed of light. Now, if the speed of the ball or the train varies; i.e., if the ball or the train accelerates or retards, Special Theory of Relativity does not apply. For an accelerating reference frame, General Theory of Relativity has to be invoked. The contraction in length and the dilation of time is special relativistic effects whereas the acceleration due to gravitation is a general relativistic effect. Therefore, Special Relativity is a particular case of General Relativity. According to General Theory of Relativity, one can cancel the effect of gravitation locally by moving in an accelerating reference frame such as a free falling lift or aircraft. But it cannot be canceled globally. We know that the universe is expanding. All the galaxies are receding from each other. Since the rate of expansion of the universe changes, the dynamic of the universe is described by General Theory of Relativity.

Special theory of relativity is one of the difficult subjects of physics to be understood by the students. The current research designed as a qualitative research aim to determine the pre-service physics teachers’ understanding level and the alternative conceptions about three core concepts of special theory of relativity, such as time dilatation, length contraction and reference frames. The data were collected through semi structured interviews and were analyzed by using content analysis. At the end of the analysis process the understanding level of the students was determined to be “complete understanding”, “incomplete understanding” and “misunderstanding”. In order to achieve this, the students’ conceptual frameworks based on the operational definitions made by the students were determined firstly. The findings obtained in this research indicate that high school teachers as well as university instructors should take special care with some points in the teaching of the subjects related with special theory of relativity. This research might be useful to other studies to be done in the future, especially for investigating the students’ mental models related to special theory of relativity. Key words: Length contraction, reference frames, special relativity, time dilatation, understanding level.

In the Einstein Centenary, it is no surprise to find a new book on relativity. But the competition is very strong, since there already exist so many texts on special relativity, and one must look for strong features in a new book. One great attraction of Giulini's book is that it succinctly contains all the essentials of special relativity within a mere 153 pages.

The mathematical treatment and graphical representation of Special Relativity (SR) are well established, yet carry deep implications that remain hard to visualize. This paper presents a new graphical interpretation of the geometry of SR that may, by complementing the standard works, aid the understanding of SR and its fundamental principles in a more intuitive way. From the axiom that the velocity of light remains constant to any inertial observer, the geodesic is presented as a line of constant angle on the complex plane across a set of diverging reference frames. The resultant curve is a logarithmic spiral, and this view of the geodesic is extended to illustrate the relativistic Doppler effect, time dilation, length contraction, the twin paradox, and relativistic radar distance in an original way, whilst retaining the essential mathematical relationships of SR. Using a computer-generated graphical representation of photon trajectories allows a visual comparison between the relativistic relationships and their classical counterparts, to visualize the consequences of SR as velocities become relativistic. The model may readily be extended to other situations, and may be found useful in presenting a fresh understanding of SR through geometric visualization.

When the German mathematician Hermann Minkowski first introduced the space-time diagrams that came to be associated with his name, the idea of picturing motion by geometric means, holding time as a fourth dimension of space, was hardly new. But the pictorial device invented by Minkowski was tailor-made for a peculiar variety of space-time: the one imposed by the kinematics of Einstein’s special theory of relativity, with its unified, non-Euclidean underlying geometric structure. By plo tting two or more reference frames in relative motion on the same picture, Minkowski managed to exhibit the geometric basis of such relativistic phenomena as time dilation, length contraction or the dislocation of simultaneity. These disconcerting effects were shown to result from arbitrary projections within four-dimensional space-time. In that respect, Minkowski diagrams are fundamentally different from ordinary space-time graphs. The best way to understand their specificity is to realize how productively ambiguous they are.

Background Tests of special relativity have been conducted over the past century with increasing accuracy and none have showed violations of Lorentz invariance. In this paper we will examine whether these tests are together sufficient to rule out theories that violate observational symmetry. Methods A variant theory is outlined where relativistic effects such as length contraction and time dilation are purely local consequences of the relative velocity between a system and its medium. The outlined theory is tested against the fundamental tests of special relativity. Results It is found that although this alteration does not align with the principle of relativity, it quantitatively aligns with the experimental results of the fundamental tests of special relativity and their modern variations, and makes diverging, testable but as of yet untested predictions concerning Doppler shift and time dilation. Conclusions These results warrant a closer theoretical inspection of the outlined theory, and could provide a direction to test for new physics. A modified Ives-Stilwell experiment is proposed to test between this model and special relativity.

This compact yet informative Guide presents an accessible route through Special Relativity, taking a modern axiomatic and geometrical approach. It begins by explaining key concepts and introducing Einstein's postulates. The consequences of the postulates – length contraction and time dilation – are unravelled qualitatively and then quantitatively. These strands are then tied together using the mathematical framework of the Lorentz transformation, before applying these ideas to kinematics and dynamics. This volume demonstrates the essential simplicity of the core ideas of Special Relativity, while acknowledging the challenges of developing new intuitions and dealing with the apparent paradoxes that arise. A valuable supplementary resource for intermediate undergraduates, as well as independent learners with some technical background, the Guide includes numerous exercises with hints and notes provided online. It lays the foundations for further study in General Relativity, which is introduced briefly in an appendix.

The Quantum Theory and Special Relativity stand apart because their authors were admittedly unclear about Wave-quantum Unity of light and light propagating medium. A single experiment shows wave-quantum unity for low intensity light and moving electrons. In our Unified Theory light is propagated as a Wave-Quantum UNITY along transverse electromagnetic wave as per Poyinting vector in the physical 'sharmon medium' contiguously via 1-spin sharmons, which do not physically move. It appears as Quantum Theory's wave-or-quantum DUALITY for observing only one of the two not both characters at a time. Sharmon comprises a positive positrino and negative negatrino, the two all-composing indivisible elements of diameter 1.6x1033cm, electric charge 1.3729x10-30 esu, mass 2.596116x10-48 gm, spin = 1/2. These compose all forms of mass, energy, energy quanta like photon and particles like quarks, leptons and hadrons. Copenhagen interpretation of Quantum Theory is reviewed. Uncertainty Principle is rejected and replaced with new ‘Principle of Null Action’ based on the conservation of massenergy, momentum and action. Since spin of light-emitter does not fall and of absorber does not rise by one, NOT the 1-spin photon but 0-spin sharmon composed energy-quantum is emitted, absorbed and propagated. 'Contraction of space' and 'dilatation of time' are unrealistic. Constancy and invariance of light velocity c are explained, as also the observed variability, superluminality and subluminality which invalidate Relativity theories. Energized 1-spin sharmon replaces conventional photon and explains photoelectric effect and bending of light under gravity. Michelson-Morley got zero fringe-shift as light velocity in the sharmon medium entrained with earth is the same for the two perpendicular interfering beams. Non-Doppler cosmological redshift supports non-expanding universe.

A critical error is found in the Special Theory of Relativity (STR): mixing up the concepts of the STR abstract time of a reference frame and the displayed time of a physical clock, which leads to use the properties of the abstract time to predict time dilation on physical clocks and all other physical processes. Actually, a clock can never directly measure the abstract time, but can only record the result of a physical process during a period of the abstract time such as the number of cycles of oscillation which is the multiplication of the abstract time and the frequency of oscillation. After Lorentz Transformation, the abstract time of a reference frame expands by a factor gamma, but the frequency of a clock decreases by the same factor gamma, and the resulting multiplication i.e. the displayed time of a moving clock remains unchanged. That is, the displayed time of any physical clock is an invariant of Lorentz Transformation. The Lorentz invariance of the displayed times of clocks can further prove within the framework of STR our earth based standard physical time is absolute, universal and independent of inertial reference frames as confirmed by both the physical fact of the universal synchronization of clocks on the GPS satellites and clocks on the earth, and the theoretical existence of the absolute and universal Galilean time in STR which has proved that time dilation and space contraction are pure illusions of STR. The existence of the absolute and universal time in STR has directly denied that the reference frame dependent abstract time of STR is the physical time, and therefore, STR is wrong and all its predictions can never happen in the physical world.

This compact yet informative Guide presents an accessible route through Special Relativity, taking a modern axiomatic and geometrical approach. It begins by explaining key concepts and introducing Einstein's postulates. The consequences of the postulates – length contraction and time dilation – are unravelled qualitatively and then quantitatively. These strands are then tied together using the mathematical framework of the Lorentz transformation, before applying these ideas to kinematics and dynamics. This volume demonstrates the essential simplicity of the core ideas of Special Relativity, while acknowledging the challenges of developing new intuitions and dealing with the apparent paradoxes that arise. A valuable supplementary resource for intermediate undergraduates, as well as independent learners with some technical background, the Guide includes numerous exercises with hints and notes provided online. It lays the foundations for further study in General Relativity, which is introduced briefly in an appendix.

Assuming the “Big Bang” theory as well as the usual axioms in the Special Theory of Relativity, the time dilations and length contractions are treated as real physical effects. This becomes possible by relating everything to the hypothetical frame,S a , at rest relative to the “Big Bang” event. This frame in many senses plays the role of the classical aether frame. A clock's real ryhthm, as opposed to its rhythm observed by restricted methods, is then a function of its velocity relative toS a (assuming a uniform gravitational field). It is further assumed that gravitational radiation is composed of “electromagnetic-like” waves. Therefore when a clock changes its velocity in a uniform gravitational field it must receive a different total energy due to the average frequency shift (Doppler effect), the time dilations are then caused by the change in energy due to this frequency shift. That is, not wo clocks can be in the “same” gravitational field unless they have no relative velocity, and therefore the Special Theory of Relativity is a special case of the General Theory from this viewpoint. Two feasible experimental tests, using the Mossbauer effect, are described that would decide on these viewpoints. The principle of equivalence and the “twin paradox” are also discussed.

This article refutes the Time Dilation Equation and Length Contraction that are derived in the Special Theory of Relativity. The conclusion reached in this article is that Time Dilation and Length Contraction cannot be characterized by simple equations due to repulsion gravity. The conclusion follows from gravity being a natural force of repulsion rather than the assumption that gravity is an attraction force. That gravity is a repulsion force follows from the Sir Arthur Eddington experiment designed to prove that gravity affects light. Few looked at that experiment as anything other than proving Einstein’s General Theory of Relativity that suggested gravity would affect light. The experiment went beyond what most imagined it accomplished. It surely verified that gravity affects light. But it did more than that. The experiment showed that gravity is a force of repulsion and not attraction as most believed. That gravity is repulsion and not an attraction force indicates that the relativity time dilation equation derived in the Special Theory of Relativity is intractably undecidable likely subject to Godels Incompleteness theorems