Relativity Notes Lecturer
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Visualising Special Relativity
Visualising Special Relativity C.M. Savage1 and A.C. Searle2 Department of Physics and Theoretical Physics, Australian National University ACT 0200, Australia Abstract We describe a graphics package we have developed for producing photo-realistic images of relativistically moving objects. The physics of relativistic images is outlined. INTRODUCTION Since Einstein’s first paper on relativity3 physicists have wondered how things would look at relativistic speeds. However it is only in the last forty years that the physics of relativistic images has become clear. The pioneering studies of Penrose4, Terrell5, and Weisskopf6 showed that there is more to relativistic images than length contraction. At relativistic speeds a rich visual environment is produced by the combined effects of the finite speed of light, aberration, the Doppler effect, time dilation, and length contraction. Computers allow us to use relativistic physics to construct realistic images of relativistic scenes. There have been two approaches to generating relativistic images: ray tracing7,8 and polygon rendering9,10,11. Ray tracing is slow but accurate, while polygon rendering is fast enough to work in real time11, but unsuitable for incorporating the full complexity of everyday scenes, such as shadows. One of us (A.S.) has developed a ray tracer, called “Backlight”12, which maximises realism and flexibility. Relativity is difficult, at least in part, because it challenges our fundamental notions of space and time in a disturbing way. Nevertheless, relativistic images, such as Figure 1, are dramatic but comprehensible. It is part of most people’s experience that curved mirrors and the like can produce strange optical distortions. -
Relativistic Effects for Time-Resolved Light Transport
Relativistic Effects for Time-Resolved Light Transport The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Jarabo, Adrian, Belen Masia, Andreas Velten, Christopher Barsi, Ramesh Raskar, and Diego Gutierrez. “Relativistic Effects for Time- Resolved Light Transport.” Computer Graphics Forum 34, no. 8 (May 13, 2015): 1–12. As Published http://dx.doi.org/10.1111/cgf.12604 Publisher John Wiley & Sons Version Author's final manuscript Citable link http://hdl.handle.net/1721.1/103759 Terms of Use Creative Commons Attribution-Noncommercial-Share Alike Detailed Terms http://creativecommons.org/licenses/by-nc-sa/4.0/ Volume 0 (1981), Number 0 pp. 1–12 COMPUTER GRAPHICS forum Relativistic Effects for Time-Resolved Light Transport Adrian Jarabo1 Belen Masia1;2;3 Andreas Velten4 Christopher Barsi2 Ramesh Raskar2 Diego Gutierrez1 1Universidad de Zaragoza 2MIT Media Lab 3I3A Institute 4Morgridge Institute for Research Abstract We present a real-time framework which allows interactive visualization of relativistic effects for time-resolved light transport. We leverage data from two different sources: real-world data acquired with an effective exposure time of less than 2 picoseconds, using an ultrafast imaging technique termed femto-photography, and a transient renderer based on ray-tracing. We explore the effects of time dilation, light aberration, frequency shift and radiance accumulation by modifying existing models of these relativistic effects to take into account the time-resolved nature of light propagation. Unlike previous works, we do not impose limiting constraints in the visualization, allowing the virtual camera to explore freely a reconstructed 3D scene depicting dynamic illumination. -
8.20 MIT Introduction to Special Relativity IAP 2005 Tentative Outline 1 Main Headings 2 More Detail
8.20 MIT Introduction to Special Relativity IAP 2005 Tentative Outline 1 Main Headings I Introduction and relativity preEinstein II Einstein’s principle of relativity and a new concept of spacetime III The great kinematic consequences of relativity IV Velocity addition and other differential transformations V Kinematics and “Paradoxes” VI Relativistic momentum and energy I: Basics VII Relativistic momentum and energy II: Four vectors and trans formation properties VIII General relativity: Einstein’s theory of gravity 2 More Detail I.0 Summary of organization I.1 Intuition and familiarity in physical law. I.2 Relativity before Einstein • Inertial frames • Galilean relativity • Form invariance of Newton’s Laws • Galilean transformation • Noninertial frames • Galilean velocity addition • Getting wet in the rain I.3 Electromagnetism, light and absolute motion. • Particle and wave interpretations of light • Measurement of c • Maxwell’s theory → electromagnetic waves • Maxwell waves ↔ light. I.4 Search for the aether • Properties of the aether • MichelsonMorley experiment • Aether drag & stellar aberration I.5 Precursors of Einstein • Lorentz and Poincar´e • Lorentz contraction • Lorentz invariance of electromagnetism II.1 Principles of relativity • Postulates • Resolution of MichelsonMorley experiment • Need for a transformation of time. II.2 Intertial systems, clock and meter sticks, reconsidered. • Setting up a frame • Synchronization • Infinite family of inertial frames II.3 Lorentz transformation • The need for a transformation between inertial frames • Derivation of the Lorentz transformation II.4 Immediate consequences • Relativity of simultaneity • Spacetime, world lines, events • Lorentz transformation of events II.5 Algebra of Lorentz transformations • β, γ, and the rapidity, η. • Analogy to rotations 2 • Inverse Lorentz transformation. -
APPARENT SIMULTANEITY Hanoch Ben-Yami
APPARENT SIMULTANEITY Hanoch Ben-Yami ABSTRACT I develop Special Relativity with backward-light-cone simultaneity, which I call, for reasons made clear in the paper, ‘Apparent Simultaneity’. In the first section I show some advantages of this approach. I then develop the kinematics in the second section. In the third section I apply the approach to the Twins Paradox: I show how it removes the paradox, and I explain why the paradox was a result of an artificial symmetry introduced to the description of the process by Einstein’s simultaneity definition. In the fourth section I discuss some aspects of dynamics. I conclude, in a fifth section, with a discussion of the nature of light, according to which transmission of light energy is a form of action at a distance. 1 Considerations Supporting Backward-Light-Cone Simultaneity ..........................1 1.1 Temporal order...............................................................................................2 1.2 The meaning of coordinates...........................................................................3 1.3 Dependence of simultaneity on location and relative motion........................4 1.4 Appearance as Reality....................................................................................5 1.5 The Time Lag Argument ...............................................................................6 1.6 The speed of light as the greatest possible speed...........................................8 1.7 More on the Apparent Simultaneity approach ...............................................9 -
Einstein's Boxes
Einstein’s Boxes Travis Norsen∗ Marlboro College, Marlboro, Vermont 05344 (Dated: February 1, 2008) At the 1927 Solvay conference, Albert Einstein presented a thought experiment intended to demon- strate the incompleteness of the quantum mechanical description of reality. In the following years, the experiment was modified by Einstein, de Broglie, and several other commentators into a simple scenario involving the splitting in half of the wave function of a single particle in a box. This paper collects together several formulations of this thought experiment from the literature, analyzes and assesses it from the point of view of the Einstein-Bohr debates, the EPR dilemma, and Bell’s the- orem, and argues for “Einstein’s Boxes” taking its rightful place alongside similar but historically better known quantum mechanical thought experiments such as EPR and Schr¨odinger’s Cat. I. INTRODUCTION to Copenhagen such as Bohm’s non-local hidden variable theory. Because of its remarkable simplicity, the Einstein Boxes thought experiment also is well-suited as an intro- It is well known that several of quantum theory’s duction to these topics for students and other interested founders were dissatisfied with the theory as interpreted non-experts. by Niels Bohr and other members of the Copenhagen school. Before about 1928, for example, Louis de Broglie The paper is organized as follows. In Sec. II we intro- duce Einstein’s Boxes by quoting a detailed description advocated what is now called a hidden variable theory: a pilot-wave version of quantum mechanics in which parti- due to de Broglie. We compare it to the EPR argument and discuss how it fares in eluding Bohr’s rebuttal of cles follow continuous trajectories, guided by a quantum wave.1 David Bohm’s2 rediscovery and completion of the EPR. -
Class 4: Space-Time Diagrams
Class 4: Space-time Diagrams In this class we will explore how space-time diagrams may be used to visualize events and causality in relativity Class 4: Space-time Diagrams At the end of this session you should be able to … • … create a space-time diagram showing events and world lines in a given reference frame • … determine geometrically which events are causally connected, via the concept of the light cone • … understand that events separated by a constant space-time interval from the origin map out a hyperbola in space-time • … use space-time diagrams to relate observations in different inertial frames, via tilted co-ordinate systems What is a space-time diagram? • A space-time diagram is a graph showing the position of objects (events) in a reference frame, as a function of time • Conventionally, space (�) is represented in the horizontal direction, and time (�) runs upwards �� Here is an event This is the path of a light ray travelling in the �- direction (i.e., � = ��), We have scaled time by a which makes an angle of factor of �, so it has the 45° with the axis same dimensions as space What would be the � path of an object moving at speed � < �? What is a space-time diagram? • The path of an object (or light ray) moving through a space- time diagram is called a world line of that object, and may be thought of as a chain of many events • Note that a world line in a space-time diagram may not be the same shape as the path of an object through space �� Rocket ship accelerating in �-direction (path in Rocket ship accelerating in �- -
EPR, Time, Irreversibility and Causality
EPR, time, irreversibility and causality Umberto Lucia 1;a & Giulia Grisolia 1;b 1 Dipartimento Energia \Galileo Ferraris", Politecnico di Torino Corso Duca degli Abruzzi 24, 10129 Torino, Italy a [email protected] b [email protected] Abstract Causality is the relationship between causes and effects. Following Relativity, any cause of an event must always be in the past light cone of the event itself, but causes and effects must always be related to some interactions. In this paper, causality is developed as a conse- quence of the analysis of the Einstein, Podolsky and Rosen paradox. Causality is interpreted as the result of the time generation, due to irreversible interactions of real systems among them. Time results as a consequence of irreversibility, so any state function of a system in its space cone, when affected by an interaction with an observer, moves into a light cone or within it, with the consequence that any cause must precede its effect in a common light cone. Keyword: EPR paradox; Irreversibility; Quantum Physics; Quan- tum thermodynamics; Time. 1 Introduction The problem of the link between entropy and time has a long story and many viewpoints. In relation to irreversibility, Franklin [1] analysed some processes, in their steady states, evaluating their entropy variation, and he highlighted that they are related only to energy transformations. Thermodynamics is the physical science which develops the study of the energy transformations , allowing the scientists and engineers to ob- tain fundamental results [2{4] in physics, chemistry, biophysics, engineering and information theory. During the development of this science, entropy 1 has generally been recognized as one of the most important thermodynamic quantities due to its interdisciplinary applications [5{8]. -
The Flow of Time in the Theory of Relativity Mario Bacelar Valente
The flow of time in the theory of relativity Mario Bacelar Valente Abstract Dennis Dieks advanced the view that the idea of flow of time is implemented in the theory of relativity. The ‘flow’ results from the successive happening/becoming of events along the time-like worldline of a material system. This leads to a view of now as local to each worldline. Each past event of the worldline has occurred once as a now- point, and we take there to be an ever-changing present now-point ‘marking’ the unfolding of a physical system. In Dieks’ approach there is no preferred worldline and only along each worldline is there a Newtonian-like linear order between successive now-points. We have a flow of time per worldline. Also there is no global temporal order of the now-points of different worldlines. There is, as much, what Dieks calls a partial order. However Dieks needs for a consistency reason to impose a limitation on the assignment of the now-points along different worldlines. In this work it is made the claim that Dieks’ consistency requirement is, in fact, inbuilt in the theory as a spatial relation between physical systems and processes. Furthermore, in this work we will consider (very) particular cases of assignments of now-points restricted by this spatial relation, in which the now-points taken to be simultaneous are not relative to the adopted inertial reference frame. 1 Introduction When we consider any experiment related to the theory of relativity,1 like the Michelson-Morley experiment (see, e.g., Møller 1955, 26-8), we can always describe it in terms of an intuitive notion of passage or flow of time: light is send through the two arms of the interferometer at a particular moment – the now of the experimenter –, and the process of light propagation takes time to occur, as can be measured by a clock calibrated to the adopted time scale. -
2.3 Twin Paradox and Light Clocks (Computational Example)
2.3 Twin paradox and light clocks (Computational example) With the Lorentz transformation, world points and world lines are transformed between coordinate systems with constant relative velocity. This can be used to study the twin paradox of the Special Theory of Relativity, or to show that, in a moving frame of reference, time slows down. clear all Lorentz transformation from chapter 2.2 load Kap02 LT LT = syms c positive 1 World lines Light cone Light propagates in all directions at the same speed. The world points appear shifted in different reference systems, but the cone's opening angle remains the same. Light cone: LK=[-2 -1, 0 1 2;2 1 0 1 2] LK = -2 -1 0 1 2 2 1 0 1 2 1 Graphic representation of world lines (WeLi see 'function' below) in reference systems with velocities subplot(3,3,4) WeLi(LK,-c/2,0.6*[1 1 0],LT) axis equal subplot(3,3,5) WeLi(LK,0,0.6*[1 1 0],LT) axis equal subplot(3,3,6) WeLi(LK,c/2,0.6*[1 1 0],LT) axis equal 2 Twin paradox One twin stays at home, the other moves away at half the speed of light, to turn back after one unit of time (in the rest frame of the one staying at home). World lines of the twins: ZW=[0 0 0 1/2 0;2 1 0 1 2]; Graphics Plo({LK ZW},{.6*[1 1 0] 'b'},LT) 2 In the rest frame of the 'home stayer', his twin returns after one time unit, but in the rest frame of the 'traveler' this happens already after 0.86 time units. -
Arxiv:2104.01492V2 [Physics.Pop-Ph] 1 May 2021 Gravitational Lensing [10], Etc
An alternative transformation factor in the framework of the relativistic aberration of light D Rold´an1,∗ R Sempertegui,† F Rold´an‡ 1University of Cuenca Abstract In the present study, we analyze in combination the principles of special relativity and the phenomenon of the aberration of light, deriving a system of equations that allows establishing the relationship between the angles commonly involved in this phenomenon. As a consequence, a transformation factor is obtained that generates two solutions, one of them with the same values as the Lorentz factor. This suggests that due to relativistic aberration, an apparent double image of celestial objects could be obtained. On the other hand, its functional form does not establish a limit to the speed of the reference frames with inertial movement. Keywords: relativistic aberration, alternative Lorentz transformations, faster than light 1 Introduction tive studies with didactic interest [3], with this study falling into the latter category. The aberration of light is a phenomenon in as- The phenomenon of the aberration of light tronomy in which the location of celestial objects can be described as the angle difference between does not indicate their real position due to the a light beam in two different inertial reference relative speed of the observer and that of light. frames. Relativistic effects can be included in James Bradley proposed an explanation for this a study to enrich its analysis [4]. Thus, for ex- phenomenon as early as 1727, considering the ample, the aberration has been related to rel- movement of Earth in relation to the Sun [1, 2]. ativistic Doppler shifts and relativistic velocity In 1905, Albert Einstein presented an analysis addition [5], light-time correction [6], simultane- of the aberration of light from the perspective ity [7], Kerr spacetime [8], light refraction [9], of the special theory of relativity, in the con- arXiv:2104.01492v2 [physics.pop-ph] 1 May 2021 gravitational lensing [10], etc. -
Doing Physics with Quaternions
Doing Physics with Quaternions Douglas B. Sweetser ©2005 doug <[email protected]> All righs reserved. 1 INDEX Introduction 2 What are Quaternions? 3 Unifying Two Views of Events 4 A Brief History of Quaternions Mathematics 6 Multiplying Quaternions the Easy Way 7 Scalars, Vectors, Tensors and All That 11 Inner and Outer Products of Quaternions 13 Quaternion Analysis 23 Topological Properties of Quaternions 28 Quaternion Algebra Tool Set Classical Mechanics 32 Newton’s Second Law 35 Oscillators and Waves 37 Four Tests of for a Conservative Force Special Relativity 40 Rotations and Dilations Create the Lorentz Group 43 An Alternative Algebra for Lorentz Boosts Electromagnetism 48 Classical Electrodynamics 51 Electromagnetic Field Gauges 53 The Maxwell Equations in the Light Gauge: QED? 56 The Lorentz Force 58 The Stress Tensor of the Electromagnetic Field Quantum Mechanics 62 A Complete Inner Product Space with Dirac’s Bracket Notation 67 Multiplying quaternions in Polar Coordinate Form 69 Commutators and the Uncertainty Principle 74 Unifying the Representations of Integral and Half−Integral Spin 79 Deriving A Quaternion Analog to the Schrödinger Equation 83 Introduction to Relativistic Quantum Mechanics 86 Time Reversal Transformations for Intervals Gravity 89 Unified Field Theory by Analogy 101 Einstein’s vision I: Classical unified field equations for gravity and electromagnetism using Riemannian quaternions 115 Einstein’s vision II: A unified force equation with constant velocity profile solutions 123 Strings and Quantum Gravity 127 Answering Prima Facie Questions in Quantum Gravity Using Quaternions 134 Length in Curved Spacetime 136 A New Idea for Metrics 138 The Gravitational Redshift 140 A Brief Summary of Important Laws in Physics Written as Quaternions 155 Conclusions 2 What Are Quaternions? Quaternions are numbers like the real numbers: they can be added, subtracted, multiplied, and divided. -
Relativistic Angle-Changes and Frequency-Changes Freq' / Freq = ( )
Relativistic angle-changes and frequency-changes Eric Baird ([email protected]) We generate a set of "relativistic" predictions for the relationship between viewing angle and apparent frequency, for each of three different non-transverse shift equations. We find that a detector aimed transversely (in the lab frame) at a moving object should report a redshift effect with two out of the three equations. 1. Introduction In the new frame, our original illuminated One of the advantages of Einstein’s special spherical surface now has to obey the condition theory [1] over other contemporary models was that the round-trip flight-time from that it provided clear and comparatively A®surface®B is the same for rays sent in any straightforward predictions for angle-changes direction, and a map of a cross-section through and frequency changes at different angles. the surface now gives us an elongated ellipse, These calculations could be rather confusing in with the emission and absorption points being the variety of aether models that were available the two ellipse foci. at the time (see e.g. Lodge, 1894 [2] ). In this paper, we derive the relativistic angle- changes and wavelength-changes that would be associated with each of the three main non- transverse Doppler equations, if used as the basis of a relativistic model. The same set of illumination-events can mark 2. Geometry out a spherical spatial surface in one frame and Wavefront shape a spheroidal spatial surface in the other. [3][4] If a “stationary” observer emits a unidirectional Ellipse dimensions pulse of light and watches the progress of the Instead of using the “fixed flat aether approach” spreading wavefront, the signal that they detect for calculating distances in our diagram (setting is not the outgoing wavefront itself, but a the distance between foci as v and the round- secondary (incoming) signal generated when trip distance as 2c), we will instead construct the outgoing wavefront illuminates dust or our map around the wavelength-distances that other objects in the surrounding region.