The Flow of Time in the Theory of Relativity Mario Bacelar Valente
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Time Dilation of Accelerated Clocks
Astrophysical Institute Neunhof Circular se07117, February 2016 1 Time Dilation of accelerated Clocks The definitions of proper time and proper length in General Rela- tivity Theory are presented. Time dilation and length contraction are explicitly computed for the example of clocks and rulers which are at rest in a rotating reference frame, and hence accelerated versus inertial reference frames. Experimental proofs of this time dilation, in particular the observation of the decay rates of acceler- ated muons, are discussed. As an illustration of the equivalence principle, we show that the general relativistic equation of motion of objects, which are at rest in a rotating reference frame, reduces to Newtons equation of motion of the same objects at rest in an inertial reference system and subject to a gravitational field. We close with some remarks on real versus ideal clocks, and the “clock hypothesis”. 1. Coordinate Diffeomorphisms In flat Minkowski space, we place a rectangular three-dimensional grid of fiducial marks in three-dimensional position space, and assign cartesian coordinate values x1, x2, x3 to each fiducial mark. The values of the xi are constants for each fiducial mark, i. e. the fiducial marks are at rest in the coordinate frame, which they define. Now we stretch and/or compress and/or skew and/or rotate the grid of fiducial marks locally and/or globally, and/or re-name the fiducial marks, e. g. change from cartesian coordinates to spherical coordinates or whatever other coordinates. All movements and renamings of the fiducial marks are subject to the constraint that the map from the initial grid to the deformed and/or renamed grid must be differentiable and invertible (i. -
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 -
COORDINATE TIME in the VICINITY of the EARTH D. W. Allan' and N
COORDINATE TIME IN THE VICINITY OF THE EARTH D. W. Allan’ and N. Ashby’ 1. Time and Frequency Division, National Bureau of Standards Boulder, Colorado 80303 2. Department of Physics, Campus Box 390 University of Colorado, Boulder, Colorado 80309 ABSTRACT. Atomic clock accuracies continue to improve rapidly, requir- ing the inclusion of general relativity for unambiguous time and fre- quency clock comparisons. Atomic clocks are now placed on space vehi- cles and there are many new applications of time and frequency metrology. This paper addresses theoretical and practical limitations in the accuracy of atomic clock comparisons arising from relativity, and demonstrates that accuracies of time and frequency comparison can approach a few picoseconds and a few parts in respectively. 1. INTRODUCTION Recent experience has shown that the accuracy of atomic clocks has improved by about an order of magnitude every seven years. It has therefore been necessary to include relativistic effects in the reali- zation of state-of-the-art time and frequency comparisons for at least the last decade. There is a growing need for agreement about proce- dures for incorporating relativistic effects in all disciplines which use modern time and frequency metrology techniques. The areas of need include sophisticated communication and navigation systems and funda- mental areas of research such as geodesy and radio astrometry. Significant progress has recently been made in arriving at defini- tions €or coordinate time that are practical, and in experimental veri- fication of the self-consistency of these procedures. International Atomic Time (TAI) and Universal Coordinated Time (UTC) have been defin- ed as coordinate time scales to assist in the unambiguous comparison of time and frequency in the vicinity of the Earth. -
Time Scales and Time Differences
SECTION 2 TIME SCALES AND TIME DIFFERENCES Contents 2.1 Introduction .....................................................................................2–3 2.2 Time Scales .......................................................................................2–3 2.2.1 Ephemeris Time (ET) .......................................................2–3 2.2.2 International Atomic Time (TAI) ...................................2–3 2.2.3 Universal Time (UT1 and UT1R)....................................2–4 2.2.4 Coordinated Universal Time (UTC)..............................2–5 2.2.5 GPS or TOPEX Master Time (GPS or TPX)...................2–5 2.2.6 Station Time (ST) ..............................................................2–5 2.3 Time Differences..............................................................................2–6 2.3.1 ET − TAI.............................................................................2–6 2.3.1.1 The Metric Tensor and the Metric...................2–6 2.3.1.2 Solar-System Barycentric Frame of Reference............................................................2–10 2.3.1.2.1 Tracking Station on Earth .................2–13 2.3.1.2.2 Earth Satellite ......................................2–16 2.3.1.2.3 Approximate Expression...................2–17 2.3.1.3 Geocentric Frame of Reference.......................2–18 2–1 SECTION 2 2.3.1.3.1 Tracking Station on Earth .................2–19 2.3.1.3.2 Earth Satellite ......................................2–20 2.3.2 TAI − UTC .........................................................................2–20 -
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. -
Observer with a Constant Proper Acceleration Cannot Be Treated Within the Theory of Special Relativity and That Theory of General Relativity Is Absolutely Necessary
Observer with a constant proper acceleration Claude Semay∗ Groupe de Physique Nucl´eaire Th´eorique, Universit´ede Mons-Hainaut, Acad´emie universitaire Wallonie-Bruxelles, Place du Parc 20, BE-7000 Mons, Belgium (Dated: February 2, 2008) Abstract Relying on the equivalence principle, a first approach of the general theory of relativity is pre- sented using the spacetime metric of an observer with a constant proper acceleration. Within this non inertial frame, the equation of motion of a freely moving object is studied and the equation of motion of a second accelerated observer with the same proper acceleration is examined. A com- parison of the metric of the accelerated observer with the metric due to a gravitational field is also performed. PACS numbers: 03.30.+p,04.20.-q arXiv:physics/0601179v1 [physics.ed-ph] 23 Jan 2006 ∗FNRS Research Associate; E-mail: [email protected] Typeset by REVTEX 1 I. INTRODUCTION The study of a motion with a constant proper acceleration is a classical exercise of special relativity that can be found in many textbooks [1, 2, 3]. With its analytical solution, it is possible to show that the limit speed of light is asymptotically reached despite the constant proper acceleration. The very prominent notion of event horizon can be introduced in a simple context and the problem of the twin paradox can also be analysed. In many articles of popularisation, it is sometimes stated that the point of view of an observer with a constant proper acceleration cannot be treated within the theory of special relativity and that theory of general relativity is absolutely necessary. -
Coordinate Systems in De Sitter Spacetime Bachelor Thesis A.C
Coordinate systems in De Sitter spacetime Bachelor thesis A.C. Ripken July 19, 2013 Radboud University Nijmegen Radboud Honours Academy Abstract The De Sitter metric is a solution for the Einstein equation with positive cosmological constant, modelling an expanding universe. The De Sitter metric has a coordinate sin- gularity corresponding to an event horizon. The physical properties of this horizon are studied. The Klein-Gordon equation is generalized for curved spacetime, and solved in various coordinate systems in De Sitter space. Contact Name Chris Ripken Email [email protected] Student number 4049489 Study Physicsandmathematics Supervisor Name prof.dr.R.H.PKleiss Email [email protected] Department Theoretical High Energy Physics Adress Heyendaalseweg 135, Nijmegen Cover illustration Projection of two-dimensional De Sitter spacetime embedded in Minkowski space. Three coordinate systems are used: global coordinates, static coordinates, and static coordinates rotated in the (x1,x2)-plane. Contents Preface 3 1 Introduction 4 1.1 TheEinsteinfieldequations . 4 1.2 Thegeodesicequations. 6 1.3 DeSitterspace ................................. 7 1.4 TheKlein-Gordonequation . 10 2 Coordinate transformations 12 2.1 Transformations to non-singular coordinates . ......... 12 2.2 Transformationsofthestaticmetric . ..... 15 2.3 Atranslationoftheorigin . 22 3 Geodesics 25 4 The Klein-Gordon equation 28 4.1 Flatspace.................................... 28 4.2 Staticcoordinates ............................... 30 4.3 Flatslicingcoordinates. 32 5 Conclusion 39 Bibliography 40 A Maple code 41 A.1 Theprocedure‘riemann’. 41 A.2 Flatslicingcoordinates. 50 A.3 Transformationsofthestaticmetric . ..... 50 A.4 Geodesics .................................... 51 A.5 TheKleinGordonequation . 52 1 Preface For ages, people have studied the motion of objects. These objects could be close to home, like marbles on a table, or far away, like planets and stars. -
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]. -
Space-Time Reference Systems
Space-Time reference systems Michael Soffel Dresden Technical University What are Space-Time Reference Systems (RS)? Science talks about events (e.g., observations): something that happens during a short interval of time in some small volume of space Both space and time are considered to be continous combined they are described by a ST manifold A Space-Time RS basically is a ST coordinate system (t,x) describing the ST position of events in a certain part of space-time In practice such a coordinate system has to be realized in nature with certain observations; the realization is then called the corresponding Reference Frame Newton‘s space and time In Newton‘s ST things are quite simple: Time is absolute as is Space -> there exists a class of preferred inertial coordinates (t, x) that have direct physical meaning, i.e., observables can be obtained directly from the coordinate picture of the physical system. Example: ∆τ = ∆t (proper) time coordinate time indicated by some clock Newton‘s absolute space Newtonian astrometry physically preferred global inertial coordinates observables are directly related to the inertial coordinates Example: observed angle θ between two incident light-rays 1 and 2 x (t) = x0 + c n (t – t ) ii i 0 cos θ = n. n 1 2 Is Newton‘s conception of Time and Space in accordance with nature? NO! The reason for that is related with properties of light propagation upon which time measurements are based Presently: The (SI) second is the duration of 9 192 631 770 periods of radiation that corresponds to a certain transition -
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. -
1 Temporal Arrows in Space-Time Temporality
Temporal Arrows in Space-Time Temporality (…) has nothing to do with mechanics. It has to do with statistical mechanics, thermodynamics (…).C. Rovelli, in Dieks, 2006, 35 Abstract The prevailing current of thought in both physics and philosophy is that relativistic space-time provides no means for the objective measurement of the passage of time. Kurt Gödel, for instance, denied the possibility of an objective lapse of time, both in the Special and the General theory of relativity. From this failure many writers have inferred that a static block universe is the only acceptable conceptual consequence of a four-dimensional world. The aim of this paper is to investigate how arrows of time could be measured objectively in space-time. In order to carry out this investigation it is proposed to consider both local and global arrows of time. In particular the investigation will focus on a) invariant thermodynamic parameters in both the Special and the General theory for local regions of space-time (passage of time); b) the evolution of the universe under appropriate boundary conditions for the whole of space-time (arrow of time), as envisaged in modern quantum cosmology. The upshot of this investigation is that a number of invariant physical indicators in space-time can be found, which would allow observers to measure the lapse of time and to infer both the existence of an objective passage and an arrow of time. Keywords Arrows of time; entropy; four-dimensional world; invariance; space-time; thermodynamics 1 I. Introduction Philosophical debates about the nature of space-time often centre on questions of its ontology, i.e.