Proper-Velocity and F≤Mα from the Metric
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Quantum Phase Space in Relativistic Theory: the Case of Charge-Invariant Observables
Proceedings of Institute of Mathematics of NAS of Ukraine 2004, Vol. 50, Part 3, 1448–1453 Quantum Phase Space in Relativistic Theory: the Case of Charge-Invariant Observables A.A. SEMENOV †, B.I. LEV † and C.V. USENKO ‡ † Institute of Physics of NAS of Ukraine, 46 Nauky Ave., 03028 Kyiv, Ukraine E-mail: [email protected], [email protected] ‡ Physics Department, Taras Shevchenko Kyiv University, 6 Academician Glushkov Ave., 03127 Kyiv, Ukraine E-mail: [email protected] Mathematical method of quantum phase space is very useful in physical applications like quantum optics and non-relativistic quantum mechanics. However, attempts to generalize it for the relativistic case lead to some difficulties. One of problems is band structure of energy spectrum for a relativistic particle. This corresponds to an internal degree of freedom, so- called charge variable. In physical problems we often deal with such dynamical variables that do not depend on this degree of freedom. These are position, momentum, and any combination of them. Restricting our consideration to this kind of observables we propose the relativistic Weyl–Wigner–Moyal formalism that contains some surprising differences from its non-relativistic counterpart. This paper is devoted to the phase space formalism that is specific representation of quan- tum mechanics. This representation is very close to classical mechanics and its basic idea is a description of quantum observables by means of functions in phase space (symbols) instead of operators in the Hilbert space of states. The first idea about this representation has been proposed in the early days of quantum mechanics in the well-known Weyl work [1]. -
Spacetime Diagrams(1D in Space)
PH300 Modern Physics SP11 Last time: • Time dilation and length contraction Today: • Spacetime • Addition of velocities • Lorentz transformations Thursday: • Relativistic momentum and energy “The only reason for time is so that HW03 due, beginning of class; HW04 assigned everything doesn’t happen at once.” 2/1 Day 6: Next week: - Albert Einstein Questions? Intro to quantum Spacetime Thursday: Exam I (in class) Addition of Velocities Relativistic Momentum & Energy Lorentz Transformations 1 2 Spacetime Diagrams (1D in space) Spacetime Diagrams (1D in space) c · t In PHYS I: v In PH300: x x x x Δx Δx v = /Δt Δt t t Recall: Lucy plays with a fire cracker in the train. (1D in space) Spacetime Diagrams Ricky watches the scene from the track. c· t In PH300: object moving with 0<v<c. ‘Worldline’ of the object L R -2 -1 0 1 2 x object moving with 0>v>-c v c·t c·t Lucy object at rest object moving with v = -c. at x=1 x=0 at time t=0 -2 -1 0 1 2 x -2 -1 0 1 2 x Ricky 1 Example: Ricky on the tracks Example: Lucy in the train ct ct Light reaches both walls at the same time. Light travels to both walls Ricky concludes: Light reaches left side first. x x L R L R Lucy concludes: Light reaches both sides at the same time In Ricky’s frame: Walls are in motion In Lucy’s frame: Walls are at rest S Frame S’ as viewed from S ... -3 -2 -1 0 1 2 3 .. -
Reflection Invariant and Symmetry Detection
1 Reflection Invariant and Symmetry Detection Erbo Li and Hua Li Abstract—Symmetry detection and discrimination are of fundamental meaning in science, technology, and engineering. This paper introduces reflection invariants and defines the directional moments(DMs) to detect symmetry for shape analysis and object recognition. And it demonstrates that detection of reflection symmetry can be done in a simple way by solving a trigonometric system derived from the DMs, and discrimination of reflection symmetry can be achieved by application of the reflection invariants in 2D and 3D. Rotation symmetry can also be determined based on that. Also, if none of reflection invariants is equal to zero, then there is no symmetry. And the experiments in 2D and 3D show that all the reflection lines or planes can be deterministically found using DMs up to order six. This result can be used to simplify the efforts of symmetry detection in research areas,such as protein structure, model retrieval, reverse engineering, and machine vision etc. Index Terms—symmetry detection, shape analysis, object recognition, directional moment, moment invariant, isometry, congruent, reflection, chirality, rotation F 1 INTRODUCTION Kazhdan et al. [1] developed a continuous measure and dis- The essence of geometric symmetry is self-evident, which cussed the properties of the reflective symmetry descriptor, can be found everywhere in nature and social lives, as which was expanded to 3D by [2] and was augmented in shown in Figure 1. It is true that we are living in a spatial distribution of the objects asymmetry by [3] . For symmetric world. Pursuing the explanation of symmetry symmetry discrimination [4] defined a symmetry distance will provide better understanding to the surrounding world of shapes. -
From Relativistic Time Dilation to Psychological Time Perception
From relativistic time dilation to psychological time perception: an approach and model, driven by the theory of relativity, to combine the physical time with the time perceived while experiencing different situations. Andrea Conte1,∗ Abstract An approach, supported by a physical model driven by the theory of relativity, is presented. This approach and model tend to conciliate the relativistic view on time dilation with the current models and conclusions on time perception. The model uses energy ratios instead of geometrical transformations to express time dilation. Brain mechanisms like the arousal mechanism and the attention mechanism are interpreted and combined using the model. Matrices of order two are generated to contain the time dilation between two observers, from the point of view of a third observer. The matrices are used to transform an observer time to another observer time. Correlations with the official time dilation equations are given in the appendix. Keywords: Time dilation, Time perception, Definition of time, Lorentz factor, Relativity, Physical time, Psychological time, Psychology of time, Internal clock, Arousal, Attention, Subjective time, Internal flux, External flux, Energy system ∗Corresponding author Email address: [email protected] (Andrea Conte) 1Declarations of interest: none Preprint submitted to PsyArXiv - version 2, revision 1 June 6, 2021 Contents 1 Introduction 3 1.1 The unit of time . 4 1.2 The Lorentz factor . 6 2 Physical model 7 2.1 Energy system . 7 2.2 Internal flux . 7 2.3 Internal flux ratio . 9 2.4 Non-isolated system interaction . 10 2.5 External flux . 11 2.6 External flux ratio . 12 2.7 Total flux . -
Arxiv:1401.0181V1 [Gr-Qc] 31 Dec 2013 † Isadpyia Infiac Fteenwcodntsaediscussed
Painlev´e-Gullstrand-type coordinates for the five-dimensional Myers-Perry black hole Tehani K. Finch† NASA Goddard Space Flight Center Greenbelt MD 20771 ABSTRACT The Painlev´e-Gullstrand coordinates provide a convenient framework for pre- senting the Schwarzschild geometry because of their flat constant-time hyper- surfaces, and the fact that they are free of coordinate singularities outside r=0. Generalizations of Painlev´e-Gullstrand coordinates suitable for the Kerr geome- try have been presented by Doran and Nat´ario. These coordinate systems feature a time coordinate identical to the proper time of zero-angular-momentum ob- servers that are dropped from infinity. Here, the methods of Doran and Nat´ario arXiv:1401.0181v1 [gr-qc] 31 Dec 2013 are extended to the five-dimensional rotating black hole found by Myers and Perry. The result is a new formulation of the Myers-Perry metric. The proper- ties and physical significance of these new coordinates are discussed. † tehani.k.finch (at) nasa.gov 1 Introduction By using the Birkhoff theorem, the Schwarzschild geometry has been shown to be the unique vacuum spherically symmetric solution of the four-dimensional Einstein equations. The Kerr geometry, on the other hand, has been shown only to be the unique stationary, rotating vacuum black hole solution of the four-dimensional Einstein equations. No distri- bution of matter is currently known to produce a Kerr exterior. Thus the Kerr geometry does not necessarily correspond to the spacetime outside a rotating star or planet [1]. This is an indication of the complications encountered upon trying to extend results found for the Schwarzschild spacetime to the Kerr spacetime. -
Physics 325: General Relativity Spring 2019 Problem Set 2
Physics 325: General Relativity Spring 2019 Problem Set 2 Due: Fri 8 Feb 2019. Reading: Please skim Chapter 3 in Hartle. Much of this should be review, but probably not all of it|be sure to read Box 3.2 on Mach's principle. Then start on Chapter 6. Problems: 1. Spacetime interval. Hartle Problem 4.13. 2. Four-vectors. Hartle Problem 5.1. 3. Lorentz transformations and hyperbolic geometry. In class, we saw that a Lorentz α0 α β transformation in 2D can be written as a = L β(#)a , that is, 0 ! ! ! a0 cosh # − sinh # a0 = ; (1) a10 − sinh # cosh # a1 where a is spacetime vector. Here, the rapidity # is given by tanh # = β; cosh # = γ; sinh # = γβ; (2) where v = βc is the velocity of frame S0 relative to frame S. (a) Show that two successive Lorentz boosts of rapidity #1 and #2 are equivalent to a single α γ α Lorentz boost of rapidity #1 +#2. In other words, check that L γ(#1)L(#2) β = L β(#1 +#2), α where L β(#) is the matrix in Eq. (1). You will need the following hyperbolic trigonometry identities: cosh(#1 + #2) = cosh #1 cosh #2 + sinh #1 sinh #2; (3) sinh(#1 + #2) = sinh #1 cosh #2 + cosh #1 sinh #2: (b) From Eq. (3), deduce the formula for tanh(#1 + #2) in terms of tanh #1 and tanh #2. For the appropriate choice of #1 and #2, use this formula to derive the special relativistic velocity tranformation rule V − v V 0 = : (4) 1 − vV=c2 Physics 325, Spring 2019: Problem Set 2 p. -
Simulating Gamma-Ray Binaries with a Relativistic Extension of RAMSES? A
A&A 560, A79 (2013) Astronomy DOI: 10.1051/0004-6361/201322266 & c ESO 2013 Astrophysics Simulating gamma-ray binaries with a relativistic extension of RAMSES? A. Lamberts1;2, S. Fromang3, G. Dubus1, and R. Teyssier3;4 1 UJF-Grenoble 1 / CNRS-INSU, Institut de Planétologie et d’Astrophysique de Grenoble (IPAG) UMR 5274, 38041 Grenoble, France e-mail: [email protected] 2 Physics Department, University of Wisconsin-Milwaukee, Milwaukee WI 53211, USA 3 Laboratoire AIM, CEA/DSM - CNRS - Université Paris 7, Irfu/Service d’Astrophysique, CEA-Saclay, 91191 Gif-sur-Yvette, France 4 Institute for Theoretical Physics, University of Zürich, Winterthurestrasse 190, 8057 Zürich, Switzerland Received 11 July 2013 / Accepted 29 September 2013 ABSTRACT Context. Gamma-ray binaries are composed of a massive star and a rotation-powered pulsar with a highly relativistic wind. The collision between the winds from both objects creates a shock structure where particles are accelerated, which results in the observed high-energy emission. Aims. We want to understand the impact of the relativistic nature of the pulsar wind on the structure and stability of the colliding wind region and highlight the differences with colliding winds from massive stars. We focus on how the structure evolves with increasing values of the Lorentz factor of the pulsar wind, keeping in mind that current simulations are unable to reach the expected values of pulsar wind Lorentz factors by orders of magnitude. Methods. We use high-resolution numerical simulations with a relativistic extension to the hydrodynamics code RAMSES we have developed. We perform two-dimensional simulations, and focus on the region close to the binary, where orbital motion can be ne- glected. -
Doppler Boosting and Cosmological Redshift on Relativistic Jets
Doppler Boosting and cosmological redshift on relativistic jets Author: Jordi Fernández Vilana Advisor: Valentí Bosch i Ramon Facultat de Física, Universitat de Barcelona, Diagonal 645, 08028 Barcelona, Spain. Abstract: In this study our goal was to identify the possible effects that influence with the Spectral Energy Distribution of a blazar, which we deducted that are Doppler Boosting effect and cosmological redshift. Then we investigated how the spectral energy distribution varies by changing intrinsic parameters of the jet like its Lorentz factor, the angle of the line of sight respect to the jet orientation or the blazar redshift. The obtained results showed a strong enhancing of the Spectral Energy Distribution for nearly 0º angles and high Lorentz factors, while the increase in redshift produced the inverse effect, reducing the normalization of the distribution and moving the peak to lower energies. These two effects compete in the observations of all known blazars and our simple model confirmed the experimental results, that only those blazars with optimal characteristics, if at very high redshift, are suitable to be measured by the actual instruments, making the study of blazars with a high redshift an issue that needs very deep observations at different wavelengths to collect enough data to properly characterize the source population. jet those particles are moving at relativistic speeds, producing I. INTRODUCTION high energy interactions in the process. This scenario produces a wide spectrum of radiation that mainly comes, as we can see in more detail in [2], from Inverse Compton We know that active galactic nuclei or AGN are galaxies scattering. Low energy photons inside the jet are scattered by with an accreting supermassive blackhole in the centre. -
(Special) Relativity
(Special) Relativity With very strong emphasis on electrodynamics and accelerators Better: How can we deal with moving charged particles ? Werner Herr, CERN Reading Material [1 ]R.P. Feynman, Feynman lectures on Physics, Vol. 1 + 2, (Basic Books, 2011). [2 ]A. Einstein, Zur Elektrodynamik bewegter K¨orper, Ann. Phys. 17, (1905). [3 ]L. Landau, E. Lifschitz, The Classical Theory of Fields, Vol2. (Butterworth-Heinemann, 1975) [4 ]J. Freund, Special Relativity, (World Scientific, 2008). [5 ]J.D. Jackson, Classical Electrodynamics (Wiley, 1998 ..) [6 ]J. Hafele and R. Keating, Science 177, (1972) 166. Why Special Relativity ? We have to deal with moving charges in accelerators Electromagnetism and fundamental laws of classical mechanics show inconsistencies Ad hoc introduction of Lorentz force Applied to moving bodies Maxwell’s equations lead to asymmetries [2] not shown in observations of electromagnetic phenomena Classical EM-theory not consistent with Quantum theory Important for beam dynamics and machine design: Longitudinal dynamics (e.g. transition, ...) Collective effects (e.g. space charge, beam-beam, ...) Dynamics and luminosity in colliders Particle lifetime and decay (e.g. µ, π, Z0, Higgs, ...) Synchrotron radiation and light sources ... We need a formalism to get all that ! OUTLINE Principle of Relativity (Newton, Galilei) - Motivation, Ideas and Terminology - Formalism, Examples Principle of Special Relativity (Einstein) - Postulates, Formalism and Consequences - Four-vectors and applications (Electromagnetism and accelerators) § ¤ some slides are for your private study and pleasure and I shall go fast there ¦ ¥ Enjoy yourself .. Setting the scene (terminology) .. To describe an observation and physics laws we use: - Space coordinates: ~x = (x, y, z) (not necessarily Cartesian) - Time: t What is a ”Frame”: - Where we observe physical phenomena and properties as function of their position ~x and time t. -
Basic Four-Momentum Kinematics As
L4:1 Basic four-momentum kinematics Rindler: Ch5: sec. 25-30, 32 Last time we intruduced the contravariant 4-vector HUB, (II.6-)II.7, p142-146 +part of I.9-1.10, 154-162 vector The world is inconsistent! and the covariant 4-vector component as implicit sum over We also introduced the scalar product For a 4-vector square we have thus spacelike timelike lightlike Today we will introduce some useful 4-vectors, but rst we introduce the proper time, which is simply the time percieved in an intertial frame (i.e. time by a clock moving with observer) If the observer is at rest, then only the time component changes but all observers agree on ✁S, therefore we have for an observer at constant speed L4:2 For a general world line, corresponding to an accelerating observer, we have Using this it makes sense to de ne the 4-velocity As transforms as a contravariant 4-vector and as a scalar indeed transforms as a contravariant 4-vector, so the notation makes sense! We also introduce the 4-acceleration Let's calculate the 4-velocity: and the 4-velocity square Multiplying the 4-velocity with the mass we get the 4-momentum Note: In Rindler m is called m and Rindler's I will always mean with . which transforms as, i.e. is, a contravariant 4-vector. Remark: in some (old) literature the factor is referred to as the relativistic mass or relativistic inertial mass. L4:3 The spatial components of the 4-momentum is the relativistic 3-momentum or simply relativistic momentum and the 0-component turns out to give the energy: Remark: Taylor expanding for small v we get: rest energy nonrelativistic kinetic energy for v=0 nonrelativistic momentum For the 4-momentum square we have: As you may expect we have conservation of 4-momentum, i.e. -
Lecture Notes 17: Proper Time, Proper Velocity, the Energy-Momentum 4-Vector, Relativistic Kinematics, Elastic/Inelastic
UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 17 Prof. Steven Errede LECTURE NOTES 17 Proper Time and Proper Velocity As you progress along your world line {moving with “ordinary” velocity u in lab frame IRF(S)} on the ct vs. x Minkowski/space-time diagram, your watch runs slow {in your rest frame IRF(S')} in comparison to clocks on the wall in the lab frame IRF(S). The clocks on the wall in the lab frame IRF(S) tick off a time interval dt, whereas in your 2 rest frame IRF( S ) the time interval is: dt dtuu1 dt n.b. this is the exact same time dilation formula that we obtained earlier, with: 2 2 uu11uc 11 and: u uc We use uurelative speed of an object as observed in an inertial reference frame {here, u = speed of you, as observed in the lab IRF(S)}. We will henceforth use vvrelative speed between two inertial systems – e.g. IRF( S ) relative to IRF(S): Because the time interval dt occurs in your rest frame IRF( S ), we give it a special name: ddt = proper time interval (in your rest frame), and: t = proper time (in your rest frame). The name “proper” is due to a mis-translation of the French word “propre”, meaning “own”. Proper time is different than “ordinary” time, t. Proper time is a Lorentz-invariant quantity, whereas “ordinary” time t depends on the choice of IRF - i.e. “ordinary” time is not a Lorentz-invariant quantity. 222222 The Lorentz-invariant interval: dI dx dx dx dx ds c dt dx dy dz Proper time interval: d dI c2222222 ds c dt dx dy dz cdtdt22 = 0 in rest frame IRF(S) 22t Proper time: ddtttt 21 t 21 11 Because d and are Lorentz-invariant quantities: dd and: {i.e. -
Newton's Laws
Newton’s Laws First Law A body moves with constant velocity unless a net force acts on the body. Second Law The rate of change of momentum of a body is equal to the net force applied to the body. Third Law If object A exerts a force on object B, then object B exerts a force on object A. These have equal magnitude but opposite direction. Newton’s second law The second law should be familiar: F = ma where m is the inertial mass (a scalar) and a is the acceleration (a vector). Note that a is the rate of change of velocity, which is in turn the rate of change of displacement. So d d2 a = v = r dt dt2 which, in simplied notation is a = v_ = r¨ The principle of relativity The principle of relativity The laws of nature are identical in all inertial frames of reference An inertial frame of reference is one in which a freely moving body proceeds with uniform velocity. The Galilean transformation - In Newtonian mechanics, the concepts of space and time are completely separable. - Time is considered an absolute quantity which is independent of the frame of reference: t0 = t. - The laws of mechanics are invariant under a transformation of the coordinate system. u y y 0 S S 0 x x0 Consider two inertial reference frames S and S0. The frame S0 moves at a velocity u relative to the frame S along the x axis. The trans- formation of the coordinates of a point is, therefore x0 = x − ut y 0 = y z0 = z The above equations constitute a Galilean transformation u y y 0 S S 0 x x0 These equations are linear (as we would hope), so we can write the same equations for changes in each of the coordinates: ∆x0 = ∆x − u∆t ∆y 0 = ∆y ∆z0 = ∆z u y y 0 S S 0 x x0 For moving particles, we need to know how to transform velocity, v, and acceleration, a, between frames.