General Relativity and Spatial Flows: I
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Advances in Quantum Field Theory
ADVANCES IN QUANTUM FIELD THEORY Edited by Sergey Ketov Advances in Quantum Field Theory Edited by Sergey Ketov Published by InTech Janeza Trdine 9, 51000 Rijeka, Croatia Copyright © 2012 InTech All chapters are Open Access distributed under the Creative Commons Attribution 3.0 license, which allows users to download, copy and build upon published articles even for commercial purposes, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. After this work has been published by InTech, authors have the right to republish it, in whole or part, in any publication of which they are the author, and to make other personal use of the work. Any republication, referencing or personal use of the work must explicitly identify the original source. As for readers, this license allows users to download, copy and build upon published chapters even for commercial purposes, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. Notice Statements and opinions expressed in the chapters are these of the individual contributors and not necessarily those of the editors or publisher. No responsibility is accepted for the accuracy of information contained in the published chapters. The publisher assumes no responsibility for any damage or injury to persons or property arising out of the use of any materials, instructions, methods or ideas contained in the book. Publishing Process Manager Romana Vukelic Technical Editor Teodora Smiljanic Cover Designer InTech Design Team First published February, 2012 Printed in Croatia A free online edition of this book is available at www.intechopen.com Additional hard copies can be obtained from [email protected] Advances in Quantum Field Theory, Edited by Sergey Ketov p. -
Relativistic Solutions 11.1 Free Particle Motion
Physics 411 Lecture 11 Relativistic Solutions Lecture 11 Physics 411 Classical Mechanics II September 21st, 2007 With our relativistic equations of motion, we can study the solutions for x(t) under a variety of different forces. The hallmark of a relativistic solution, as compared with a classical one, is the bound on velocity for massive particles. We shall see this in the context of a constant force, a spring force, and a one-dimensional Coulomb force. This tour of potential interactions leads us to the question of what types of force are even allowed { we will see changes in the dynamics of a particle, but what about the relativistic viability of the mechanism causing the forces? A real spring, for example, would break long before a mass on the end of it was accelerated to speeds nearing c. So we are thinking of the spring force, for example, as an approximation to a well in a more realistic potential. To the extent that effective potentials are uninteresting (or even allowed in general), we really only have one classical, relativistic force { the Lorentz force of electrodynamics. 11.1 Free Particle Motion For the equations of motion in proper time parametrization, we have x¨µ = 0 −! xµ = Aµ τ + Bµ: (11.1) Suppose we rewrite this solution in terms of t { inverting the x0 = c t equation with the initial conditions that t(τ = 0) = 0, we have c t τ = (11.2) A0 and then the rest of the equations read: c t c t c t x = A1 + B1 y = A2 + B2 z = A3 + B3; (11.3) A0 A0 A0 1 of 8 11.2. -
Physics/0409010V1 [Physics.Gen-Ph] 1 Sep 2004 Ri Ihteclsilshr Sntfound
Physical Interpretations of Relativity Theory Conference IX London, Imperial College, September, 2004 ............... Mach’s Principle II by James G. Gilson∗ School of Mathematical Sciences Queen Mary College University of London October 29, 2018 Abstract 1 Introduction The question of the validity or otherwise of Mach’s The meaning and significance of Mach’s Principle Principle[5] has been with us for almost a century and and its dependence on ideas about relativistic rotat- has been vigorously analysed and discussed for all of ing frame theory and the celestial sphere is explained that time and is, up to date, still not resolved. One and discussed. Two new relativistic rotation trans- contentious issue is, does general[7] relativity theory formations are introduced by using a linear simula- imply that Mach’s Principle is operative in that theo- tion for the rotating disc situation. The accepted retical structure or does it not? A deduction[4] from formula for centrifugal acceleration in general rela- general relativity that the classical Newtonian cen- tivity is then analysed with the use of one of these trifugal acceleration formula should be modified by transformations. It is shown that for this general rel- an additional relativistic γ2(v) factor, see equations ativity formula to be valid throughout all space-time (2.11) and (2.27), will help us in the analysis of that there has to be everywhere a local standard of ab- question. Here I shall attempt to throw some light solutely zero rotation. It is then concluded that the on the nature and technical basis of the collection of field off all possible space-time null geodesics or pho- arXiv:physics/0409010v1 [physics.gen-ph] 1 Sep 2004 technical and philosophical problems generated from ton paths unify the absolute local non-rotation stan- Mach’s Principle and give some suggestions about dard throughout space-time. -
On Relativistic Theory of Spinning and Deformable Particles
On relativistic theory of spinning and deformable particles A.N. Tarakanov ∗ Minsk State High Radiotechnical College Independence Avenue 62, 220005, Minsk, Belarus Abstract A model of relativistic extended particle is considered with the help of gener- alization of space-time interval. Ten additional dimensions are connected with six rotational and four deformational degrees of freedom. An obtained 14-dimensional space is assumed to be an embedding one both for usual space-time and for 10- dimensional internal space of rotational and deformational variables. To describe such an internal space relativistic generalizations of inertia and deformation tensors are given. Independence of internal and external motions from each other gives rise to splitting the equation of motion and some conditions for 14-dimensional metric. Using the 14-dimensional ideology makes possible to assign a unique proper time for all points of extended object, if the metric will be degenerate. Properties of an internal space are discussed in details in the case of absence of spatial rotations. arXiv:hep-th/0703159v1 17 Mar 2007 1 Introduction More and more attention is spared to relativistic description of extended objects, which could serve a basis for construction of dynamics of interacting particles. Necessity of introduction extended objects to elementary particle theory is out of doubt. Therefore, since H.A.Lorentz attempts to introduce particles of finite size were undertaken. However a relativization of extended body is found prove to be a difficult problem as at once there was a contradiction to Einstein’s relativity principle. Even for simplest model of absolutely rigid body [1] it is impossible for all points of a body to attribute the same proper time. -
Arxiv:1007.1861V1 [Gr-Qc] 12 Jul 2010 Feulms N Hp.Temaueeto Uhagravito-Mag a Such of Measurement 10 the Is Shape
A laser gyroscope system to detect the Gravito-Magnetic effect on Earth A. Di Virgilio∗ INFN Sez. di Pisa, Pisa, Italy K. U. Schreiber and A. Gebauer† Technische Universitaet Muenchen, Forschungseinrichtung Satellitengeodaesie Fundamentalstation Wettzell, 93444 Bad K¨otzting, Germany J-P. R. Wells‡ Department of Physics and Astronomy, University of Canterbury, PB4800, Christchurch 8020, New Zealand A. Tartaglia§ Polit. of Torino and INFN, Torino, Italy J. Belfi and N. Beverini¶ Univ. of Pisa and CNISM, Pisa, Italy A.Ortolan∗∗ Laboratori Nazionali di Legnaro, INFN Legnaro (Padova), Italy Large scale square ring laser gyros with a length of four meters on each side are approaching a sensitivity of 1 10−11rad/s/√Hz. This is about the regime required to measure the gravito- magnetic effect (Lense× Thirring) of the Earth. For an ensemble of linearly independent gyros each measurement signal depends upon the orientation of each single axis gyro with respect to the rotational axis of the Earth. Therefore at least 3 gyros are necessary to reconstruct the complete angular orientation of the apparatus. In general, the setup consists of several laser gyroscopes (we would prefer more than 3 for sufficient redundancy), rigidly referenced to each other. Adding more gyros for one plane of observation provides a cross-check against intra-system biases and furthermore has the advantage of improving the signal to noise ratio by the square root of the number of gyros. In this paper we analyze a system of two pairs of identical gyros (twins) with a slightly different orientation with respect to the Earth axis. The twin gyro configuration has several interesting properties. -
An Essay on the Coriolis Force
James F. Price Woods Hole Oceanographic Institution Woods Hole, Massachusetts, 02543 http:// www.whoi.edu/science/PO/people/jprice [email protected] Version 3.3 January 10, 2006 Summary: An Earth-attached and thus rotating reference frame is almost always used for the analysis of geophysical flows. The equation of motion transformed into a steadily rotating reference frame includes two terms that involve the rotation vector; a centrifugal term and a Coriolis term. In the special case of an Earth-attached reference frame, the centrifugal term is exactly canceled by gravitational mass attraction and drops out of the equation of motion. When we solve for the acceleration seen from an Earth-attached frame, the Coriolis term is interpreted as a force. The rotating frame perspective gives up the properties of global momentum conservation and invariance to Galilean transformation. Nevertheless, it leads to a greatly simplified analysis of geophysical flows since only the comparatively small relative velocity, i.e., winds and currents, need be considered. The Coriolis force has a simple mathematical form, 2˝ V 0M , where ˝ is Earth’s rotation vector, V 0 is the velocity observed from the rotating frame and M is the particle mass. The Coriolis force is perpendicular to the velocity and can do no work. It tends to cause a deflection of velocity, and gives rise to two important modes of motion: (1) If the Coriolis force is the only force acting on a moving particle, then the velocity vector of the particle will be continually deflected and rotate clockwise in the northern hemisphere and anticlockwise in the southern hemisphere. -
Relativistic Thermodynamics
C. MØLLER RELATIVISTIC THERMODYNAMICS A Strange Incident in the History of Physics Det Kongelige Danske Videnskabernes Selskab Matematisk-fysiske Meddelelser 36, 1 Kommissionær: Munksgaard København 1967 Synopsis In view of the confusion which has arisen in the later years regarding the correct formulation of relativistic thermodynamics, the case of arbitrary reversible and irreversible thermodynamic processes in a fluid is reconsidered from the point of view of observers in different systems of inertia. Although the total momentum and energy of the fluid do not transform as the components of a 4-vector in this case, it is shown that the momentum and energy of the heat supplied in any process form a 4-vector. For reversible processes this four-momentum of supplied heat is shown to be proportional to the four-velocity of the matter, which leads to Otts transformation formula for the temperature in contrast to the old for- mula of Planck. PRINTED IN DENMARK BIANCO LUNOS BOGTRYKKERI A-S Introduction n the years following Einsteins fundamental paper from 1905, in which I he founded the theory of relativity, physicists were engaged in reformu- lating the classical laws of physics in order to bring them in accordance with the (special) principle of relativity. According to this principle the fundamental laws of physics must have the same form in all Lorentz systems of coordinates or, more precisely, they must be expressed by equations which are form-invariant under Lorentz transformations. In some cases, like in the case of Maxwells equations, these laws had already the appropriate form, in other cases, they had to be slightly changed in order to make them covariant under Lorentz transformations. -
The Dynamical Velocity Superposition Effect in the Quantum-Foam In
The Dynamical Velocity Superposition Effect in the Quantum-Foam In-Flow Theory of Gravity Reginald T. Cahill School of Chemistry, Physics and Earth Sciences Flinders University GPO Box 2100, Adelaide 5001, Australia (Reg.Cahill@flinders.edu.au) arXiv:physics/0407133 July 26, 2004 To be published in Relativity, Gravitation, Cosmology Abstract The new ‘quantum-foam in-flow’ theory of gravity has explained numerous so-called gravitational anomalies, particularly the ‘dark matter’ effect which is now seen to be a dynamical effect of space itself, and whose strength is determined by the fine structure constant, and not by Newton’s gravitational constant G. Here we show an experimen- tally significant approximate dynamical effect, namely a vector superposition effect which arises under certain dynamical conditions when we have absolute motion and gravitational in-flows: the velocities for these processes are shown to be approximately vectorially additive under these conditions. This effect plays a key role in interpreting the data from the numerous experiments that detected the absolute linear motion of the earth. The violations of this superposition effect lead to observable effects, such as the generation of turbulence. The flow theory also leads to vorticity effects that the Grav- ity Probe B gyroscope experiment will soon begin observing. As previously reported General Relativity predicts a smaller vorticity effect (therein called the Lense-Thirring ‘frame-dragging’ effect) than the new theory of gravity. arXiv:physics/0407133v3 [physics.gen-ph] 10 Nov -
Kepler's Orbits and Special Relativity in Introductory Classical Mechanics
Kepler's Orbits and Special Relativity in Introductory Classical Mechanics Tyler J. Lemmon∗ and Antonio R. Mondragony (Dated: April 21, 2016) Kepler's orbits with corrections due to Special Relativity are explored using the Lagrangian formalism. A very simple model includes only relativistic kinetic energy by defining a Lagrangian that is consistent with both the relativistic momentum of Special Relativity and Newtonian gravity. The corresponding equations of motion are solved in a Keplerian limit, resulting in an approximate relativistic orbit equation that has the same form as that derived from General Relativity in the same limit and clearly describes three characteristics of relativistic Keplerian orbits: precession of perihelion; reduced radius of circular orbit; and increased eccentricity. The prediction for the rate of precession of perihelion is in agreement with established calculations using only Special Relativity. All three characteristics are qualitatively correct, though suppressed when compared to more accurate general-relativistic calculations. This model is improved upon by including relativistic gravitational potential energy. The resulting approximate relativistic orbit equation has the same form and symmetry as that derived using the very simple model, and more accurately describes characteristics of relativistic orbits. For example, the prediction for the rate of precession of perihelion of Mercury is one-third that derived from General Relativity. These Lagrangian formulations of the special-relativistic Kepler problem are equivalent to the familiar vector calculus formulations. In this Keplerian limit, these models are supposed to be physical based on the likeness of the equations of motion to those derived using General Relativity. The resulting approximate relativistic orbit equations are useful for a qualitative understanding of general-relativistic corrections to Keplerian orbits. -
The Curious Case of Sherlock Holmes and Albert Einstein
Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3787493/9781560803140_frontmatter.pdf by guest on 28 September 2021 GEOPHYSICAL MONOGRAPH SERIES NUMBER 18 REMOTE SENSING IN ACTION: THE CURIOUS CASE OF SHERLOCK HOLMES AND ALBERT EINSTEIN Enders A. Robinson Dean Clark Rebecca B. Latimer, managing editor Joel Greer, volume editor Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3787493/9781560803140_frontmatter.pdf by guest on 28 September 2021 0510_01_FM_v4--SCS.indd 1 8/15/2014 10:50:08 AM ISBN 978-0-931830-56-3 (Series) ISBN 978-1-56080-313-3 (Volume) Society of Exploration Geophysicists P.O. Box 702740 Tulsa, OK 74170-2740 © 2014 by Society of Exploration Geophysicists All rights reserved. This book or parts hereof may not be reproduced in any form without written permission from the publisher. Published 2014 Printed in the United States of America Grateful acknowledgment to Conan Doyle Estate Ltd. for permission to use the Sherlock Holmes characters created by the late sir Arthur Conan Doyle. Library of Congress Cataloging-in-Publication Data Robinson, Enders A. Remote sensing in action : the curious case of Sherlock Holmes and Albert Einstein / Enders A. Robinson, Dean Clark ; Rebecca B. Latimer, managing editor ; Joel Greer, volume editor. pages cm. -- (Geophysical monograph series ; number 18) Includes index. ISBN 978-1-56080-313-3 (pbk.) -- ISBN 978-0-931830-56-3 (series) 1. Remote sensing. 2. Geophysics--Remote sensing. 3. General relativ- ity (Physics) 4. Holmes, Sherlock. 5. Einstein, Albert, 1879-1955. I. Clark, Dean, 1944- II. Latimer, Rebecca B., editor. III. Title. G70.4.R63 2014 550.28’7--dc23 2014017009 Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3787493/9781560803140_frontmatter.pdf by guest on 28 September 2021 0510_01_FM_v4--SCS.indd 2 8/15/2014 10:50:08 AM Contents About the authors .......................................iv Preface ...............................................viii Acknowledgments. -
Introduction to Relativistic Mechanics and the Concept of Mass
Introduction to Relativistic Mechanics and the Concept of Mass Gron Tudor Jones CERN HST2014 University of Birmingham Introduction to relativistic kinematics and the concept of mass Mass is one of the most fundamental concepts in physics. When a new particle is discovered (e.g. the Higgs boson), the first question physicists will ask is, ‘What is its mass?’ Classical physics (v << c) T = mv2/2 m = 2T/v2 Knowing any 2 of T, p and v, p = mv m = p/v one can calculate m. 2 T = p2/2m m = p /2T Same is true in relativity but we need the generalised formulae. Before that: a brief discussion of ‘E = mc2’ Einstein’s equation: ‘E = mc2’ 2 2 2 2 E0 = m c E = m c E0 = m0 c E = m0 c where c = velocity of light in vacuo E = total energy of free body E0 = rest energy of free body m0 = rest mass m = mass Q1: Which equation most rationally follows from special relativity and expresses one of its main consequences and predictions? Q2: Which of these equations was first written by Einstein and was considered by him a consequence of special relativity? 2 The correct answer to both questions is: E0 = mc (Poll carried out by Lev Okun among professional physicists in 1980s showed that the majority preferred 2 or 3 as the answer to both questions.) ‘This choice is caused by the confusing terminology widely used in the popular science literature and in many textbooks. According to this terminology a body at rest has a proper mass or rest mass m0, whereas a body moving with speed v has a relativistic mass or mass m, given by E m0 m 2 c v2 1 c2 ‘ … this terminology had some historical justification at the start of our century, but it has no justification today. -
Relativistic Mechanics
Relativistic mechanics Further information: Mass in special relativity and relativistic center of mass for details. Conservation of energy The equations become more complicated in the more fa- miliar three-dimensional vector calculus formalism, due In physics, relativistic mechanics refers to mechanics to the nonlinearity in the Lorentz factor, which accu- compatible with special relativity (SR) and general rel- rately accounts for relativistic velocity dependence and ativity (GR). It provides a non-quantum mechanical de- the speed limit of all particles and fields. However, scription of a system of particles, or of a fluid, in cases they have a simpler and elegant form in four-dimensional where the velocities of moving objects are comparable spacetime, which includes flat Minkowski space (SR) and to the speed of light c. As a result, classical mechanics curved spacetime (GR), because three-dimensional vec- is extended correctly to particles traveling at high veloc- tors derived from space and scalars derived from time ities and energies, and provides a consistent inclusion of can be collected into four vectors, or four-dimensional electromagnetism with the mechanics of particles. This tensors. However, the six component angular momentum was not possible in Galilean relativity, where it would be tensor is sometimes called a bivector because in the 3D permitted for particles and light to travel at any speed, in- viewpoint it is two vectors (one of these, the conventional cluding faster than light. The foundations of relativistic angular momentum, being an axial vector). mechanics are the postulates of special relativity and gen- eral relativity. The unification of SR with quantum me- chanics is relativistic quantum mechanics, while attempts 1 Relativistic kinematics for that of GR is quantum gravity, an unsolved problem in physics.