DOCSLIB.ORG

General Relativity Derivation of Beam Rest-Frame Hamiltonian

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
General Relativity Derivation of Beam Rest-Frame Hamiltonian Proceedings of the 2001 Particle Accelerator Conference, Chicago GENERAL RELATIVITY DERIVATION OF BEAM REST-FRAME HAMILTONIAN £ J. Wei Ý , Brookhaven National Laboratory, New York, USA Ù = dÜ Abstract where =dØ are the contravariant components of the spatial velocity, the Greek letter indicates the three spatial Analysis of particle interaction in the laboratory frame of = d c components, dØ , is the speed of light, and storage rings is often complicated by the fact that particle ½ µ motion is relativistic, and that reference particle trajectory ´ ¾ ¾ ¾ ½ g Ù Ù 4 is curved. Rest frame of the reference particle is a conve- ¾ Ô = ´ g µ 44 (1) ¾ c g c nient coordinate system to work with, within which particle 44 motion is non-relativistic. We have derived the equations of motion in the beam rest frame from the general relativ- is the generalized Lorentz factor. The contravariant com- i i È = Ñ Ñ Í ¼ ity formalism, and have successfully applied them to the ponents of the four-momentum are ¼ , where analysis of crystalline beams [1]. is the proper mass of the particle. fF g Acted upon by a non-gravitational four-force i , the 1 DERIVATION equations of motion of the particle can be written as The motion of charged particles under Coulomb interac- i DÈ ik i = g F ; F Í ¼; i tion and external electromagnetic (EM) forces can be most k (2) conveniently described in the rotating rest frame of the ref- d ij 4 ¢ 4 fg g fg g erence particle of which the orientation of the axes are con- where the matrix is the inverse of ij , and stantly aligned to the radial, tangential, and vertical direc- the covariant differentiation is defined as tion of the motion. In this frame, particle motion within the i ÐÑ i dÈ @g @g g DÈ @g Ñk ik Ñi Ð k Ð Ð · Í · È ; beam bunch is non-relativistic. = ik ik k i Ñ d ¾ @Ü @Ü @Ü We derive the equations of motion using the general rel- d (3) ativity formalism. First, we express the equations of mo- Among the non-gravitational forces, the external EM force tion in a general tensor formalism. The Lorentz force ex- acting upon the particles is expressed by means of the EM perienced by the particle is constructed as a product of the fF g field tensor ij as EM field tensor and the four-velocity. Starting from the e laboratory frame, the EM field tensor is written by means k F = F Í ; i ik (4) of the components of the EM fields. Then, tensor algebra c is used to transform this field tensor into the rest frame. e fF g where is the electric charge, and ij is anti-symmetric. With a similar transformation, the metric tensor of the rest The Maxwell’s equations are given by frame is also obtained. The equations of motion can thus 8 @F @F be constructed in the rest frame which include centrifugal @F ik kÐ Ði > > · · =¼; < i k force, Coriolis force, time-dependent external EM forces, Ð @Ü @Ü @Ü i Ô (5) @ ½ Í ¼ and electrostatic Coulomb forces. Finally, these equations ik > Ô > ´ jg jF ; µ= : i c are re-scaled in terms of dimensionless quantities for the @Ü jg j convenience of computer simulation and analysis. ij iÐ jÑ F = g g F jg j where ÐÑ , is the absolute value of the de- fg g ¼ 1.1 Tensor formalism terminant of the metric ij , and is the charge density measured in the system of inertia. Adopting the formalism used by M6 oller,[2] we consider the motion of a particle in an arbitrary system of coordi- i Ü µ i = ½; ¾; ¿; 4 nates ´ , where indicate the space-time 1.2 Laboratory frame i fg g components. The metric tensor ij is defined in terms X µ In the Cartesian coordinate system of inertia ´ , the ¾ i j d× d× = g dÜ dÜ of the differential line element as ij , so-called laboratory frame, Eq. 2 can be written in the con- where the summation is performed over the four indices. ventional vector form. In this frame, the only non-zero i i Ü = Ü µ Let ´ be the equation of the time track of the components of the metric tensor are motion, being the proper time of the particle. The con- g = g = g =½; aÒd g ½: travariant components of the four-velocity are = ½½ ¾¾ ¿¿ 44 (6) i dÜ i In terms of the conventionally defined electric and mag- Í =´ Ù ; cµ; d netic fields £ Work performed under the auspices of the US Department of Energy E =´E ;E ;E µ; B =´B ;B ;B µ; Ý ¾ ¿ ½ ¾ ¿ [email protected], on joint appointment at ORNL and BNL ½ (7) 0-7803-7191-7/01/$10.00 ©2001 IEEE. 1678 Proceedings of the 2001 Particle Accelerator Conference, Chicago ¼ fg fg g g the EM field tensor is of the rest frame is related to ij of the laboratory ij ¼ Ð Ñ ¼ ½ g = « « g : frame by the relation ÐÑ Using the transfor- ij i j ¼ B B E ¾ ½ ¿ mation matrix, the contravariant components of a vector B C B ¼ B E ¿ ½ ¾ i B C fa g fF g = : ij (8) in the laboratory frame are transformed into the rest @ A ¼ B B ¼ E ¾ ½ ¿ i i k a = « a frame as : On the other hand, the covariant com- k E E E ¼ ¾ ¿ ½ ponents of a tensor of rank 2, e.g. the EM field tensor, are ¼ Ð Ñ F = « « F It is straightforward with these expressions to verify that transformed as ÐÑ . With these relations, the ij i j Eq. 2 is equivalent to the equations equation of motion in the beam rest frame of fixed orienta- tion can be obtained from Eq. 2 by an explicit evaluation. e d d´ Ùµ ¾ Ñ = eE · Ù ¢ B; Ñ c = eÙ ¡ E; ¼ ¼ (9) c dØ dØ 1.4 Rotating beam rest frame ¡ ½=¾ ¾ ¾ dØ = d = ½ Ù where , and =c is in this case The equations of motion can be greatly simplified in the Lorentz factor. terms of the variables tangential and normal to the direc- tion of the motion of the reference particle. We thus seek 1.3 Beam rest frame of fixed orientation for another transformation into the rotating beam rest frame Consider a reference particle circulating around the ver- of which the orientation of the spatial axes are constantly ¾ Ü Þ Ý ! tical axis X with a uniform angular velocity at a radius aligned to the radial ( ), tangential ( ), and vertical ( ) di- i ff g Ê . Its time track may be described in the laboratory rection of the motion of the reference particle. frame in terms of the proper time Define the rotating beam frame as the rest frame of Þ which the orientations of the spatial axes Ü and rotate i ½ ¿ f ´ µ=´Ê cÓ× ; ¼;Ê×iÒ ; ­ µ ; Ü (10) relative to those of the fixed orientation Ü and ¼ ½ ¼ ½ ¼ ½ ¡ ½=¾ ½ ~ ~ ¾ Ü cÓ× ¼ ×iÒ ¼ Ü = !­ ­ = ¬ where is the revolution angle, ½ , ¾ B C B C B C Ý ¼ ½ ¼ ¼ Ü = Ê! and ¬c is the velocity. Introduce a rigid system B C B C B C = : (14) ¿ i @ A @ A @ A ~ ~ Þ Ü ×iÒ ¼ cÓ× ¼ Ü µ of coordinates ´ which follows the reference particle in 4 Ü ¼ ¼ ½ its motion, so that the particle is constantly situated at the ¼ origin of this frame of reference, and that the spatial axes Express the electric and magnetic fields in the laboratory have constant orientations. The transformation connecting Þ i i frame in terms of the tangential component in and the X µ ´Ü µ the variables ´ and are normal component in Ü i i 4 X = f ´ µ·« Ü ; aÒd Ü = ; i (11) E = E cÓ× · E ×iÒ ; B = B cÓ× · B ×iÒ Ü ½ ¿ Ü ½ ¿ = E ; B = B where the check ( ) denotes the inverse operation, the sum- E Ý ¾ Ý ¾ ½; ¾; ¿ = E ×iÒ · E cÓ× ; B = B ×iÒ · B cÓ× mation over is on spatial components . The coef- E Þ ½ ¿ Þ ½ ¿ « ficients ij are obtained by means of successive infinitesi- (15) mal Lorentz transformations without rotation of the spatial The equations of motion can be obtained from Eqs. 2 and axes,[2] 3 by using Eq. 8 and the transformation relations as ¼ ½ 8 ¡ Ñ ~ ¼ « ¼ « ­¬ ×iÒ ¾ 4 ¾ ¾ ¾ ½½ ½¿ > Ü ¾­ ! Þ_ ­ ! Ü Ñ ­ ! Ê ´½ µ= ¼ > B C > ¼ ½ ¼ ¼ > B C > ¢ £ @Î e f« g = ; > ij ¼ ¼ ¾ ¼ > @ A ~ ; B Ý_ B ´_Þ · ­ !ܵ = e´½ µE · > « ¼ « ­¬ cÓ× ¿½ ¿¿ Þ Ý Ü > > c @Ü > Ñ > ­¬ ×iÒ ¼ ­¬ cÓ× ­ ¼ ¼ > Ý = e´½ µE · < Ý (12) ¢ £ @Î e ~ ~ ~ ¼ ¾ ¼ ¾ « = cÓ× cÓ× · ­ ×iÒ ×iÒ « = ×iÒ cÓ× > ½¿ where ½½ , B ´_Þ · ­ !Ü B ´_Ü ­ !Þ µ ; · > Ü Þ > ~ ~ ~ > c @Ý ­ cÓ× ×iÒ « = cÓ× ×iÒ ­ ×iÒ cÓ× « = > ¿¿ , ¿½ , ¡ > Ñ ¼ > ¾ 4 ¾ ¾ ~ ~ ~ > Þ ·¾­ ! Ü_ ­ ! Þ = ×iÒ ×iÒ · ­ cÓ× cÓ× = !­ = ­: , and The fact > > > ~ > ¢ £ > @Î that differs from the revolution angle by the Lorentz e ¾ ¼ ¼ ¼ : ; Ý_ ´_Ü ­ !Þ µ B B · = e´½ µE Ü Ý Þ ­ c @Þ factor is the consequence of the Thomas precession. To obtain the metric tensor of the rest frame, we need (16) to derive the relation between the line elements of the two where coordinate systems. Differentiation of Eq. 11 gives ¼ ¼ E = ­ ´E ¬B µ; B = ­ ´B · ¬E µ Ü Ý Ü Ý Ü Ü ¼ ¼ i E = ­ ´E · ¬B µ; B = ­ ´B ¬E µ Ý Ü Ü Ý (17) Ý Ý « ´ µ f ´ µ i i i j ¼ ¼ dX · Ü =« dÜ · d « dÜ ; i j E = E ; B = B Þ Þ Þ Þ d d (13) i i ½ f« g = f« where g is the transformation matrix be- are the electric and magnetic fields after a Lorentz transfor- j j tween the line elements of the laboratory and the rest frame.
Load more

Recommended publications
  • Physics 9Hb: Special Relativity & Thermal/Statistical Physics
    PHYSICS 9HB: SPECIAL RELATIVITY & THERMAL/STATISTICAL PHYSICS Tom Weideman University of California, Davis UCD: Physics 9HB – Special Relativity and Thermal/Statistical Physics This text is disseminated via the Open Education Resource (OER) LibreTexts Project (https://LibreTexts.org) and like the hundreds of other texts available within this powerful platform, it freely available for reading, printing and "consuming." Most, but not all, pages in the library have licenses that may allow individuals to make changes, save, and print this book. Carefully consult the applicable license(s) before pursuing such effects. Instructors can adopt existing LibreTexts texts or Remix them to quickly build course-specific resources to meet the needs of their students. Unlike traditional textbooks, LibreTexts’ web based origins allow powerful integration of advanced features and new technologies to support learning. The LibreTexts mission is to unite students, faculty and scholars in a cooperative effort to develop an easy-to-use online platform for the construction, customization, and dissemination of OER content to reduce the burdens of unreasonable textbook costs to our students and society. The LibreTexts project is a multi-institutional collaborative venture to develop the next generation of open-access texts to improve postsecondary education at all levels of higher learning by developing an Open Access Resource environment. The project currently consists of 13 independently operating and interconnected libraries that are constantly being optimized by students, faculty, and outside experts to supplant conventional paper-based books. These free textbook alternatives are organized within a central environment that is both vertically (from advance to basic level) and horizontally (across different fields) integrated.
    [Show full text]
  • Physics 200 Problem Set 7 Solution Quick Overview: Although Relativity Can Be a Little Bewildering, This Problem Set Uses Just A
    Physics 200 Problem Set 7 Solution Quick overview: Although relativity can be a little bewildering, this problem set uses just a few ideas over and over again, namely 1. Coordinates (x; t) in one frame are related to coordinates (x0; t0) in another frame by the Lorentz transformation formulas. 2. Similarly, space and time intervals (¢x; ¢t) in one frame are related to inter- vals (¢x0; ¢t0) in another frame by the same Lorentz transformation formu- las. Note that time dilation and length contraction are just special cases: it is time-dilation if ¢x = 0 and length contraction if ¢t = 0. 3. The spacetime interval (¢s)2 = (c¢t)2 ¡ (¢x)2 between two events is the same in every frame. 4. Energy and momentum are always conserved, and we can make e±cient use of this fact by writing them together in an energy-momentum vector P = (E=c; p) with the property P 2 = m2c2. In particular, if the mass is zero then P 2 = 0. 1. The earth and sun are 8.3 light-minutes apart. Ignore their relative motion for this problem and assume they live in a single inertial frame, the Earth-Sun frame. Events A and B occur at t = 0 on the earth and at 2 minutes on the sun respectively. Find the time di®erence between the events according to an observer moving at u = 0:8c from Earth to Sun. Repeat if observer is moving in the opposite direction at u = 0:8c. Answer: According to the formula for a Lorentz transformation, ³ u ´ 1 ¢tobserver = γ ¢tEarth-Sun ¡ ¢xEarth-Sun ; γ = p : c2 1 ¡ (u=c)2 Plugging in the numbers gives (notice that the c implicit in \light-minute" cancels the extra factor of c, which is why it's nice to measure distances in terms of the speed of light) 2 min ¡ 0:8(8:3 min) ¢tobserver = p = ¡7:7 min; 1 ¡ 0:82 which means that according to the observer, event B happened before event A! If we reverse the sign of u then 2 min + 0:8(8:3 min) ¢tobserver 2 = p = 14 min: 1 ¡ 0:82 2.
    [Show full text]
  • 8. Special Relativity
    8. Special Relativity Although Newtonian mechanics gives an excellent description of Nature, it is not uni- versally valid. When we reach extreme conditions — the very small, the very heavy or the very fast — the Newtonian Universe that we’re used to needs replacing. You could say that Newtonian mechanics encapsulates our common sense view of the world. One of the major themes of twentieth century physics is that when you look away from our everyday world, common sense is not much use. One such extreme is when particles travel very fast. The theory that replaces New- tonian mechanics is due to Einstein. It is called special relativity. The effects of special relativity become apparent only when the speeds of particles become comparable to the speed of light in the vacuum. Universally denoted as c, the speed of light is c = 299792458 ms−1 This value of c is exact. In fact, it would be more precise to say that this is the definition of what we mean by a meter: it is the distance travelled by light in 1/299792458 seconds. For the purposes of this course, we’ll be quite happy with the approximation c 3 108 ms−1. ≈ × The first thing to say is that the speed of light is fast. Really fast. The speed of sound is around 300 ms−1; escape velocity from the Earth is around 104 ms−1; the orbital speed of our solar system in the Milky Way galaxy is around 105 ms−1. As we shall soon see, nothing travels faster than c.
    [Show full text]
  • (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.
    [Show full text]
  • Relativistic Kinematics of Particle Interactions Introduction
    le kin rel.tex Relativistic Kinematics of Particle Interactions byW von Schlipp e, March2002 1. Notation; 4-vectors, covariant and contravariant comp onents, metric tensor, invariants. 2. Lorentz transformation; frequently used reference frames: Lab frame, centre-of-mass frame; Minkowski metric, rapidity. 3. Two-b o dy decays. 4. Three-b o dy decays. 5. Particle collisions. 6. Elastic collisions. 7. Inelastic collisions: quasi-elastic collisions, particle creation. 8. Deep inelastic scattering. 9. Phase space integrals. Intro duction These notes are intended to provide a summary of the essentials of relativistic kinematics of particle reactions. A basic familiarity with the sp ecial theory of relativity is assumed. Most derivations are omitted: it is assumed that the interested reader will b e able to verify the results, which usually requires no more than elementary algebra. Only the phase space calculations are done in some detail since we recognise that they are frequently a bit of a struggle. For a deep er study of this sub ject the reader should consult the monograph on particle kinematics byByckling and Ka jantie. Section 1 sets the scene with an intro duction of the notation used here. Although other notations and conventions are used elsewhere, I present only one version which I b elieveto b e the one most frequently encountered in the literature on particle physics, notably in such widely used textb o oks as Relativistic Quantum Mechanics by Bjorken and Drell and in the b o oks listed in the bibliography. This is followed in section 2 by a brief discussion of the Lorentz transformation.
    [Show full text]
  • A Re-Interpretation of the Concept of Mass and of the Relativistic Mass-Energy Relation
    A re-interpretation of the concept of mass and of the relativistic mass-energy relation 1 Stefano Re Fiorentin Summary . For over a century the definitions of mass and derivations of its relation with energy continue to be elaborated, demonstrating that the concept of mass is still not satisfactorily understood. The aim of this study is to show that, starting from the properties of Minkowski spacetime and from the principle of least action, energy expresses the property of inertia of a body. This implies that inertial mass can only be the object of a definition – the so called mass-energy relation - aimed at measuring energy in different units, more suitable to describe the huge amount of it enclosed in what we call the “rest-energy” of a body. Likewise, the concept of gravitational mass becomes unnecessary, being replaceable by energy, thus making the weak equivalence principle intrinsically verified. In dealing with mass, a new unit of measurement is foretold for it, which relies on the de Broglie frequency of atoms, the value of which can today be measured with an accuracy of a few parts in 10 9. Keywords Classical Force Fields; Special Relativity; Classical General Relativity; Units and Standards 1. Introduction Albert Einstein in the conclusions of his 1905 paper “Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?” (“Does the inertia of a body depend upon its energy content? ”) [1] says “The mass of a body is a measure of its energy-content ”. Despite of some claims that the reasoning that brought Einstein to the mass-energy relation was somehow approximated or even defective (starting from Max Planck in 1907 [2], to Herbert E.
    [Show full text]
  • Special Relativity: a Centenary Perspective
    Special Relativity: A Centenary Perspective Clifford M. Will McDonnell Center for the Space Sciences and Department of Physics Washington University, St. Louis MO 63130 USA Contents 1 Introduction 1 2 Fundamentals of special relativity 2 2.1 Einstein’s postulates and insights . ....... 2 2.2 Timeoutofjoint ................................. 3 2.3 SpacetimeandLorentzinvariance . ..... 5 2.4 Special relativistic dynamics . ...... 7 3 Classic tests of special relativity 7 3.1 The Michelson-Morley experiment . ..... 7 3.2 Invariance of c ................................... 9 3.3 Timedilation ................................... 9 3.4 Lorentz invariance and quantum mechanics . ....... 10 3.5 Consistency tests of special relativity . ......... 10 4 Special relativity and curved spacetime 12 4.1 Einstein’s equivalence principle . ....... 13 4.2 Metrictheoriesofgravity. .... 14 4.3 Effective violations of local Lorentz invariance . ........... 14 5 Is gravity Lorentz invariant? 17 arXiv:gr-qc/0504085v1 19 Apr 2005 6 Tests of local Lorentz invariance at the centenary 18 6.1 Frameworks for Lorentz symmetry violations . ........ 18 6.2 Modern searches for Lorentz symmetry violation . ......... 20 7 Concluding remarks 21 References........................................ 21 1 Introduction A hundred years ago, Einstein laid the foundation for a revolution in our conception of time and space, matter and energy. In his remarkable 1905 paper “On the Electrodynamics of Moving Bodies” [1], and the follow-up note “Does the Inertia of a Body Depend upon its Energy-Content?” [2], he established what we now call special relativity as one of the two pillars on which virtually all of physics of the 20th century would be built (the other pillar being quantum mechanics). The first new theory to be built on this framework was general relativity [3], and the successful measurement of the predicted deflection of light in 1919 made both Einstein the person and relativity the theory internationally famous.
    [Show full text]
  • Special Relativity (Pdf)
    Special relativity Thomas DeGrand February 18, 2012 Relativity 2 Contents 1 A top-down view of relativity 5 1.1 Whatthesenotesareallabout . .. 6 1.2 Thegeometryofspace-time . .. 7 2 Relativity for clocks and rulers 11 2.1 Non-relativisticrelativity. ....... 12 2.2 The Michaelson-Morley experiment . ..... 15 2.3 Introductiontorelativity. ..... 18 2.4 Space-timediagrams .............................. 27 2.5 Revisitingsomeexamples . .. 29 2.6 Velocityaddition ................................ 31 2.7 Dopplershift ................................... 33 3 Relativistic kinematics 37 3.1 Energy and momentum in special relativity . ...... 38 3.2 Summarizing results, and some useful formulas . ......... 41 3.3 Simpleexamples ................................. 42 3.4 Simple homework-style examples . .... 44 4 The end of special relativity 51 3 Relativity 4 4.1 Summary of the important concepts and results . ....... 52 4.2 What’sleft .................................... 52 Chapter 1 A top-down view of relativity 5 Relativity 6 “The non-mathematician is seized by a mysterious shuddering when he hears of ‘four- dimensional’ things, by a feeling not unlike that awakened by thoughts of the occult. And yet there is no more commonplace statement, that the world in which we live is a four- dimensional space-time continuum.”–A. Einstein 1.1 What these notes are all about I am a theorist who studies elementary particle physics. The phenomena I study take place on an energy scale where the effects of special relativity are so important that they dominate everything else. This means that, right from the beginning, everything I do has to be made consistent with the rules of special relativity, or what I compute will simply be wrong. This forces me to think in a particular way about special relativity.
    [Show full text]
  • Neutrino Proper Time?
    Neutrino Proper Time? Richard Shurtleff ∗ March 8, 2000 Abstract An electron neutrino can have the quantum phase of an electron, i.e. share its internal clock, if the neutrino takes a path in space-time that is not in the direction of its energy-momentum. Each flavor neutrino would then have a different internal clock; a muon neutrino would have a muon clock and a tau neutrino would have a tau clock. Perhaps surprisingly, there is some evidence suggesting neutrinos have such clocks. If muon neutrinos travel on space-like paths then some atmospheric muon neutrinos would take such paths backwards into outer space and not be observed. These are lost at the source and have nothing to do with oscillations or flavor-changing in flight. The expected depletion of source muon neutrinos is shown here to be 9%, which accounts for half of the missing muon neutrino source flux reported by Super- Kamiokande. Since there is no depletion in the electron neutrino flux source reported at SK and SN1987A electron neutrinos seem to have traveled at the speed of light, the electron neutrino travels on a light-like path. Accelerator-based experiments could be arranged to confirm the reverse motion of muon neutrinos. arXiv:hep-ph/0003071v1 8 Mar 2000 PACS numbers: 03.65.Bz, 14.60.Lm, 96.40.Tv ∗affiliation and mailing address: Department of Mathematics and Applied Sciences, Wentworth Institute of Technology, 550 Huntington Avenue, Boston, MA, USA, ZIP 02115, telephone number: (617) 989-4338, fax number: (617) 989-4591 , e-mail address: shurtleff[email protected] 1 1 INTRODUCTION 2 1 Introduction The quantum phase of a free electron on a given path Γ is the product of its mass and the proper time along Γ, divided by the constanth.
    [Show full text]
  • 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.
    [Show full text]
  • AST1100 Lecture Notes
    AST1100 Lecture Notes 9–10 The special theory of relativity: Four vectors and relativistic dynamics 1 Worldlines In the spacetime diagram in figure 1 we see the path of a particle (or any object) through spacetime. We see the different positions (x, t) in space and time that the particle has passed through. Such a path showing the points in spacetime that an object passed is called a worldline. We will now study two events A and B (on the worldline of a particle) which are separated by a small spacetime interval ∆s. These events could be the particle emitting two flashes of light or the particle passing through two specific points in space. The corresponding space and time intervals between these two events in the laboratory frame are called ∆t and ∆x. From the figure you see that ∆t > ∆x. You can see that this also holds for every small spacetime interval along the path. This has to be this way: The speed of the particle at a given instant is v = ∆x/∆t. If ∆x = ∆t then v = 1 and the particle travels at the speed of light. That ∆t > ∆x simply means that the particle travels at a speed v < c which it must. The worldline of a photon would thus be a line at 45◦ with the coordinate axes. The worldline of any material particle will therefore always make less than 45◦ with the time axis. Events which are separated by spacetime distances such that ∆t > ∆x are called timelike events. Timelike events may be causally connected since a particle with velocity v < c would have the possibility to travel from one of the events to the other event.
    [Show full text]
  • Relativity Chap 1.Pdf
    Chapter 1 Kinematics, Part 1 Special Relativity, For the Enthusiastic Beginner (Draft version, December 2016) Copyright 2016, David Morin, [email protected] TO THE READER: This book is available as both a paperback and an eBook. I have made the first chapter available on the web, but it is possible (based on past experience) that a pirated version of the complete book will eventually appear on file-sharing sites. In the event that you are reading such a version, I have a request: If you don’t find this book useful (in which case you probably would have returned it, if you had bought it), or if you do find it useful but aren’t able to afford it, then no worries; carry on. However, if you do find it useful and are able to afford the Kindle eBook (priced below $10), then please consider purchasing it (available on Amazon). If you don’t already have the Kindle reading app for your computer, you can download it free from Amazon. I chose to self-publish this book so that I could keep the cost low. The resulting eBook price of around $10, which is very inexpensive for a 250-page physics book, is less than a movie and a bag of popcorn, with the added bonus that the book lasts for more than two hours and has zero calories (if used properly!). – David Morin Special relativity is an extremely counterintuitive subject, and in this chapter we will see how its bizarre features come about. We will build up the theory from scratch, starting with the postulates of relativity, of which there are only two.
    [Show full text]