Relativistic Mechanics
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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. -
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. -
Decay Rates and Cross Section
Decay rates and Cross section Ashfaq Ahmad National Centre for Physics Outlines Introduction Basics variables used in Exp. HEP Analysis Decay rates and Cross section calculations Summary 11/17/2014 Ashfaq Ahmad 2 Standard Model With these particles we can explain the entire matter, from atoms to galaxies In fact all visible stable matter is made of the first family, So Simple! Many Nobel prizes have been awarded (both theory/Exp. side) 11/17/2014 Ashfaq Ahmad 3 Standard Model Why Higgs Particle, the only missing piece until July 2012? In Standard Model particles are massless =>To explain the non-zero mass of W and Z bosons and fermions masses are generated by the so called Higgs mechanism: Quarks and leptons acquire masses by interacting with the scalar Higgs field (amount coupling strength) 11/17/2014 Ashfaq Ahmad 4 Fundamental Fermions 1st generation 2nd generation 3rd generation Dynamics of fermions described by Dirac Equation 11/17/2014 Ashfaq Ahmad 5 Experiment and Theory It doesn’t matter how beautiful your theory is, it doesn’t matter how smart you are. If it doesn’t agree with experiment, it’s wrong. Richard P. Feynman A theory is something nobody believes except the person who made it, An experiment is something everybody believes except the person who made it. Albert Einstein 11/17/2014 Ashfaq Ahmad 6 Some Basics Mandelstam Variables In a two body scattering process of the form 1 + 2→ 3 + 4, there are 4 four-vectors involved, namely pi (i =1,2,3,4) = (Ei, pi) Three Lorentz Invariant variables namely s, t and u are defined. -
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. -
Relativistic Kinematics & Dynamics Momentum and Energy
Physics 106a, Caltech | 3 December, 2019 Lecture 18 { Relativistic kinematics & dynamics Momentum and energy Our discussion of 4-vectors in general led us to the energy-momentum 4-vector p with components in some inertial frame (E; ~p) = (mγu; mγu~u) with ~u the velocity of the particle in that inertial 2 −1=2 1 2 frame and γu = (1 − u ) . The length-squared of the 4-vector is (since u = 1) p2 = m2 = E2 − p2 (1) where the last expression is the evaluation in some inertial frame and p = j~pj. In conventional units this is E2 = p2c2 + m2c4 : (2) For a light particle (photon) m = 0 and so E = p (going back to c = 1). Thus the energy-momentum 4-vector for a photon is E(1; n^) withn ^ the direction of propagation. This is consistent with the quantum expressions E = hν; ~p = (h/λ)^n with ν the frequency and λ the wave length, and νλ = 1 (the speed of light) We are led to the expression for the relativistic 3-momentum and energy m~u m ~p = p ;E = p : (3) 1 − u2 1 − u2 Note that ~u = ~p=E. The frame independent statement that the energy-momentum 4-vector is conserved leads to the conservation of this 3-momentum and energy in each inertial frame, with the new implication that mass can be converted to and from energy. Since (E; ~p) form the components of a 4-vector they transform between inertial frames in the same way as (t; ~x), i.e. in our standard configuration 0 px = γ(px − vE) (4) 0 py = py (5) 0 pz = pz (6) 0 E = γ(E − vpx) (7) and the inverse 0 0 px = γ(px + vE ) (8) 0 py = py (9) 0 pz = pz (10) 0 0 E = γ(E + vpx) (11) These reduce to the Galilean expressions for small particle speeds and small transformation speeds. -
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. -
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. -
General Relativity and Spatial Flows: I
1 GENERAL RELATIVITY AND SPATIAL FLOWS: I. ABSOLUTE RELATIVISTIC DYNAMICS* Tom Martin Gravity Research Institute Boulder, Colorado 80306-1258 [email protected] Abstract Two complementary and equally important approaches to relativistic physics are explained. One is the standard approach, and the other is based on a study of the flows of an underlying physical substratum. Previous results concerning the substratum flow approach are reviewed, expanded, and more closely related to the formalism of General Relativity. An absolute relativistic dynamics is derived in which energy and momentum take on absolute significance with respect to the substratum. Possible new effects on satellites are described. 1. Introduction There are two fundamentally different ways to approach relativistic physics. The first approach, which was Einstein's way [1], and which is the standard way it has been practiced in modern times, recognizes the measurement reality of the impossibility of detecting the absolute translational motion of physical systems through the underlying physical substratum and the measurement reality of the limitations imposed by the finite speed of light with respect to clock synchronization procedures. The second approach, which was Lorentz's way [2] (at least for Special Relativity), recognizes the conceptual superiority of retaining the physical substratum as an important element of the physical theory and of using conceptually useful frames of reference for the understanding of underlying physical principles. Whether one does relativistic physics the Einsteinian way or the Lorentzian way really depends on one's motives. The Einsteinian approach is primarily concerned with * http://xxx.lanl.gov/ftp/gr-qc/papers/0006/0006029.pdf 2 being able to carry out practical space-time experiments and to relate the results of these experiments among variously moving observers in as efficient and uncomplicated manner as possible. -
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. -
Measurement of the Mass of the Higgs Boson in the Two Photon Decay Channel with the CMS Experiment
Facolt`adi Scienze Matematiche Fisiche e Naturali Ph.D. Thesis Measurement of the mass of the Higgs Boson in the two photon decay channel with the CMS experiment. Candidate Thesis advisors Shervin Nourbakhsh Dott. Riccardo Paramatti Dott. Paolo Meridiani Matricola 1106418 Year 2012/2013 Contents Introduction i 1 Standard Model Higgs and LHC physics 1 1.1 StandardModelHiggsboson.......................... 1 1.1.1 Overview ................................ 2 1.1.2 Spontaneous symmetry breaking and the Higgs boson . 3 1.1.3 Higgsbosonproduction ........................ 5 1.1.4 Higgsbosondecays ........................... 5 1.1.5 Higgs boson property study in the two photon final state . 6 1.2 LHC ....................................... 11 1.2.1 The LHC layout............................. 12 1.2.2 Machineoperation ........................... 13 1.2.3 LHC physics (proton-proton collision) . 15 2 Compact Muon Solenoid Experiment 17 2.1 Detectoroverview................................ 17 2.1.1 Thecoordinatesystem . 18 2.1.2 Themagnet ............................... 18 2.1.3 Trackersystem ............................. 19 2.1.4 HadronCalorimeter. 21 2.1.5 Themuonsystem............................ 22 2.2 Electromagnetic Calorimeter . 24 2.2.1 Physics requirements and design goals . 24 I Contents II 2.2.2 ECAL design............................... 24 2.3 Triggeranddataacquisition . 26 2.3.1 CalorimetricTrigger . 27 2.4 CMS simulation ................................. 28 3 Electrons and photons reconstruction and identification 32 3.1 Energy measurement in ECAL ......................... 32 3.2 Energycalibration ............................... 33 3.2.1 Responsetimevariation . 34 3.2.2 Intercalibrations. 36 3.3 Clusteringandenergycorrections . 38 3.4 Algorithmic corrections to electron and photon energies ........... 39 3.4.1 Parametric electron and photon energy corrections . ....... 40 3.4.2 MultiVariate (MVA) electron and photon energy corrections . 41 3.5 Photonreconstruction ............................ -
Invariant Mass the Invariant Mass Is Also Called the "Rest Mass", and Is Characteristic of a Particle
INTRODUCTION The LHC – The Large Hadron Collider and ATLAS The accelerator is located at CERN, in a tunnel at a depth of around 100m beneath the French-Swiss border near Geneva. The protons used in the experiments are accelerated in opposite directions in the underground circular accelerator which has a circumference of 27 km. Given the wide variety of research objectives of the LHC, 6 different types of detectors are placed in four caves around the four points of intersection of the beams, Two of them (ATLAS and CMS) are large general purpose detectors, the other two (ALICE and LHCb) have specific goals, while the TOTEM and LHCf are much smaller and have much more specific purposes. The ATLAS detector (A Toroidal LHC ApparatuS), which is the focus of this exercise, was designed to cover the widest possible range of investigations. Its main objectives are to discover the Higgs boson, to confirm (or not) the theory of supersymmetry, to study matter and antimatter, the formation of dark matter and to search for extra dimensions and new phenomena. Involved in the experiment are 3000 physicists from 170 universities and laboratories in 37 countries The detector with 44m length and 25m height is the largest in volume for any detector ever built (see Figure 1). The size of the detector stems from the need for great accuracy in detecting the nature and trajectory of the particles resulting from the particle collisions. Figure 1 : The ATLAS detector The sections of ATLAS The ATLAS detector consists of a series of concentric cylinders mounted so that the beams collide in the center. -
A Geometric Introduction to Spacetime and Special Relativity
A GEOMETRIC INTRODUCTION TO SPACETIME AND SPECIAL RELATIVITY. WILLIAM K. ZIEMER Abstract. A narrative of special relativity meant for graduate students in mathematics or physics. The presentation builds upon the geometry of space- time; not the explicit axioms of Einstein, which are consequences of the geom- etry. 1. Introduction Einstein was deeply intuitive, and used many thought experiments to derive the behavior of relativity. Most introductions to special relativity follow this path; taking the reader down the same road Einstein travelled, using his axioms and modifying Newtonian physics. The problem with this approach is that the reader falls into the same pits that Einstein fell into. There is a large difference in the way Einstein approached relativity in 1905 versus 1912. I will use the 1912 version, a geometric spacetime approach, where the differences between Newtonian physics and relativity are encoded into the geometry of how space and time are modeled. I believe that understanding the differences in the underlying geometries gives a more direct path to understanding relativity. Comparing Newtonian physics with relativity (the physics of Einstein), there is essentially one difference in their basic axioms, but they have far-reaching im- plications in how the theories describe the rules by which the world works. The difference is the treatment of time. The question, \Which is farther away from you: a ball 1 foot away from your hand right now, or a ball that is in your hand 1 minute from now?" has no answer in Newtonian physics, since there is no mechanism for contrasting spatial distance with temporal distance.