Classical Trajectories in Rindler Space and Restricted Structure of Phase Space with PT-Symmetric Hamiltonian. Abstract
|
|
- Felix Newton
- 6 years ago
- Views:
Transcription
1 Classial Trajetories in Rindler Spae and Restrited Struture of Phase Spae with PT-Symmetri Hamiltonian Soma Mitra 1 and Somenath Chakrabarty 2 Department of Physis, Visva-Bharati, Santiniketan , West Bengal, India 1 somaphysis@gmail.om 2 somenath.hakrabarty@visva-bharati.a.in Abstrat The nature of single partile lassial phase spae trajetories in Rindler spae with the PTsymmetri hamiltonian have been studied. It has been shown that only a small portion of the phase spae is aessible to the partiles, whereas the major part of the phase spae region is ompletely forbidden. It has also been notied that the area / volume of the forbidden region of phase spae inreases with the inrease in the strength of gravitational field. The physial signifiane of suh squeezing of phase spae in strong gravitational field has been disussed. PACS numbers: Ge,03.65.Pm, p, q 1
2 It is well known that in the ase of a system of inertial frame of referenes, the spaetime transformations in the relativisti senario are given by the Lorentz transformations [1, 2]. However, for a frame undergoing a uniform aelerated motion in an otherwise flat Minkowski spae-time geometry, the Lorentz type transformation an also be derived to relate the oordinate sets in aelerated frame with that of inertial set of oordinates [3 7]. These are the so alled Rindler oordinate transformations and are given by: ( ) ( ) 2 αt t = α + x sinh and ( ) ( ) 2 αt x = α + x osh (1) Hene one an show that the inverse transformations an be written as: ( ) t = 2 x + t 2α ln x t and x = (x 2 (t) 2 ) 1/2 2 α (2) Where α is the uniform aeleration of the frame. The Rindler spae-time oordinates as shown above (eqns.(1) and (2)) are then just an aelerated frame transformation of the Minkowski metri of speial relativity. It an very easily be shown that the Rindler oordinate transformation hanges the Minkowski line element ds 2 = d(t) 2 dx 2 dy 2 dz 2 to ( ) 2 ds 2 = 1 + αx d(t ) 2 dx 2 dy 2 dz 2 The general form of metri tensor may then be written as ( ( g µν = diag 1 + αx ) ) 2, 1, 1, 1 (3) (4) whereas in the ase of onventional flat Minkowski spae-time we have g µν = diag(+1, 1, 1, 1) (5) A survey of literature shows that the onept of priniple of equivalene is essential to obtain the Rindler oordinate transformation. Our study is therefore based on the priniple of equivalene, aording to whih a frame of referene undergoing an aelerated motion in absene of gravitational field is equivalent to a frame at rest in presene of a gravitational field. Therefore a frame undergoing a uniform aeleration motion is equivalent to a rest frame in presene of a onstant gravitational field of strength equal to the magnitude of the 2
3 aeleration. Then α in the present senario may be treated as the onstant gravitational field for a frame at rest. Now following the onept of relativisti dynamis of speial theory of relativity [1], the ation integral may be written as b b S = α 0 ds Ldt (6) a a Then putting α 0 = m 0, as has been done in speial theory of relativity, where m 0 is the rest mass of the partile and is the veloity of light, the Lagrangian of the partile is given by L = m 0 [ ( 1 + αx ) 2 v 2 ] where v is the three veloity of the partile. The three momentum of the partile is then given by Hene the Hamiltonian of the partile may be written as p = L, or v (8) m 0 v p = [ (1 ) ] 2 1/2 + αx v 2 2 (9) (7) H = p. v L or (10) ( H = m αx ) ( ) 1/2 1 + p2 (11) m 2 0 In the lassial level, the quantities H, x and p are treated as dynamial variables. Further, it an very easily be verified that the Hamiltonian H eqn.(11) is not hermitian. However the energy spetum has been observed to be real [8]. This is found to be solely beause of the fat that H is PT-invarient. Now it is well know that PxP 1 = x, PpP 1 = p, whereas, TpT 1 = p and PαP 1 = α but TαT 1 = α, therefore it is a matter of simple algebra to show that PT H (PT) 1 = H PT = H. As has been shown by several authors [9] that if H PT = H, then the energy spetrum will be real. Here P and T are respetively the parity and the time reversal operators. Whih has also been reported in one of our publiations [8]. Of ourse with the replaement of hermitiity of the Hamiltonian with the P T-symmetry, we have not disarded the important quantum mehanial key features of the system desribed this Hamiltonian (eqn.(11)). This point was also disussed in an elaborate manner in referene [9] and in some of the referenes ited there. 3
4 The lassial Hamilton s equation of motion for the partile is then given by [10] ẋ = [H, x] p.x and ṗ = [H, p] p,x (12) where [H, f] p,x for f = x or p is the Poisson braket and is defined by [10] [f, g] p,x = f g p x f g x p (13) Here dot indiates the derivative with respet to time. Now using the expression for Hamiltonian from eqn.(11), the expliit form of the equation of motions are given by ( ẋ = 1 + αx ) p (p 2 + m ) 1/2 and ṗ = α (p2 + m ) 1/2 (14) The parametri form of x and p whih give the time evolution of spae oordinate and the orresponding anonial momentum an be obtained after integrating the above oupled equations and are given by x = 2 α [C 0 os(ωt) 1] and p = m 0 tan(ωt nπ) (15) where C 0 is some integration onstant, ω = α/ is the frequeny defined for some kind of quanta in [8] and n is an integer inluding zero. Now for x = 0 at t = 0, we have C 0 = 1. Hene we an write x = [os(ωt) 1] (16) ω Again it is very easy to show that for any integer n, inluding zero, we have p = m 0 tan(ωt) (17) To draw the trajetories for the partile in the phase spae, we start with the relation Integrating over x and p, we have dp dx = ṗ ẋ = ω p 2 + m 2 ( ) xω p 2 (p 2 + m ) 1/2 = C 0 ( 1 + xω (18) ) 1 (19) Now assuming that for x = 0, p = 0, the stable ritial points, the integration onstant beomes C 0 = m 0. Then we an rewrite the above equation in the form [ ( p = m xω ) ] (20)
5 This is the mathematial form of lassial trajetory of the partile in the phase spae. It is quite obvious from the above expression that x an not be positive. The positive value of x will make the momentum imaginary, whih is unphysial. Therefore the part of the phase spae with x > 0 are ompletely forbidden for the partile under onsideration in Rindler spae. Further, one an easily verify that as x /ω, the momentum p ±. Beyond whih the momentum will again beome imaginary. Therefore in the Rindler spae only the portion of phase spae with /ω < x 0 will be aessible to the partiles. Sine p is the x-omponent of momentum, it an be negative as well. It is also obvious that if the strength of gravitational field is strong enough the range of x will derease. Therefore near the event horizon of a blak hole sine the gravitational field strength α is extremely high, the aessible phase spae volume / area will aordingly be low enough. In the extreme ase, when the gravitational field strength is infinitely large, the area / volume of the phase spae will be of infinitesimally small in the negative x diretion. In fig.(1) we have shematially shown the appearane of the trajetories. For the sake of illustration we have used /ω = 1 and m 0 = 0.1, 9.5 and 1.0. The upper symmetri pair is for m 0 = 1 and the other two pairs are for 0.5 and 0.1 respetively. It is also obvious from the appearane of the trajetories that all of them are approahing from asymptotially p ± and x 1 to (0, 0) point. Now from the knowledge of statistial mehanis one an infer that with the derease in phase spae volume (or area in low dimension), the abundane of partiles will derease. Therefore the physial signifiane of derease in phase spae volume near the event horizon of a blak hole is that there will be more and more redution in number of partiles at that region. This is of ourse not orret for the eletromagneti waves with semi-lassial quantized form. This is beause of the zero rest mass of the photons. Massless partiles an not exist in pure lassial idea. The derease in abundane of massive partiles with the redution is phase spae area / volume indiretly indiates the absorption of partiles by the blak holes instead of their emission. This is exatly what we expet from the lassial model of a blak hole [11, 12]. Therefore our final onlusion is that at the surfae of event horizon, the phase spae volume or area is vanishingly small and as a onsequene the aomodated partile number is also negligibly small. In other wards, no partile an exist in stable onfiguration very lose to 5
6 the event horizon of a lassial blak hole. [1] Landau L.D. and Lifshitz E.M., The Classial Theory of Fields, Butterworth-Heimenann, Oxford, (1975). [2] W.G. Rosser, Contemporary Physis, 1, 453, (1960). [3] N.D. Birrell and P.C.W. Davies, Quantum Field Theory in Curved Spae, Cambridge University Press, Cambridge, (1982). [4] C.G. Huang and J.R. Sun, arxiv:gr-q/ , (2007). [5] Domingo J Louis-Martinez, Class. Quantum Grav., 28, , (2011). [6] D. Peroo and V.M. Villaba, Class. Quantum Grav., 9, 307, (1992). [7] S. De, S. Ghosh and S. Chakrabarty, Astrophys and Spae Si. (in press, 2015). [8] S. De, S. Ghosh and S. Chakrabarty, Mod. Phys. Lett. A (in press 2015). [9] Carl M. Bender, arxiv:quant-ph/ (and referenes therein) [10] Classial Mehanis, H. Goldstein, Addision Wesley (1972). [11] S.L. Shapiro and S.A. Teukolsky, Blak Holes, White Dwarfs and Neutron Stars, John Wiley and Sons, New York, (1983). [12] S. Weinberg, Gravitation and Cosmology, John Wiley & Sons, New York, (1972). 6
7 FIG. 1: Shemati diagram for the phase spae trajetories for the typial values of m 0 = 0.1, 0.5 and 1.0 7
The Electromagnetic Radiation and Gravity
International Journal of Theoretial and Mathematial Physis 016, 6(3): 93-98 DOI: 10.593/j.ijtmp.0160603.01 The Eletromagneti Radiation and Gravity Bratianu Daniel Str. Teiului Nr. 16, Ploiesti, Romania
More informationThe gravitational phenomena without the curved spacetime
The gravitational phenomena without the urved spaetime Mirosław J. Kubiak Abstrat: In this paper was presented a desription of the gravitational phenomena in the new medium, different than the urved spaetime,
More informationSaha Equation for Partially Ionized Relativistic Hydrogen Plasma in Rindler Space
Open Access Journal of Physics Volume, Issue 3, 018, PP 5-9 Saha Equation for Partially Ionized Relativistic Hydrogen Plasma in Rindler Space Sanchita Das 1, Somenath Chakrabarty 1 Department of physics,visva
More informationA EUCLIDEAN ALTERNATIVE TO MINKOWSKI SPACETIME DIAGRAM.
A EUCLIDEAN ALTERNATIVE TO MINKOWSKI SPACETIME DIAGRAM. S. Kanagaraj Eulidean Relativity s.kana.raj@gmail.om (1 August 009) Abstrat By re-interpreting the speial relativity (SR) postulates based on Eulidean
More informationThe Unified Geometrical Theory of Fields and Particles
Applied Mathematis, 014, 5, 347-351 Published Online February 014 (http://www.sirp.org/journal/am) http://dx.doi.org/10.436/am.014.53036 The Unified Geometrial Theory of Fields and Partiles Amagh Nduka
More informationGravitation is a Gradient in the Velocity of Light ABSTRACT
1 Gravitation is a Gradient in the Veloity of Light D.T. Froedge V5115 @ http://www.arxdtf.org Formerly Auburn University Phys-dtfroedge@glasgow-ky.om ABSTRACT It has long been known that a photon entering
More informationarxiv:gr-qc/ v7 14 Dec 2003
Propagation of light in non-inertial referene frames Vesselin Petkov Siene College, Conordia University 1455 De Maisonneuve Boulevard West Montreal, Quebe, Canada H3G 1M8 vpetkov@alor.onordia.a arxiv:gr-q/9909081v7
More informationName Solutions to Test 1 September 23, 2016
Name Solutions to Test 1 September 3, 016 This test onsists of three parts. Please note that in parts II and III, you an skip one question of those offered. Possibly useful formulas: F qequb x xvt E Evpx
More informationDerivation of Non-Einsteinian Relativistic Equations from Momentum Conservation Law
Asian Journal of Applied Siene and Engineering, Volue, No 1/13 ISSN 35-915X(p); 37-9584(e) Derivation of Non-Einsteinian Relativisti Equations fro Moentu Conservation Law M.O.G. Talukder Varendra University,
More informationThe Concept of Mass as Interfering Photons, and the Originating Mechanism of Gravitation D.T. Froedge
The Conept of Mass as Interfering Photons, and the Originating Mehanism of Gravitation D.T. Froedge V04 Formerly Auburn University Phys-dtfroedge@glasgow-ky.om Abstrat For most purposes in physis the onept
More information). In accordance with the Lorentz transformations for the space-time coordinates of the same event, the space coordinates become
Relativity and quantum mehanis: Jorgensen 1 revisited 1. Introdution Bernhard Rothenstein, Politehnia University of Timisoara, Physis Department, Timisoara, Romania. brothenstein@gmail.om Abstrat. We first
More informationarxiv:gr-qc/ v2 6 Feb 2004
Hubble Red Shift and the Anomalous Aeleration of Pioneer 0 and arxiv:gr-q/0402024v2 6 Feb 2004 Kostadin Trenčevski Faulty of Natural Sienes and Mathematis, P.O.Box 62, 000 Skopje, Maedonia Abstrat It this
More informationSpinning Charged Bodies and the Linearized Kerr Metric. Abstract
Spinning Charged Bodies and the Linearized Kerr Metri J. Franklin Department of Physis, Reed College, Portland, OR 97202, USA. Abstrat The physis of the Kerr metri of general relativity (GR) an be understood
More informationAharonov-Bohm effect. Dan Solomon.
Aharonov-Bohm effet. Dan Solomon. In the figure the magneti field is onfined to a solenoid of radius r 0 and is direted in the z- diretion, out of the paper. The solenoid is surrounded by a barrier that
More informationRelativity in Classical Physics
Relativity in Classial Physis Main Points Introdution Galilean (Newtonian) Relativity Relativity & Eletromagnetism Mihelson-Morley Experiment Introdution The theory of relativity deals with the study of
More informationELECTROMAGNETIC WAVES WITH NONLINEAR DISPERSION LAW. P. М. Меdnis
ELECTROMAGNETIC WAVES WITH NONLINEAR DISPERSION LAW P. М. Меdnis Novosibirs State Pedagogial University, Chair of the General and Theoretial Physis, Russia, 636, Novosibirs,Viljujsy, 8 e-mail: pmednis@inbox.ru
More informationarxiv:physics/ v1 [physics.class-ph] 8 Aug 2003
arxiv:physis/0308036v1 [physis.lass-ph] 8 Aug 003 On the meaning of Lorentz ovariane Lszl E. Szab Theoretial Physis Researh Group of the Hungarian Aademy of Sienes Department of History and Philosophy
More informationFig 1: Variables in constant (1+1)D acceleration. speed of time. p-velocity & c-time. velocities (e.g. v/c) & times (e.g.
Proper veloity and frame-invariant aeleration in speial relativity P. Fraundorf Department of Physis & Astronomy University of Missouri-StL, St. Louis MO (November, 99) We examine here a possible endpoint
More informationSimple Considerations on the Cosmological Redshift
Apeiron, Vol. 5, No. 3, July 8 35 Simple Considerations on the Cosmologial Redshift José Franiso Garía Juliá C/ Dr. Maro Mereniano, 65, 5. 465 Valenia (Spain) E-mail: jose.garia@dival.es Generally, the
More informationThe Concept of the Effective Mass Tensor in GR. The Gravitational Waves
The Conept of the Effetive Mass Tensor in GR The Gravitational Waves Mirosław J. Kubiak Zespół Szkół Tehniznyh, Grudziądz, Poland Abstrat: In the paper [] we presented the onept of the effetive mass tensor
More informationarxiv: v1 [physics.class-ph] 14 Dec 2010
Classial relativisti ideal gas in thermodynami equilibrium in a uniformly aelerated referene frame arxiv:11.363v1 [physis.lass-ph] 14 De 1 Domingo J. Louis-Martinez Department of Physis and Astronomy,
More informationParticle-wave symmetry in Quantum Mechanics And Special Relativity Theory
Partile-wave symmetry in Quantum Mehanis And Speial Relativity Theory Author one: XiaoLin Li,Chongqing,China,hidebrain@hotmail.om Corresponding author: XiaoLin Li, Chongqing,China,hidebrain@hotmail.om
More informationQUANTUM MECHANICS II PHYS 517. Solutions to Problem Set # 1
QUANTUM MECHANICS II PHYS 57 Solutions to Problem Set #. The hamiltonian for a lassial harmoni osillator an be written in many different forms, suh as use ω = k/m H = p m + kx H = P + Q hω a. Find a anonial
More informationRelativistic Dynamics
Chapter 7 Relativisti Dynamis 7.1 General Priniples of Dynamis 7.2 Relativisti Ation As stated in Setion A.2, all of dynamis is derived from the priniple of least ation. Thus it is our hore to find a suitable
More informationHamiltonian with z as the Independent Variable
Hamiltonian with z as the Independent Variable 1 Problem Kirk T. MDonald Joseph Henry Laboratories, Prineton University, Prineton, NJ 08544 Marh 19, 2011; updated June 19, 2015) Dedue the form of the Hamiltonian
More informationGravitomagnetic Effects in the Kerr-Newman Spacetime
Advaned Studies in Theoretial Physis Vol. 10, 2016, no. 2, 81-87 HIKARI Ltd, www.m-hikari.om http://dx.doi.org/10.12988/astp.2016.512114 Gravitomagneti Effets in the Kerr-Newman Spaetime A. Barros Centro
More informationRelativistic Addition of Velocities *
OpenStax-CNX module: m42540 1 Relativisti Addition of Veloities * OpenStax This work is produed by OpenStax-CNX and liensed under the Creative Commons Attribution Liense 3.0 Abstrat Calulate relativisti
More informationFour-dimensional equation of motion for viscous compressible substance with regard to the acceleration field, pressure field and dissipation field
Four-dimensional equation of motion for visous ompressible substane with regard to the aeleration field, pressure field and dissipation field Sergey G. Fedosin PO box 6488, Sviazeva str. -79, Perm, Russia
More informationCollinear Equilibrium Points in the Relativistic R3BP when the Bigger Primary is a Triaxial Rigid Body Nakone Bello 1,a and Aminu Abubakar Hussain 2,b
International Frontier Siene Letters Submitted: 6-- ISSN: 9-8, Vol., pp -6 Aepted: -- doi:.8/www.sipress.om/ifsl.. Online: --8 SiPress Ltd., Switzerland Collinear Equilibrium Points in the Relativisti
More informationChapter 26 Lecture Notes
Chapter 26 Leture Notes Physis 2424 - Strauss Formulas: t = t0 1 v L = L0 1 v m = m0 1 v E = m 0 2 + KE = m 2 KE = m 2 -m 0 2 mv 0 p= mv = 1 v E 2 = p 2 2 + m 2 0 4 v + u u = 2 1 + vu There were two revolutions
More informationTHE TWIN PARADOX A RELATIVISTIC DOMAIN RESOLUTION
THE TWIN PARADOX A RELATIVISTIC DOMAIN RESOLUTION Peter G.Bass P.G.Bass www.relativitydomains.om January 0 ABSTRACT This short paper shows that the so alled "Twin Paradox" of Speial Relativity, is in fat
More informationCHAPTER 26 The Special Theory of Relativity
CHAPTER 6 The Speial Theory of Relativity Units Galilean-Newtonian Relativity Postulates of the Speial Theory of Relativity Simultaneity Time Dilation and the Twin Paradox Length Contration Four-Dimensional
More informationCritical Reflections on the Hafele and Keating Experiment
Critial Refletions on the Hafele and Keating Experiment W.Nawrot In 1971 Hafele and Keating performed their famous experiment whih onfirmed the time dilation predited by SRT by use of marosopi loks. As
More information(a) We desribe physics as a sequence of events labelled by their space time coordinates: x µ = (x 0, x 1, x 2 x 3 ) = (c t, x) (12.
2 Relativity Postulates (a) All inertial observers have the same equations of motion and the same physial laws. Relativity explains how to translate the measurements and events aording to one inertial
More informationarxiv: v1 [physics.gen-ph] 5 Jan 2018
The Real Quaternion Relativity Viktor Ariel arxiv:1801.03393v1 [physis.gen-ph] 5 Jan 2018 In this work, we use real quaternions and the basi onept of the final speed of light in an attempt to enhane the
More informationThe Laws of Acceleration
The Laws of Aeleration The Relationships between Time, Veloity, and Rate of Aeleration Copyright 2001 Joseph A. Rybzyk Abstrat Presented is a theory in fundamental theoretial physis that establishes the
More informationThe Dirac Equation in a Gravitational Field
8/28/09, 8:52 PM San Franiso, p. 1 of 7 sarfatti@pabell.net The Dira Equation in a Gravitational Field Jak Sarfatti Einstein s equivalene priniple implies that Newton s gravity fore has no loal objetive
More information1 sin 2 r = 1 n 2 sin 2 i
Physis 505 Fall 005 Homework Assignment #11 Solutions Textbook problems: Ch. 7: 7.3, 7.5, 7.8, 7.16 7.3 Two plane semi-infinite slabs of the same uniform, isotropi, nonpermeable, lossless dieletri with
More informationGreen s function for the wave equation
Green s funtion for the wave equation Non-relativisti ase January 2019 1 The wave equations In the Lorentz Gauge, the wave equations for the potentials are (Notes 1 eqns 43 and 44): 1 2 A 2 2 2 A = µ 0
More informationEinstein s Three Mistakes in Special Relativity Revealed. Copyright Joseph A. Rybczyk
Einstein s Three Mistakes in Speial Relativity Revealed Copyright Joseph A. Rybzyk Abstrat When the evidene supported priniples of eletromagneti propagation are properly applied, the derived theory is
More informationGeneration of EM waves
Generation of EM waves Susan Lea Spring 015 1 The Green s funtion In Lorentz gauge, we obtained the wave equation: A 4π J 1 The orresponding Green s funtion for the problem satisfies the simpler differential
More informationTENSOR FORM OF SPECIAL RELATIVITY
TENSOR FORM OF SPECIAL RELATIVITY We begin by realling that the fundamental priniple of Speial Relativity is that all physial laws must look the same to all inertial observers. This is easiest done by
More informationChapter 9. The excitation process
Chapter 9 The exitation proess qualitative explanation of the formation of negative ion states Ne and He in He-Ne ollisions an be given by using a state orrelation diagram. state orrelation diagram is
More informationTowards an Absolute Cosmic Distance Gauge by using Redshift Spectra from Light Fatigue.
Towards an Absolute Cosmi Distane Gauge by using Redshift Spetra from Light Fatigue. Desribed by using the Maxwell Analogy for Gravitation. T. De Mees - thierrydemees @ pandora.be Abstrat Light is an eletromagneti
More informationAddition of velocities. Taking differentials of the Lorentz transformation, relative velocities may be calculated:
Addition of veloities Taking differentials of the Lorentz transformation, relative veloities may be allated: So that defining veloities as: x dx/dt, y dy/dt, x dx /dt, et. it is easily shown that: With
More informationChapter Outline The Relativity of Time and Time Dilation The Relativistic Addition of Velocities Relativistic Energy and E= mc 2
Chapter 9 Relativeity Chapter Outline 9-1 The Postulate t of Speial Relativity it 9- The Relativity of Time and Time Dilation 9-3 The Relativity of Length and Length Contration 9-4 The Relativisti Addition
More informationVector Field Theory (E&M)
Physis 4 Leture 2 Vetor Field Theory (E&M) Leture 2 Physis 4 Classial Mehanis II Otober 22nd, 2007 We now move from first-order salar field Lagrange densities to the equivalent form for a vetor field.
More informationThe Hanging Chain. John McCuan. January 19, 2006
The Hanging Chain John MCuan January 19, 2006 1 Introdution We onsider a hain of length L attahed to two points (a, u a and (b, u b in the plane. It is assumed that the hain hangs in the plane under a
More informationSpecial and General Relativity
9/16/009 Speial and General Relativity Inertial referene frame: a referene frame in whih an aeleration is the result of a fore. Examples of Inertial Referene Frames 1. This room. Experiment: Drop a ball.
More informationEinstein s Road Not Taken
Einstein s Road Not Taken Robert D. Bok R-DEX Systems, In. May 25, 2017 Abstrat When onfronted with the hallenge of defining distant simultaneity Einstein looked down two roads that seemingly diverged.
More informationWave Propagation through Random Media
Chapter 3. Wave Propagation through Random Media 3. Charateristis of Wave Behavior Sound propagation through random media is the entral part of this investigation. This hapter presents a frame of referene
More informationCasimir self-energy of a free electron
Casimir self-energy of a free eletron Allan Rosenwaig* Arist Instruments, In. Fremont, CA 94538 Abstrat We derive the eletromagneti self-energy and the radiative orretion to the gyromagneti ratio of a
More information12.1 Events at the same proper distance from some event
Chapter 1 Uniform Aeleration 1.1 Events at the same proper distane from some event Consider the set of events that are at a fixed proper distane from some event. Loating the origin of spae-time at this
More informationMillennium Relativity Acceleration Composition. The Relativistic Relationship between Acceleration and Uniform Motion
Millennium Relativity Aeleration Composition he Relativisti Relationship between Aeleration and niform Motion Copyright 003 Joseph A. Rybzyk Abstrat he relativisti priniples developed throughout the six
More informationAn Effective Photon Momentum in a Dielectric Medium: A Relativistic Approach. Abstract
An Effetive Photon Momentum in a Dieletri Medium: A Relativisti Approah Bradley W. Carroll, Farhang Amiri, and J. Ronald Galli Department of Physis, Weber State University, Ogden, UT 84408 Dated: August
More informationPhysical Laws, Absolutes, Relative Absolutes and Relativistic Time Phenomena
Page 1 of 10 Physial Laws, Absolutes, Relative Absolutes and Relativisti Time Phenomena Antonio Ruggeri modexp@iafria.om Sine in the field of knowledge we deal with absolutes, there are absolute laws that
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.286: The Early Universe December 21, 2013 Prof. Alan Guth QUIZ 3 SOLUTIONS
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physis Department Physis 8.286: The Early Universe Deember 2, 203 Prof. Alan Guth QUIZ 3 SOLUTIONS Quiz Date: Deember 5, 203 PROBLEM : DID YOU DO THE READING? (35
More informationarxiv:physics/ v3 22 Dec 1996
A one-map two-lok approah to teahing relativity in introdutory physis P. Fraundorf Department of Physis & Astronomy University of Missouri-StL, St. Louis MO 632 (January 7, 2002) arxiv:physis/960 v3 22
More information( x vt) m (0.80)(3 10 m/s)( s) 1200 m m/s m/s m s 330 s c. 3.
Solutions to HW 10 Problems and Exerises: 37.. Visualize: At t t t 0 s, the origins of the S, S, and S referene frames oinide. Solve: We have 1 ( v/ ) 1 (0.0) 1.667. (a) Using the Lorentz transformations,
More informationGravity from the Uncertainty Principle.
Gravity from the Unertainty Priniple. M.E. MCulloh Otober 29, 2013 Abstrat It is shown here that Newton's gravity law an be derived from the unertainty priniple. The idea is that as the distane between
More informationA 4 4 diagonal matrix Schrödinger equation from relativistic total energy with a 2 2 Lorentz invariant solution.
A 4 4 diagonal matrix Shrödinger equation from relativisti total energy with a 2 2 Lorentz invariant solution. Han Geurdes 1 and Koji Nagata 2 1 Geurdes datasiene, 2593 NN, 164, Den Haag, Netherlands E-mail:
More informationThe Thomas Precession Factor in Spin-Orbit Interaction
p. The Thomas Preession Fator in Spin-Orbit Interation Herbert Kroemer * Department of Eletrial and Computer Engineering, Uniersity of California, Santa Barbara, CA 9306 The origin of the Thomas fator
More informationThe Exact Solution of the Pioneer Anomaly and Flyby Anomaly and the Interpretation of Inertia from an asymmetric Casimir effect
The Exat Solution of the Pioneer Anomaly and Flyby Anomaly and the Interpretation of Inertia from an asymmetri Casimir effet Abstrat Azzam Almosallami Zurih, Switzerland a.almosallami71@gmail.om In this
More informationBerry s phase for coherent states of Landau levels
Berry s phase for oherent states of Landau levels Wen-Long Yang 1 and Jing-Ling Chen 1, 1 Theoretial Physis Division, Chern Institute of Mathematis, Nankai University, Tianjin 300071, P.R.China Adiabati
More informationDr G. I. Ogilvie Lent Term 2005
Aretion Diss Mathematial Tripos, Part III Dr G. I. Ogilvie Lent Term 2005 1.4. Visous evolution of an aretion dis 1.4.1. Introdution The evolution of an aretion dis is regulated by two onservation laws:
More informationTWO WAYS TO DISTINGUISH ONE INERTIAL FRAME FROM ANOTHER
TWO WAYS TO DISTINGUISH ONE INERTIAL FRAME FROM ANOTHER (WHY IS THE SPEED OF LIGHT CONSTANT?) Dr. Tamas Lajtner Correspondene via web site: www.lajtnemahine.om. ABSTRACT... 2 2. SPACETIME CONTINUUM BY
More informationTime and Energy, Inertia and Gravity
Time and Energy, Inertia and Gravity The Relationship between Time, Aeleration, and Veloity and its Affet on Energy, and the Relationship between Inertia and Gravity Copyright 00 Joseph A. Rybzyk Abstrat
More informationWhere as discussed previously we interpret solutions to this partial differential equation in the weak sense: b
Consider the pure initial value problem for a homogeneous system of onservation laws with no soure terms in one spae dimension: Where as disussed previously we interpret solutions to this partial differential
More informationClassical Diamagnetism and the Satellite Paradox
Classial Diamagnetism and the Satellite Paradox 1 Problem Kirk T. MDonald Joseph Henry Laboratories, Prineton University, Prineton, NJ 08544 (November 1, 008) In typial models of lassial diamagnetism (see,
More informationHidden Momentum in a Spinning Sphere
Hidden Momentum in a Spinning Sphere 1 Problem Kirk T. MDonald Joseph Henry Laboratories, Prineton University, Prineton, NJ 8544 (August 16, 212; updated June 3, 217 A spinning sphere at rest has zero
More informationarxiv:physics/ v1 14 May 2002
arxiv:physis/0205041 v1 14 May 2002 REPLY TO CRITICISM OF NECESSITY OF SIMULTANEOUS CO-EXISTENCE OF INSTANTANEOUS AND RETARDED INTERACTIONS IN CLASSICAL ELECTRODYNAMICS by J.D.Jakson ANDREW E. CHUBYKALO
More informationExtending LMR for anisotropic unconventional reservoirs
Extending LMR for anisotropi unonventional reservoirs Maro A. Perez Apahe Canada Ltd Summary It has beome inreasingly advantageous to haraterize rok in unonventional reservoirs within an anisotropi framework.
More informationLecture Notes 4 MORE DYNAMICS OF NEWTONIAN COSMOLOGY
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physis Department Physis 8.286: The Early Universe Otober 1, 218 Prof. Alan Guth Leture Notes 4 MORE DYNAMICS OF NEWTONIAN COSMOLOGY THE AGE OF A FLAT UNIVERSE: We
More informationNuclear Shell Structure Evolution Theory
Nulear Shell Struture Evolution Theory Zhengda Wang (1) Xiaobin Wang () Xiaodong Zhang () Xiaohun Wang () (1) Institute of Modern physis Chinese Aademy of SienesLan Zhou P. R. China 70000 () Seagate Tehnology
More informationZero-energy space cancels the need for dark energy. Mathematics, Physics and Philosophy in the Interpretations of Relativity Theory
Zero-energy spae anels the need for dark energy Tuomo Suntola, www.si.fi/~suntola/, Finland Mathematis, Physis and Philosophy in the Interpretations of Relativity Theory 1 Latest PhysisWeb Summaries 20.7.2007:
More informationPhysicsAndMathsTutor.com 1
PhysisAndMathsTutor.om. (a (i beam splitter [or semi-silvered mirror] (ii a ompensator [or a glass blok] allows for the thikness of the (semi-silvered mirror to obtain equal optial path lengths in the
More informationLecture 3 - Lorentz Transformations
Leture - Lorentz Transformations A Puzzle... Example A ruler is positioned perpendiular to a wall. A stik of length L flies by at speed v. It travels in front of the ruler, so that it obsures part of the
More informationThe First Principle of Thermodynamics under Relativistic Conditions and Temperature
New Horizons in Mathematial Physis, Vol., No., September 7 https://dx.doi.org/.66/nhmp.7. 37 he First Priniple of hermodynamis under Relativisti Conditions and emperature Emil Veitsman Independent Researher
More informationBrazilian Journal of Physics, vol. 29, no. 3, September, Classical and Quantum Mechanics of a Charged Particle
Brazilian Journal of Physis, vol. 9, no. 3, September, 1999 51 Classial and Quantum Mehanis of a Charged Partile in Osillating Eletri and Magneti Fields V.L.B. de Jesus, A.P. Guimar~aes, and I.S. Oliveira
More informationMatter-light duality and speed greater than light
Matter-light duality and speed greater than light Shalender Singh* and Vishnu Priya Singh Parmar Priza Tehnologies In. R&D, 155 MCarthy Blvd, Ste 1111, Milpitas, California, USA 95035 Email: shalender@prizateh.om
More informationEnergy Gaps in a Spacetime Crystal
Energy Gaps in a Spaetime Crystal L.P. Horwitz a,b, and E.Z. Engelberg a Shool of Physis, Tel Aviv University, Ramat Aviv 69978, Israel b Department of Physis, Ariel University Center of Samaria, Ariel
More informationA No-Shape-Substance is the foundation. all Physics laws depend on
A No-Shape-Substane is the foundation all Physis laws depend on The Seond Part of New Physis Ji Qi,Yinling Jiang Department of physis, Shool of Eletroni Engineering, Northeast Petroleum University, No.
More informationA note on a variational formulation of electrodynamics
Proeedings of the XV International Workshop on Geometry and Physis Puerto de la Cruz, Tenerife, Canary Islands, Spain September 11 16, 006 Publ. de la RSME, Vol. 11 (007), 314 31 A note on a variational
More informationOn the Geometrical Conditions to Determine the Flat Behaviour of the Rotational Curves in Galaxies
On the Geometrial Conditions to Determine the Flat Behaviour of the Rotational Curves in Galaxies Departamento de Físia, Universidade Estadual de Londrina, Londrina, PR, Brazil E-mail: andrenaves@gmail.om
More informationElectromagnetic radiation of the travelling spin wave propagating in an antiferromagnetic plate. Exact solution.
arxiv:physis/99536v1 [physis.lass-ph] 15 May 1999 Eletromagneti radiation of the travelling spin wave propagating in an antiferromagneti plate. Exat solution. A.A.Zhmudsky November 19, 16 Abstrat The exat
More informationPHYS-3301 Lecture 4. Chapter 2. Announcement. Sep. 7, Special Relativity. Course webpage Textbook
Announement Course webage htt://www.hys.ttu.edu/~slee/330/ Textbook PHYS-330 Leture 4 HW (due 9/4 Chater 0, 6, 36, 4, 45, 50, 5, 55, 58 Se. 7, 07 Chater Seial Relativity. Basi Ideas. Consequenes of Einstein
More information[Khalid, 5(3): March 2018] ISSN DOI /zenodo Impact Factor
GLOBAL JOURNAL OF ENGINEERING SCIENCE AND RESEARCHES LORENZ TRANSFORMATION FOR FREE SPACE AND FIELDS USING MAXWELL S EQUATIONS AND NEWTON'S LAWS Nuha Abdelrahman Khalid*, Mubarak Dirar Abdallah, Zoalnoon
More informationthe following action R of T on T n+1 : for each θ T, R θ : T n+1 T n+1 is defined by stated, we assume that all the curves in this paper are defined
How should a snake turn on ie: A ase study of the asymptoti isoholonomi problem Jianghai Hu, Slobodan N. Simić, and Shankar Sastry Department of Eletrial Engineering and Computer Sienes University of California
More informationThe Lorenz Transform
The Lorenz Transform Flameno Chuk Keyser Part I The Einstein/Bergmann deriation of the Lorentz Transform I follow the deriation of the Lorentz Transform, following Peter S Bergmann in Introdution to the
More informationA model for measurement of the states in a coupled-dot qubit
A model for measurement of the states in a oupled-dot qubit H B Sun and H M Wiseman Centre for Quantum Computer Tehnology Centre for Quantum Dynamis Griffith University Brisbane 4 QLD Australia E-mail:
More informationPhys 561 Classical Electrodynamics. Midterm
Phys 56 Classial Eletrodynamis Midterm Taner Akgün Department of Astronomy and Spae Sienes Cornell University Otober 3, Problem An eletri dipole of dipole moment p, fixed in diretion, is loated at a position
More informationarxiv: v1 [physics.class-ph] 12 Mar 2012
Relativisti Dynamis of a Charged Partile in an Eletrosalar Field D.V. Podgainy 1, O.A. Zaimidoroga 2 arxiv:1203.2490v1 [physis.lass-ph] 12 Mar 2012 Joint Institute for Nulear Researh 141980, Dubna, Russia
More informationClassical Field Theory
Preprint typeset in JHEP style - HYPER VERSION Classial Field Theory Gleb Arutyunov a a Institute for Theoretial Physis and Spinoza Institute, Utreht University, 3508 TD Utreht, The Netherlands Abstrat:
More informationRecapitulate. Prof. Shiva Prasad, Department of Physics, IIT Bombay
18 1 Reapitulate We disussed how light an be thought of onsisting of partiles known as photons. Compton Effet demonstrated that they an be treated as a partile with zero rest mass having nonzero energy
More informationMaximum Entropy and Exponential Families
Maximum Entropy and Exponential Families April 9, 209 Abstrat The goal of this note is to derive the exponential form of probability distribution from more basi onsiderations, in partiular Entropy. It
More informationphysics/ Nov 1999
Do Gravitational Fields Have Mass? Or on the Nature of Dark Matter Ernst Karl Kunst As has been shown before (a brief omment will be given in the text) relativisti mass and relativisti time dilation of
More informationRelativity fundamentals explained well (I hope) Walter F. Smith, Haverford College
Relativity fundamentals explained well (I hope) Walter F. Smith, Haverford College 3-14-06 1 Propagation of waves through a medium As you ll reall from last semester, when the speed of sound is measured
More informationThe Corpuscular Structure of Matter, the Interaction of Material Particles, and Quantum Phenomena as a Consequence of Selfvariations.
The Corpusular Struture of Matter, the Interation of Material Partiles, and Quantum Phenomena as a Consequene of Selfvariations. Emmanuil Manousos APM Institute for the Advanement of Physis and Mathematis,
More informationAstr 5465 Mar. 29, 2018 Galactic Dynamics I: Disks
Galati Dynamis Overview Astr 5465 Mar. 29, 2018 Subjet is omplex but we will hit the highlights Our goal is to develop an appreiation of the subjet whih we an use to interpret observational data See Binney
More informationDO PHYSICS ONLINE. SPECIAL RELATIVITY Frames of Reference
DO PHYSICS ONLINE SPACE SPECIAL RELATIVITY Frames of Referene Spae travel Apollo 11 spaeraft: Earth Moon v ~ 40x10 3 km.h -1 Voyager spaeraft: v ~ 60x10 3 km.h -1 (no sling shot effet) Ulysses spaeraft:
More information