DOCSLIB.ORG

Quaternions and Special Relativity

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Quaternions and Special Relativity View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server August, 1995 hep-th/95mmnnn Quaternions and Sp ecial Relativity a) Stefano De Leo UniversitadiLecce, Dipartimento di Fisica Instituto di Fisica Nucleare, Sezione di Lecce Lecce, 73100, ITALY Abstract We reformulate Sp ecial Relativityby a quaternionic algebra on reals. Using real linear quaternions,we show that previous diculties, concerning the appropriate transformations on the 3 + 1 space-time, maybeovercome. This implies that a complexi ed quaternionic version of Sp ecial Relativityisachoice and not a necessity. a) e-mail: [email protected] 1 Intro duction \The most remarkable formula in mathematics is: i e = cos + i sin : (1) This is our jewel. Wemay relate the geometry to the algebra by representing complex numb ers in a plane i x + iy = re : This is the uni cation of algebra and geometry."-Feynman [1]. i We know that a rotation of -angle around the z -axis, can b e represented by e , in fact i i(+ ) e (x + iy )=re : In 1843, Hamilton in the attempt to generalize the complex eld in order to describ e 1 the rotation in the three-dimensional space, discovered quaternions . Today a rotation ab out an axis passing trough the origin and parallel to a given uni- tary vector ~u (u ;u;u)by an angle , can b e obtained taking the transformation x y z (iu +ju +ku ) (iu +ju +ku ) x y z x y z 2 2 e (ix + jy + kz)e : (2) Therefore if we wish to represent rotations in the three-dimensional space and com- plete \the uni cation of algebra and geometry"we need quaternions. The quaternionic algebra has b een exp ounded in a series of pap ers [2] and b o oks [3] with particular reference to quantum mechanics, the reader may refer to these for further details. For convenience we rep eat and develop the relevant p oints in the following section, where the terminology is also de ned. Nothing that U (1;q) is algebraically isomorphic to SU (2;c), the imaginary units i; j; k can b e realized by means of the 2 2Pauli matrices through (i; j; k ) $ (i ; i ; i ) : 3 2 1 (this particular representation of the imaginary units i; j; k has b een intro duced in ref. [4]). So a quaternion q can b e represented bya 22 complex matrix ! z z 1 2 q $ Q = ; (3) z z 2 1 1 Quaternions, as used in this pap er, will always mean \real quaternions" q = a + ib + jc + kd ; a; b; c; d 2R: 1 where z = a + ib ; z = c id 2 C (1;i) : 1 2 It follows that a quaternion with unitary norm is identi ed by a unitary 2 2 matrix 2 with unit determinant, this gives the corresp ondence b etween unitary quaternions U (1;q) and SU (2;c). Let us consider the transformation law of a spinor (two-dimensional representations of the rotation group) 0 = U ; (4) where ! z 1 = ; U2SU (2;c) : z 2 We can immediately verify that ! z 2 ~ = z 1 transforms as follows 0 ~ ~ = U ; (5) so ! ! 0 z z z z 1 1 2 2 = U z z z z 2 2 1 1 represents again the transformation law of a spinor. Thanks to the identi cation (3) we can write the previous transformations by real quaternions as follows 0 q = U q; + with q = z + jz and U quaternion with unitary norm (N (U )=U U =1). Note that 1 2 we don't need right op erators to indicate the transformation law of a spinor. Nowwe can obtain the transformation law of a three-dimensional vector ~r (x; y ; z )by pro duct of spinors, in fact if we consider the purely imaginary quaternion + ! = qiq = ix + jy + kz or the corresp onding traceless 2 2 complex matrix ! ix y iz + = i = ; y iz ix 3 a rotation in the three-dimensional space can b e written as follows 2 In a recent pap er [5] the representation theory of the group U (1;q) has b een discussed in detail. 3 \No such `trick' works to relate the full four-vector (ct; x; y ; z ) with real quaternions." - Penrose [6]. 2 0 + ! = U ! U (quaternions) , 0 + = U U (2 2 complex matrices) . ~ ~ For in nitesimal transformations, U =1+Q,we nd 0 ~ ~ ~ ~ ~ ~ ~ ~ Q ~r = Q ~r + Q Q ~r Q ~r Q ; where ~ ~ Q (i; j; k ) ; ( ; ; ) : 4 If we rewrite the mentioned ab ove transformation in the following form 0 ~ ~ ~ ~ ~ Q ~r =[1+(Q1 j Q)] Q ~r; (6) we identify i 1 j i j 1 j j k 1 j k ; ; ; 2 2 2 5 as the generators for rotations in the three-dimensional space . Up till now, wehave considered only particular op erations on quaternions. A quaternion q can also b e multiplied by unitary quaternions V from the right. A p ossible transformation which preserves the norm is given by 0 q = U q V ; (7) + + ( U U = V V =1) : Since left and rightmultiplications commute, the group is lo cally isomorphic to SU (2) SU (2), and so to O (4), the four-dimensional Euclidean rotation group. As far as here we can recognize only particular real linear quaternions, namely 1 ; i; j; k; 1ji; 1jj; 1jk: Wehaven't hop e to describ e the Lorentz group if we use only previous ob jects. An- alyzing the most general transformation on quaternions (see section 4), weintro duce new real linear quaternions which allow ustoovercome the ab ove diculty and so obtain a quaternionic version of the Lorentz group, without the use of complexi ed 4 Barred op erators Ojq act on quaternionic ob jects as in (Ojq) = O q: 1 5 The factor guarantees that our generators satisfy the usual algebra: 2 [A ;A ]= A m; n; p =1; 2; 3 : m n mnp p 3 quaternions. This result app ears, to the b est of our knowledge, for the rst time in print. First of that we brie y recall the standard way to rewrite sp ecial relativitybya quaternionic algebra on complex (see section 3). In section 5, we present a quaternionic version of the sp ecial group SL(2;c), which is as well-known collected to the Lorentz group. Our conclusions are drawn in the nal section. 2 Quaternionic algebras A quaternionic algebra over a eld F is a set H = f + i + j + k j ; ; ; 2Fg with multiplication op erations de ned by following rules for imaginary units i; j; k 2 2 2 i = j = k = 1 ; jk = kj = i; ki = ik = j; ij = ji = k: In our pap er we will work with quaternionic algebras de ned on reals and complex, so in this section we give a panoramic review of such algebras. We start with a quaternionic algebra on reals H = f + i + j + k j ; ; ; 2Rg: R + Weintro duce the quaternion conjugation denoted by and de ned by + q = i j k : The previous de nition implies + + + ( ') = ' ; for , ' quaternionic functions. A conjugation op eration which do es not reverse the order of , ' factors is given, for example, by q~ = i + j k : An imp ortant di erence b etween quaternions and complexi ed quaternions is based on the concept of division algebra, which is a nite dimensional algebra for which a 6=0,b6= 0 implies ab 6= 0, in others words, which has not nonzero divisors of zero. A classical theorem [7] states that the only division algebras over the reals are 4 algebras of dimension 1, 2, 4 and 8; the only asso ciative algebras over the reals are R, C and H (Frob enius [8]); the nonasso ciative division algebras include the o ctonions R O (but there are others as well; see Okub o [9]). A simple example of a nondivision algebra is provided by the algebra of complexi ed quaternions H = f + i + j + k j ; ; ; 2C(1; I )g ; C [I ;i]= [I;j]= [I;k]=0 : In fact since (1 + iI )(1 iI )=0 ; there are nonzero divisors of zero. For complexi ed quaternions wehave di erent opp ortunities to de ne conjugation op erations, we shall use the following terminology: 1. The complex conjugate of q is C q = + i + j + k : C Under this op eration (I ;i;j;k) ! (I ;i;j;k) ; and (q p ) = q p : C C C C 2. The quaternion conjugate of q is C ? q = i j k : C Here (I ;i;j;k) ! (I;i; j; k ) ; and ? ? ? (q p ) = p q : C C C C 3. In the absence of standard terminology,we call that formed by combining these op erations, the full conjugate + = i j k : q C Under this op eration (I ;i;j;k) ! (I;i;j;k) ; and + + + (q p ) = p q : C C C C 5 3 Complexi ed Quaternions and Sp ecial Relativi- ty We b egin this section by recalling a sentence of Anderson and Joshi [10] ab out the quaternionic reformulation of sp ecial relativity: \There has b een a long tradition of using quaternions for Sp ecial Relativity ... The use of quaternions in sp ecial relativity,however, is not entirely straightforward. Since the eld of quaternions is a four-dimensional Euclidean space, complex comp onents for the quaternions are required for the 3+1 space-time of sp ecial relativity." In the following section, we will demonstrate that a reformulation of sp ecial rela- tivityby a quaternionic algebra on reals is p ossible.
Load more

Recommended publications
  • A Mathematical Derivation of the General Relativistic Schwarzschild
    A Mathematical Derivation of the General Relativistic Schwarzschild Metric An Honors thesis presented to the faculty of the Departments of Physics and Mathematics East Tennessee State University In partial fulfillment of the requirements for the Honors Scholar and Honors-in-Discipline Programs for a Bachelor of Science in Physics and Mathematics by David Simpson April 2007 Robert Gardner, Ph.D. Mark Giroux, Ph.D. Keywords: differential geometry, general relativity, Schwarzschild metric, black holes ABSTRACT The Mathematical Derivation of the General Relativistic Schwarzschild Metric by David Simpson We briefly discuss some underlying principles of special and general relativity with the focus on a more geometric interpretation. We outline Einstein’s Equations which describes the geometry of spacetime due to the influence of mass, and from there derive the Schwarzschild metric. The metric relies on the curvature of spacetime to provide a means of measuring invariant spacetime intervals around an isolated, static, and spherically symmetric mass M, which could represent a star or a black hole. In the derivation, we suggest a concise mathematical line of reasoning to evaluate the large number of cumbersome equations involved which was not found elsewhere in our survey of the literature. 2 CONTENTS ABSTRACT ................................. 2 1 Introduction to Relativity ...................... 4 1.1 Minkowski Space ....................... 6 1.2 What is a black hole? ..................... 11 1.3 Geodesics and Christoffel Symbols ............. 14 2 Einstein’s Field Equations and Requirements for a Solution .17 2.1 Einstein’s Field Equations .................. 20 3 Derivation of the Schwarzschild Metric .............. 21 3.1 Evaluation of the Christoffel Symbols .......... 25 3.2 Ricci Tensor Components .................
    [Show full text]
  • Hypercomplex Algebras and Their Application to the Mathematical
    Hypercomplex Algebras and their application to the mathematical formulation of Quantum Theory Torsten Hertig I1, Philip H¨ohmann II2, Ralf Otte I3 I tecData AG Bahnhofsstrasse 114, CH-9240 Uzwil, Schweiz 1 [email protected] 3 [email protected] II info-key GmbH & Co. KG Heinz-Fangman-Straße 2, DE-42287 Wuppertal, Deutschland 2 [email protected] March 31, 2014 Abstract Quantum theory (QT) which is one of the basic theories of physics, namely in terms of ERWIN SCHRODINGER¨ ’s 1926 wave functions in general requires the field C of the complex numbers to be formulated. However, even the complex-valued description soon turned out to be insufficient. Incorporating EINSTEIN’s theory of Special Relativity (SR) (SCHRODINGER¨ , OSKAR KLEIN, WALTER GORDON, 1926, PAUL DIRAC 1928) leads to an equation which requires some coefficients which can neither be real nor complex but rather must be hypercomplex. It is conventional to write down the DIRAC equation using pairwise anti-commuting matrices. However, a unitary ring of square matrices is a hypercomplex algebra by definition, namely an associative one. However, it is the algebraic properties of the elements and their relations to one another, rather than their precise form as matrices which is important. This encourages us to replace the matrix formulation by a more symbolic one of the single elements as linear combinations of some basis elements. In the case of the DIRAC equation, these elements are called biquaternions, also known as quaternions over the complex numbers. As an algebra over R, the biquaternions are eight-dimensional; as subalgebras, this algebra contains the division ring H of the quaternions at one hand and the algebra C ⊗ C of the bicomplex numbers at the other, the latter being commutative in contrast to H.
    [Show full text]
  • JOHN EARMAN* and CLARK GL YMUURT the GRAVITATIONAL RED SHIFT AS a TEST of GENERAL RELATIVITY: HISTORY and ANALYSIS
    JOHN EARMAN* and CLARK GL YMUURT THE GRAVITATIONAL RED SHIFT AS A TEST OF GENERAL RELATIVITY: HISTORY AND ANALYSIS CHARLES St. John, who was in 1921 the most widely respected student of the Fraunhofer lines in the solar spectra, began his contribution to a symposium in Nncure on Einstein’s theories of relativity with the following statement: The agreement of the observed advance of Mercury’s perihelion and of the eclipse results of the British expeditions of 1919 with the deductions from the Einstein law of gravitation gives an increased importance to the observations on the displacements of the absorption lines in the solar spectrum relative to terrestrial sources, as the evidence on this deduction from the Einstein theory is at present contradictory. Particular interest, moreover, attaches to such observations, inasmuch as the mathematical physicists are not in agreement as to the validity of this deduction, and solar observations must eventually furnish the criterion.’ St. John’s statement touches on some of the reasons why the history of the red shift provides such a fascinating case study for those interested in the scientific reception of Einstein’s general theory of relativity. In contrast to the other two ‘classical tests’, the weight of the early observations was not in favor of Einstein’s red shift formula, and the reaction of the scientific community to the threat of disconfirmation reveals much more about the contemporary scientific views of Einstein’s theory. The last sentence of St. John’s statement points to another factor that both complicates and heightens the interest of the situation: in contrast to Einstein’s deductions of the advance of Mercury’s perihelion and of the bending of light, considerable doubt existed as to whether or not the general theory did entail a red shift for the solar spectrum.
    [Show full text]
  • The Theory of Relativity and Applications: a Simple Introduction
    The Downtown Review Volume 5 Issue 1 Article 3 December 2018 The Theory of Relativity and Applications: A Simple Introduction Ellen Rea Cleveland State University Follow this and additional works at: https://engagedscholarship.csuohio.edu/tdr Part of the Engineering Commons, and the Physical Sciences and Mathematics Commons How does access to this work benefit ou?y Let us know! Recommended Citation Rea, Ellen. "The Theory of Relativity and Applications: A Simple Introduction." The Downtown Review. Vol. 5. Iss. 1 (2018) . Available at: https://engagedscholarship.csuohio.edu/tdr/vol5/iss1/3 This Article is brought to you for free and open access by the Student Scholarship at EngagedScholarship@CSU. It has been accepted for inclusion in The Downtown Review by an authorized editor of EngagedScholarship@CSU. For more information, please contact [email protected]. Rea: The Theory of Relativity and Applications What if I told you that time can speed up and slow down? What if I told you that everything you think you know about gravity is a lie? When Albert Einstein presented his theory of relativity to the world in the early 20th century, he was proposing just that. And what’s more? He’s been proven correct. Einstein’s theory has two parts: special relativity, which deals with inertial reference frames and general relativity, which deals with the curvature of space- time. A surface level study of the theory and its consequences followed by a look at some of its applications will provide an introduction to one of the most influential scientific discoveries of the last century.
    [Show full text]
  • Albert Einstein and Relativity
    The Himalayan Physics, Vol.1, No.1, May 2010 Albert Einstein and Relativity Kamal B Khatri Department of Physics, PN Campus, Pokhara, Email: [email protected] Albert Einstein was born in Germany in 1879.In his the follower of Mach and his nascent concept helped life, Einstein spent his most time in Germany, Italy, him to enter the world of relativity. Switzerland and USA.He is also a Nobel laureate and worked mostly in theoretical physics. Einstein In 1905, Einstein propounded the “Theory of is best known for his theories of special and general Special Relativity”. This theory shows the observed relativity. I will not be wrong if I say Einstein a independence of the speed of light on the observer’s deep thinker, a philosopher and a real physicist. The state of motion. Einstein deduced from his concept philosophies of Henri Poincare, Ernst Mach and of special relativity the twentieth century’s best David Hume infl uenced Einstein’s scientifi c and known equation, E = m c2.This equation suggests philosophical outlook. that tiny amounts of mass can be converted into huge amounts of energy which in deed, became the Einstein at the age of 4, his father showed him a boon for the development of nuclear power. pocket compass, and Einstein realized that there must be something causing the needle to move, despite Einstein realized that the principle of special relativity the apparent ‘empty space’. This shows Einstein’s could be extended to gravitational fi elds. Since curiosity to the space from his childhood. The space Einstein believed that the laws of physics were local, of our intuitive understanding is the 3-dimensional described by local fi elds, he concluded from this that Euclidean space.
    [Show full text]
  • Derivation of Generalized Einstein's Equations of Gravitation in Some
    Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 5 February 2021 doi:10.20944/preprints202102.0157.v1 Derivation of generalized Einstein's equations of gravitation in some non-inertial reference frames based on the theory of vacuum mechanics Xiao-Song Wang Institute of Mechanical and Power Engineering, Henan Polytechnic University, Jiaozuo, Henan Province, 454000, China (Dated: Dec. 15, 2020) When solving the Einstein's equations for an isolated system of masses, V. Fock introduces har- monic reference frame and obtains an unambiguous solution. Further, he concludes that there exists a harmonic reference frame which is determined uniquely apart from a Lorentz transformation if suitable supplementary conditions are imposed. It is known that wave equations keep the same form under Lorentz transformations. Thus, we speculate that Fock's special harmonic reference frames may have provided us a clue to derive the Einstein's equations in some special class of non-inertial reference frames. Following this clue, generalized Einstein's equations in some special non-inertial reference frames are derived based on the theory of vacuum mechanics. If the field is weak and the reference frame is quasi-inertial, these generalized Einstein's equations reduce to Einstein's equa- tions. Thus, this theory may also explain all the experiments which support the theory of general relativity. There exist some differences between this theory and the theory of general relativity. Keywords: Einstein's equations; gravitation; general relativity; principle of equivalence; gravitational aether; vacuum mechanics. I. INTRODUCTION p. 411). Theoretical interpretation of the small value of Λ is still open [6]. The Einstein's field equations of gravitation are valid 3.
    [Show full text]
  • Voigt Transformations in Retrospect: Missed Opportunities?
    Voigt transformations in retrospect: missed opportunities? Olga Chashchina Ecole´ Polytechnique, Palaiseau, France∗ Natalya Dudisheva Novosibirsk State University, 630 090, Novosibirsk, Russia† Zurab K. Silagadze Novosibirsk State University and Budker Institute of Nuclear Physics, 630 090, Novosibirsk, Russia.‡ The teaching of modern physics often uses the history of physics as a didactic tool. However, as in this process the history of physics is not something studied but used, there is a danger that the history itself will be distorted in, as Butterfield calls it, a “Whiggish” way, when the present becomes the measure of the past. It is not surprising that reading today a paper written more than a hundred years ago, we can extract much more of it than was actually thought or dreamed by the author himself. We demonstrate this Whiggish approach on the example of Woldemar Voigt’s 1887 paper. From the modern perspective, it may appear that this paper opens a way to both the special relativity and to its anisotropic Finslerian generalization which came into the focus only recently, in relation with the Cohen and Glashow’s very special relativity proposal. With a little imagination, one can connect Voigt’s paper to the notorious Einstein-Poincar´epri- ority dispute, which we believe is a Whiggish late time artifact. We use the related historical circumstances to give a broader view on special relativity, than it is usually anticipated. PACS numbers: 03.30.+p; 1.65.+g Keywords: Special relativity, Very special relativity, Voigt transformations, Einstein-Poincar´epriority dispute I. INTRODUCTION Sometimes Woldemar Voigt, a German physicist, is considered as “Relativity’s forgotten figure” [1].
    [Show full text]
  • + Gravity Probe B
    NATIONAL AERONAUTICS AND SPACE ADMINISTRATION Gravity Probe B Experiment “Testing Einstein’s Universe” Press Kit April 2004 2- Media Contacts Donald Savage Policy/Program Management 202/358-1547 Headquarters [email protected] Washington, D.C. Steve Roy Program Management/Science 256/544-6535 Marshall Space Flight Center steve.roy @msfc.nasa.gov Huntsville, AL Bob Kahn Science/Technology & Mission 650/723-2540 Stanford University Operations [email protected] Stanford, CA Tom Langenstein Science/Technology & Mission 650/725-4108 Stanford University Operations [email protected] Stanford, CA Buddy Nelson Space Vehicle & Payload 510/797-0349 Lockheed Martin [email protected] Palo Alto, CA George Diller Launch Operations 321/867-2468 Kennedy Space Center [email protected] Cape Canaveral, FL Contents GENERAL RELEASE & MEDIA SERVICES INFORMATION .............................5 GRAVITY PROBE B IN A NUTSHELL ................................................................9 GENERAL RELATIVITY — A BRIEF INTRODUCTION ....................................17 THE GP-B EXPERIMENT ..................................................................................27 THE SPACE VEHICLE.......................................................................................31 THE MISSION.....................................................................................................39 THE AMAZING TECHNOLOGY OF GP-B.........................................................49 SEVEN NEAR ZEROES.....................................................................................58
    [Show full text]
  • Lecture 17 Relativity A2020 Prof. Tom Megeath What Are the Major Ideas
    4/1/10 Lecture 17 Relativity What are the major ideas of A2020 Prof. Tom Megeath special relativity? Einstein in 1921 (born 1879 - died 1955) Einstein’s Theories of Relativity Key Ideas of Special Relativity • Special Theory of Relativity (1905) • No material object can travel faster than light – Usual notions of space and time must be • If you observe something moving near light speed: revised for speeds approaching light speed (c) – Its time slows down – Its length contracts in direction of motion – E = mc2 – Its mass increases • Whether or not two events are simultaneous depends on • General Theory of Relativity (1915) your perspective – Expands the ideas of special theory to include a surprising new view of gravity 1 4/1/10 Inertial Reference Frames Galilean Relativity Imagine two spaceships passing. The astronaut on each spaceship thinks that he is stationary and that the other spaceship is moving. http://faraday.physics.utoronto.ca/PVB/Harrison/Flash/ Which one is right? Both. ClassMechanics/Relativity/Relativity.html Each one is an inertial reference frame. Any non-rotating reference frame is an inertial reference frame (space shuttle, space station). Each reference Speed limit sign posted on spacestation. frame is equally valid. How fast is that man moving? In contrast, you can tell if a The Solar System is orbiting our Galaxy at reference frame is rotating. 220 km/s. Do you feel this? Absolute Time Absolutes of Relativity 1. The laws of nature are the same for everyone In the Newtonian universe, time is absolute. Thus, for any two people, reference frames, planets, etc, 2.
    [Show full text]
  • A Solution of the Interpretation Problem of Lorentz Transformations
    Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 30 July 2020 doi:10.20944/preprints202007.0705.v1 Article A Solution of the Interpretation Problem of Lorentz Transformations Grit Kalies* HTW University of Applied Sciences Dresden; 1 Friedrich-List-Platz, D-01069 Dresden, [email protected] * Correspondence: [email protected], Tel.: +49-351-462-2552 Abstract: For more than one hundred years, scientists dispute the correct interpretation of Lorentz transformations within the framework of the special theory of relativity of Albert Einstein. On the one hand, the changes in length, time and mass with increasing velocity are interpreted as apparent due to the observer dependence within special relativity. On the other hand, real changes are described corresponding to the experimental evidence of mass increase in particle accelerators or of clock delay. This ambiguity is accompanied by an ongoing controversy about valid Lorentz-transformed thermodynamic quantities such as entropy, pressure and temperature. In this paper is shown that the interpretation problem of the Lorentz transformations is genuinely anchored within the postulates of special relativity and can be solved on the basis of the thermodynamic approach of matter-energy equivalence, i.e. an energetic distinction between matter and mass. It is suggested that the velocity-dependent changes in state quantities are real in each case, in full agreement with the experimental evidence. Keywords: interpretation problem; Lorentz transformation; special relativity; thermodynamics; potential energy; space; time; entropy; non-mechanistic ether theory © 2020 by the author(s). Distributed under a Creative Commons CC BY license. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 30 July 2020 doi:10.20944/preprints202007.0705.v1 2 of 25 1.
    [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]
  • 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.
    [Show full text]