DOCSLIB.ORG

Algebrodynamics Over Complex Space and Phase Extension of the Minkowski Geometry*

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Algebrodynamics Over Complex Space and Phase Extension of the Minkowski Geometry* ISSN 1063-7788, Physics of Atomic Nuclei, 2009, Vol. 72, No. 5, pp. 813–827. c Pleiades Publishing, Ltd., 2009. ELEMENTARY PARTICLES AND FIELDS Theory Algebrodynamics Over Complex Space and Phase Extension of the Minkowski Geometry* V. V. Kassandrov ** Institute of Gravitation and Cosmology, Peoples’ Friendship University, Moscow, Russia Received October 14, 2008 Abstract—First principles should predetermine physical geometry and dynamics both together. In the “algebrodynamics” they follow solely from the properties of biquaternion algebra B and the analysis over B. We briefly present the algebrodynamics over Minkowski background based on a nonlinear generalization to B of the Cauchi–Riemann analyticity conditions. Further, we consider the effective real geometry uniquely resulting from the structure of B multiplication and found it to be of the Minkowski type, with an additional phase invariant. Then we pass to study the primordial dynamics that takes place in the complex B space and brings into consideration a number of remarkable structures: an ensemble of identical correlated matter pre-elements (“duplicons”), caustic-like signals (interaction carriers), a concept of random complex time resulting in irreversibility of physical time at macrolevel, etc. In partucular, the concept of “dimerous electron” naturally arises in the framework of complex algebrodynamics and, together with the above- mentioned phase invariant, allows for a novel approach to explanation of quantum interference phenomena alternative to recently accepted wave–particle dualism paradigm. PACS num b e r s : 02.30.-f, 03.30.+p, 03.50.-z, 03.65.Vf DOI: 10.1134/S106377880905010X 1. STATUS OF MINKOWSKI GEOMETRY additional “hidden” dimensions (in the Kaluza– AND THE ALGEBRODYNAMICAL Klein formalism). There have been considered also PA RA D I G M the models of discrete space–time, the challenging scheme of causal sets [1] among them. A whole century after German Minkowski intro- However, none of modified space–time geometries duced his famous conception of the 4D space–time has become generally accepted and able to replace the continuum, we come to realize the restricted nature Minkowski geometry. Indeed, especial significance of this conception and the necessity of its revision, and reliability of the latter is stipulated by its orig- supplement and derivation from some general and ination from trustworthy physical principles of STR fundamental principle. and, particularly, from the structure of experimentally verificated Maxwell equations. None of its subsequent Indeed, formalism of the 4D space–time geometry modifications can boast of such a firm and uniquely was indispensable to ultimately formulate the Spe- interpreted experimental ground. cial Theory of Relativity (STR), to ascertain basic From the epoch of Minkowski we did not get better symmetries of fundamental physical equations and comprehension of the true geometry of our World, its related conservation laws. It was also the Minkowski hidden structure and origination. In fact, we are not geometry that served as a base for formulation of the even aware whether physical geometry is Riemannian concept of curved space–time in the framework of the or flat, has four dimensions or more, etc. Essentially, Einstein’s General Theory of Relativity (GTR). we can say nothing definite about the topology of Subsequently, Minkowski geometry and its space (both global and at microscale). And, of course, pseudo-Riemannian analog have been generalized we still have no satisfactory answer to sacramental via introduction of effective geometries related to question: “Why is the space three dimensional (at correspondent field dynamics (in the formalism of least, at macrolevel)?” Finally, an “eternal” question about the sense and origin of physical time stands as fiber bundles), or via exchange of Riemannnian before on the agenda. manifold for spaces with torsion, nonmetricity or Meanwhile, the Minkowski geometry suffers itself ∗The text was submitted by the author in English. from grave shortcomings, both from phenomenolog- **E-mail: [email protected] ical and generic viewpoints. To be concrete, complex 813 814 KASSANDROV structure of field equations accepted in quantum the- At a still more fundamental level of consid- ory results, generally, in string-like structure of field eration, one assumes to derive the geometry of singularities (perhaps, it was first noticed by Dirac [2]) physical space–time from some primordial principle and, moreover, these strings are unstable and, as a encoding it (perhaps, together with physical dy- rule, radiate themselves to infinity (see, e.g., [3] and namics). One can try to relate such an elementary the example in Section 2). Code of Nature with some exceptional symme- Another drawback (exactly, insufficiency) of the try (theory of physical structures of Kulakov [11] Minkowski geometry is the absence of fundamental and binary geometrophysics of Vladimirov [12]), distinction of temporal and spatial coordinates within group or algebra (quaternionic theory of relativ- its framework. Time enters the Minkowski metrical ity of Yefremov [13] and algebrodynamics of Kas- form on an equal footing with ordinary coordinates sandrov [14, 15]), with algebraically distinguished though with opposite sign. In other words, in the geometry (Finslerian anisotropic geometry of Bo- framework of the STR geometry time does not reveal goslovsky [16] and geometry of polynumbers of itself as an evolution parameter as it was even in Pavlov [17]) as well as with some special “World the antecedent Newton’s picture of the World. At function” (metrical geometry of Rylov [18]). a pragmatic level this results, in particular, in the Generally, all the above-mentioned and similar difficulty to coordinate “times” of various interacting approaches affecting the very foundations of physics (entangled) particles in an ensemble, in impossibility differ essentially one from another in the charac- to introduce universal global time and to adjust the ter of the first principle (being either purely physical latter to proper times of different observers, or in or abstract in nature), in the degree of confidence the absence of clear comprehension of the passage of of their authors to recently predominant paradigms local time and dependence of its rate on matter. All (Lorentz invariance, Standard model, etc.) and in these problems are widely discussed in physical liter- their attitude towards the necessity to reproduce, in ature (see, e.g., [4]) but are still far from resolution. the framework of the original approach, the principal notions and mathematical insrumentation of canon- However, the main discontent with generally ac- ical schemes (of Lagrangian formalism, quantization cepted Minkowski geometry is related to the fact that procedure, Minkowski space itself, etc.). In this re- this geometry does not follow from some deep log- spect the neo-Pythagorean philosophical paradigm ical premises or exceptional numerical structures. professing by the author [19–21] seems most consis- This is still more valid with respect to generaliza- tent and promising, though difficult in realization. tions of space–time structure arising, in particular, in the superstring theories (11D spaces) and in other Accordingly, under construction of an algebraic approaches for purely phenomenological, “technical” (logical, numerical) “Theory of Everything” one reasons which in no way can replace the transparent should forget all of the known physical theories and general physical principles of STR, of relativity and even experimental facts and to unprejudicely and of universal velocity of interaction propagation. read out the laws of physical World in the internal properties of some exceptional abstract primordial At present, physics and mathematics are mature structure, adding and changing nothing in the course enough for construction of multidimensional geome- of this for “better correspondence with experiment”. tries with different number of spatial and temporal In this connection, one should be ready that physical dimensions. Moreover, they aim to create a general picture of the World arising at the output could have unified conception from which it would follow definite little in common with recently accepted one and that conclusions on the true geometry of physical space the real language of Nature might be quite different and on the properties and meaning of physical time, from that we have thought out for better description on the dynamics of Time itself! of observable phenomena. In this situation none In most of approaches of such kind the Minkowski principle of correspondence with former theories space does not reproduce itself in its canonical form could be applied. but is either deformed through some parameter (say, We have no opportunity to go into details of the fundamental length and mass in the paradigm of neo-Pythagorean philosophy, quite novel and radi- Kadyshevsky [5]) under correspondence with canon- cal for modern science, sending the reader to [19– ical scheme, or changes its structure in a radical 21]. Instead, in Section 2 we briefly present its re- way. The latter takes place, in particular, in the the- alization in the framework of the “old” version of ory of Euclidean time developed by Pestov [6] (in algebrodynamics developed during the period 1980– this connection, see also [7]), in the concept of Clif- 2005 [14, 15]. Therein an attempt has been under- ford space–time of Hestenes–Pavsic (see, e.g., [8, taken to obtain the principal equations of physical
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]
  • Chapter 5 the Relativistic Point Particle
    Chapter 5 The Relativistic Point Particle To formulate the dynamics of a system we can write either the equations of motion, or alternatively, an action. In the case of the relativistic point par- ticle, it is rather easy to write the equations of motion. But the action is so physical and geometrical that it is worth pursuing in its own right. More importantly, while it is difficult to guess the equations of motion for the rela- tivistic string, the action is a natural generalization of the relativistic particle action that we will study in this chapter. We conclude with a discussion of the charged relativistic particle. 5.1 Action for a relativistic point particle How can we find the action S that governs the dynamics of a free relativis- tic particle? To get started we first think about units. The action is the Lagrangian integrated over time, so the units of action are just the units of the Lagrangian multiplied by the units of time. The Lagrangian has units of energy, so the units of action are L2 ML2 [S]=M T = . (5.1.1) T 2 T Recall that the action Snr for a free non-relativistic particle is given by the time integral of the kinetic energy: 1 dx S = mv2(t) dt , v2 ≡ v · v, v = . (5.1.2) nr 2 dt 105 106 CHAPTER 5. THE RELATIVISTIC POINT PARTICLE The equation of motion following by Hamilton’s principle is dv =0. (5.1.3) dt The free particle moves with constant velocity and that is the end of the story.
    [Show full text]
  • K-Theory and Algebraic Geometry
    http://dx.doi.org/10.1090/pspum/058.2 Recent Titles in This Series 58 Bill Jacob and Alex Rosenberg, editors, ^-theory and algebraic geometry: Connections with quadratic forms and division algebras (University of California, Santa Barbara) 57 Michael C. Cranston and Mark A. Pinsky, editors, Stochastic analysis (Cornell University, Ithaca) 56 William J. Haboush and Brian J. Parshall, editors, Algebraic groups and their generalizations (Pennsylvania State University, University Park, July 1991) 55 Uwe Jannsen, Steven L. Kleiman, and Jean-Pierre Serre, editors, Motives (University of Washington, Seattle, July/August 1991) 54 Robert Greene and S. T. Yau, editors, Differential geometry (University of California, Los Angeles, July 1990) 53 James A. Carlson, C. Herbert Clemens, and David R. Morrison, editors, Complex geometry and Lie theory (Sundance, Utah, May 1989) 52 Eric Bedford, John P. D'Angelo, Robert E. Greene, and Steven G. Krantz, editors, Several complex variables and complex geometry (University of California, Santa Cruz, July 1989) 51 William B. Arveson and Ronald G. Douglas, editors, Operator theory/operator algebras and applications (University of New Hampshire, July 1988) 50 James Glimm, John Impagliazzo, and Isadore Singer, editors, The legacy of John von Neumann (Hofstra University, Hempstead, New York, May/June 1988) 49 Robert C. Gunning and Leon Ehrenpreis, editors, Theta functions - Bowdoin 1987 (Bowdoin College, Brunswick, Maine, July 1987) 48 R. O. Wells, Jr., editor, The mathematical heritage of Hermann Weyl (Duke University, Durham, May 1987) 47 Paul Fong, editor, The Areata conference on representations of finite groups (Humboldt State University, Areata, California, July 1986) 46 Spencer J. Bloch, editor, Algebraic geometry - Bowdoin 1985 (Bowdoin College, Brunswick, Maine, July 1985) 45 Felix E.
    [Show full text]
  • Supplemental Lecture 5 Precession in Special and General Relativity
    Supplemental Lecture 5 Thomas Precession and Fermi-Walker Transport in Special Relativity and Geodesic Precession in General Relativity Abstract This lecture considers two topics in the motion of tops, spins and gyroscopes: 1. Thomas Precession and Fermi-Walker Transport in Special Relativity, and 2. Geodesic Precession in General Relativity. A gyroscope (“spin”) attached to an accelerating particle traveling in Minkowski space-time is seen to precess even in a torque-free environment. The angular rate of the precession is given by the famous Thomas precession frequency. We introduce the notion of Fermi-Walker transport to discuss this phenomenon in a systematic fashion and apply it to a particle propagating on a circular orbit. In a separate discussion we consider the geodesic motion of a particle with spin in a curved space-time. The spin necessarily precesses as a consequence of the space-time curvature. We illustrate the phenomenon for a circular orbit in a Schwarzschild metric. Accurate experimental measurements of this effect have been accomplished using earth satellites carrying gyroscopes. This lecture supplements material in the textbook: Special Relativity, Electrodynamics and General Relativity: From Newton to Einstein (ISBN: 978-0-12-813720-8). The term “textbook” in these Supplemental Lectures will refer to that work. Keywords: Fermi-Walker transport, Thomas precession, Geodesic precession, Schwarzschild metric, Gravity Probe B (GP-B). Introduction In this supplementary lecture we discuss two instances of precession: 1. Thomas precession in special relativity and 2. Geodesic precession in general relativity. To set the stage, let’s recall a few things about precession in classical mechanics, Newton’s world, as we say in the textbook.
    [Show full text]
  • APPARENT SIMULTANEITY Hanoch Ben-Yami
    APPARENT SIMULTANEITY Hanoch Ben-Yami ABSTRACT I develop Special Relativity with backward-light-cone simultaneity, which I call, for reasons made clear in the paper, ‘Apparent Simultaneity’. In the first section I show some advantages of this approach. I then develop the kinematics in the second section. In the third section I apply the approach to the Twins Paradox: I show how it removes the paradox, and I explain why the paradox was a result of an artificial symmetry introduced to the description of the process by Einstein’s simultaneity definition. In the fourth section I discuss some aspects of dynamics. I conclude, in a fifth section, with a discussion of the nature of light, according to which transmission of light energy is a form of action at a distance. 1 Considerations Supporting Backward-Light-Cone Simultaneity ..........................1 1.1 Temporal order...............................................................................................2 1.2 The meaning of coordinates...........................................................................3 1.3 Dependence of simultaneity on location and relative motion........................4 1.4 Appearance as Reality....................................................................................5 1.5 The Time Lag Argument ...............................................................................6 1.6 The speed of light as the greatest possible speed...........................................8 1.7 More on the Apparent Simultaneity approach ...............................................9
    [Show full text]
  • Hyperbolicity of Hermitian Forms Over Biquaternion Algebras
    HYPERBOLICITY OF HERMITIAN FORMS OVER BIQUATERNION ALGEBRAS NIKITA A. KARPENKO Abstract. We show that a non-hyperbolic hermitian form over a biquaternion algebra over a field of characteristic 6= 2 remains non-hyperbolic over a generic splitting field of the algebra. Contents 1. Introduction 1 2. Notation 2 3. Krull-Schmidt principle 3 4. Splitting off a motivic summand 5 5. Rost correspondences 7 6. Motivic decompositions of some isotropic varieties 12 7. Proof of Main Theorem 14 References 16 1. Introduction Throughout this note (besides of x3 and x4) F is a field of characteristic 6= 2. The basic reference for the staff related to involutions on central simple algebras is [12].p The degree deg A of a (finite dimensional) central simple F -algebra A is the integer dimF A; the index ind A of A is the degree of a central division algebra Brauer-equivalent to A. Conjecture 1.1. Let A be a central simple F -algebra endowed with an orthogonal invo- lution σ. If σ becomes hyperbolic over the function field of the Severi-Brauer variety of A, then σ is hyperbolic (over F ). In a stronger version of Conjecture 1.1, each of two words \hyperbolic" is replaced by \isotropic", cf. [10, Conjecture 5.2]. Here is the complete list of indices ind A and coindices coind A = deg A= ind A of A for which Conjecture 1.1 is known (over an arbitrary field of characteristic 6= 2), given in the chronological order: • ind A = 1 | trivial; Date: January 2008. Key words and phrases.
    [Show full text]
  • Einstein's Boxes
    Einstein’s Boxes Travis Norsen∗ Marlboro College, Marlboro, Vermont 05344 (Dated: February 1, 2008) At the 1927 Solvay conference, Albert Einstein presented a thought experiment intended to demon- strate the incompleteness of the quantum mechanical description of reality. In the following years, the experiment was modified by Einstein, de Broglie, and several other commentators into a simple scenario involving the splitting in half of the wave function of a single particle in a box. This paper collects together several formulations of this thought experiment from the literature, analyzes and assesses it from the point of view of the Einstein-Bohr debates, the EPR dilemma, and Bell’s the- orem, and argues for “Einstein’s Boxes” taking its rightful place alongside similar but historically better known quantum mechanical thought experiments such as EPR and Schr¨odinger’s Cat. I. INTRODUCTION to Copenhagen such as Bohm’s non-local hidden variable theory. Because of its remarkable simplicity, the Einstein Boxes thought experiment also is well-suited as an intro- It is well known that several of quantum theory’s duction to these topics for students and other interested founders were dissatisfied with the theory as interpreted non-experts. by Niels Bohr and other members of the Copenhagen school. Before about 1928, for example, Louis de Broglie The paper is organized as follows. In Sec. II we intro- duce Einstein’s Boxes by quoting a detailed description advocated what is now called a hidden variable theory: a pilot-wave version of quantum mechanics in which parti- due to de Broglie. We compare it to the EPR argument and discuss how it fares in eluding Bohr’s rebuttal of cles follow continuous trajectories, guided by a quantum wave.1 David Bohm’s2 rediscovery and completion of the EPR.
    [Show full text]
  • Einstein's Gravitational Field
    Einstein’s gravitational field Abstract: There exists some confusion, as evidenced in the literature, regarding the nature the gravitational field in Einstein’s General Theory of Relativity. It is argued here that this confusion is a result of a change in interpretation of the gravitational field. Einstein identified the existence of gravity with the inertial motion of accelerating bodies (i.e. bodies in free-fall) whereas contemporary physicists identify the existence of gravity with space-time curvature (i.e. tidal forces). The interpretation of gravity as a curvature in space-time is an interpretation Einstein did not agree with. 1 Author: Peter M. Brown e-mail: [email protected] 2 INTRODUCTION Einstein’s General Theory of Relativity (EGR) has been credited as the greatest intellectual achievement of the 20th Century. This accomplishment is reflected in Time Magazine’s December 31, 1999 issue 1, which declares Einstein the Person of the Century. Indeed, Einstein is often taken as the model of genius for his work in relativity. It is widely assumed that, according to Einstein’s general theory of relativity, gravitation is a curvature in space-time. There is a well- accepted definition of space-time curvature. As stated by Thorne 2 space-time curvature and tidal gravity are the same thing expressed in different languages, the former in the language of relativity, the later in the language of Newtonian gravity. However one of the main tenants of general relativity is the Principle of Equivalence: A uniform gravitational field is equivalent to a uniformly accelerating frame of reference. This implies that one can create a uniform gravitational field simply by changing one’s frame of reference from an inertial frame of reference to an accelerating frame, which is rather difficult idea to accept.
    [Show full text]
  • Theory of Relativity
    Christian Bar¨ Theory of Relativity Summer Term 2013 OTSDAM P EOMETRY IN G Version of August 26, 2013 Contents Preface iii 1 Special Relativity 1 1.1 Classical Kinematics ............................... 1 1.2 Electrodynamics ................................. 5 1.3 The Lorentz group and Minkowski geometry .................. 7 1.4 Relativistic Kinematics .............................. 20 1.5 Mass and Energy ................................. 36 1.6 Closing Remarks about Special Relativity .................... 41 2 General Relativity 43 2.1 Classical theory of gravitation .......................... 43 2.2 Equivalence Principle and the Einstein Field Equations ............. 49 2.3 Robertson-Walker spacetime ........................... 55 2.4 The Schwarzschild solution ............................ 66 Bibliography 77 Index 79 i Preface These are the lecture notes of an introductory course on relativity theory that I gave in 2013. Ihe course was designed such that no prior knowledge of differential geometry was required. The course itself also did not introduce differential geometry (as it is often done in relativity classes). Instead, students unfamiliar with differential geometry had the opportunity to learn the subject in another course precisely set up for this purpose. This way, the relativity course could concentrate on its own topic. Of course, there is a price to pay; the first half of the course was dedicated to Special Relativity which does not require much mathematical background. Only the second half then deals with General Relativity. This gave the students time to acquire the geometric concepts. The part on Special Relativity briefly recalls classical kinematics and electrodynamics empha- sizing their conceptual incompatibility. It is then shown how Minkowski geometry is used to unite the two theories and to obtain what we nowadays call Special Relativity.
    [Show full text]
  • Mach's Principle: Exact Frame
    MACH’S PRINCIPLE: EXACT FRAME-DRAGGING BY ENERGY CURRENTS IN THE UNIVERSE CHRISTOPH SCHMID Institute of Theoretical Physics, ETH Zurich CH-8093 Zurich, Switzerland We show that the dragging of axis directions of local inertial frames by a weighted average of the energy currents in the universe (Mach’s postulate) is exact for all linear perturbations of all Friedmann-Robertson-Walker universes. 1 Mach’s Principle 1.1 The Observational Fact: ’Mach zero’ The time-evolution of local inertial axes, i.e. the local non-rotating frame is experimentally determined by the spin axes of gyroscopes, as in inertial guidance systems in airplanes and satellites. This is true both in Newtonian physics (Foucault 1852) and in General Relativity. It is an observational fact within present-day accuracy that the spin axes of gyroscopes do not precess relative to quasars. This observational fact has been named ’Mach zero’, where ’zero’ designates that this fact is not yet Mach’s principle, it is just the observational starting point. — There is an extremely small dragging effect by the rotating Earth on the spin axes of gyroscopes, the Lense - Thirring effect, which makes the spin axes of gyroscopes precess relative to quasars by 43 milli-arc-sec per year. It is hoped that one will be able to detect this effect by further analysis of the data which have been taken by Gravity Probe B. 1.2 The Question What physical cause explains the observational fact ’Mach zero’ ? Equivalently: What physical cause determines the time-evolution of gyroscope axes ? In the words of John A.
    [Show full text]
  • Class 4: Space-Time Diagrams
    Class 4: Space-time Diagrams In this class we will explore how space-time diagrams may be used to visualize events and causality in relativity Class 4: Space-time Diagrams At the end of this session you should be able to … • … create a space-time diagram showing events and world lines in a given reference frame • … determine geometrically which events are causally connected, via the concept of the light cone • … understand that events separated by a constant space-time interval from the origin map out a hyperbola in space-time • … use space-time diagrams to relate observations in different inertial frames, via tilted co-ordinate systems What is a space-time diagram? • A space-time diagram is a graph showing the position of objects (events) in a reference frame, as a function of time • Conventionally, space (�) is represented in the horizontal direction, and time (�) runs upwards �� Here is an event This is the path of a light ray travelling in the �- direction (i.e., � = ��), We have scaled time by a which makes an angle of factor of �, so it has the 45° with the axis same dimensions as space What would be the � path of an object moving at speed � < �? What is a space-time diagram? • The path of an object (or light ray) moving through a space- time diagram is called a world line of that object, and may be thought of as a chain of many events • Note that a world line in a space-time diagram may not be the same shape as the path of an object through space �� Rocket ship accelerating in �-direction (path in Rocket ship accelerating in �-
    [Show full text]