DOCSLIB.ORG

Invariants and Conservation Laws 107 Conserved

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Invariants and Conservation Laws 107 Conserved © Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. 106 II. Concepts 2 which, for a sufficiently fine partition, produces the integral enclosure ∈ 66 I 0.347400172649. Thus, it turns out that the result from simpson was completely wrong! This is one example of the impor- tance of rigorous computations. 0 3 Recent Developments There is currently an ongoing effort within the IEEE community to standardize the implementation of inter- val arithmetic. The hope is that we will enable com- puter manufacturers to incorporate these types of com- –2 putations at the hardware level. This would remove –5 0 5 the large computational penalty incurred by repeat- Figure 1 Successively tighter enclosures of a graph. edly having to switch rounding modes—a task that central processing units were not designed to perform 2 Interval Analysis efficiently. The inclusion principle (2) enables us to capture contin- Further Reading uous properties of a function, using only a finite num- Alefeld, G., and J. Herzberger. 1983. Introduction to Interval ber of operations. Its most important use is to explic- Computations. New York: Academic Press. itly bound discretization errors that naturally arise in Kulisch, U. W., and W. L. Miranker. 1981. Computer Arith- numerical algorithms. metic in Theory and Practice. New York: Academic Press. 3 As an example, consider the function f(x)= cos x+ Lerch, M., G. Tischler, J. Wolff von Gudenberg, W. Hof- sin x on the domain x = [−5, 5]. For any decompo- schuster, and W. Krämer. 2006. filib++, a fast inter- sition of the domain x into a finite set of subinter- val library supporting containment computations. ACM = n Transactions on Mathematical Software 32(2):299–324. vals x i=1 xi, we can form the set-valued graph Moore, R. E., R. B. Kearfott, and M. J. Cloud. 2009. Introduc- consisting of the pairs (x1,F(x1)),...,(xn,F(xn)).As tion to Interval Analysis. Philadelphia, PA: SIAM. the partition is made finer (that is, as maxi diam(xi) is Rump, S. M. 1999. INTLAB—INTerval LABoratory. In Devel- made smaller), the set-valued graph tends to the graph opments in Reliable Computing, edited by T. Csendes, of f (see figure 1). And, most importantly, every such pp. 77–104. Dordrecht: Kluwer Academic. set-valued graph contains the graph of f . Tucker, W. 2011. Validated Numerics: A Short Introduc- This way of incorporating the discretization errors tion to Rigorous Computations. Princeton, NJ: Princeton is extremely useful for quadrature, optimization, and University Press. equation solving. As one example, suppose we wish to = 8 + x compute the definite integral I 0 sin(x e ) dx. A MATLAB function simpson that implements a sim- II.21 Invariants and Conservation ple textbook adaptive Simpson quadrature algorithm Laws produces the following result. Mark R. Dennis % Compute integral I with tolerance 1e-6. As important as the study of change in the mathemati- >> I = simpson(@(x) sin(x + exp(x)), 0, 8) cal representation of physical phenomena is the study I= of invariants. Physical laws often depend only on the 0.251102722027180 relative positions and times between phenomena, so certain physical quantities do not change; i.e., they are A(very naive) set-valued approach to quadrature is to invariant, under continuous translation or rotation of enclose the integral I via the spatial axes. Furthermore, as the spatial configura- n tion of a system evolves with time, quantities such as I ∈ F(x ) diam(x ), i i total energy may remain unchanged; that is, they are i=1 For general queries, contact [email protected] © Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. II.21. Invariants and Conservation Laws 107 conserved. The study of invariants has been a remark- that physical phenomena do not depend on the overall ably successful approach to the mathematical formu- phase (argument) of this vector. Extension of Noether’s lation of physical laws, and the study of continuous theorem here leads to the conservation of electric symmetries and conservation laws—which are related charge, and extension to Yang–Mills theories provides by the result known as Noether’s theorem—has become other conserved quantities associated with the nuclear a systematic part of our description of physics over the forces studied in contemporary fundamental particle last century, from the atomic scale to the cosmic scale. physics. Other phenomena, such as the Higgs mecha- As an example of so-called Galilean invariance, New- nism (leading to the Higgs boson recently discovered ton’s force law keeps the same form when the velocity in high-energy experiments), are a consequence of the of the frame of reference (i.e., the coordinate system breaking of certain quantum symmetries in certain low- specified by x-, y-, and z-axes) is changed by adding energy regimes. Symmetry and symmetry breaking in a constant; this is equivalent to adding the same con- quantum theory are discussed briefly at the end of this stant velocity to all the particles in a mechanical sys- article. tem. Other quantities do change under such a veloc- Spatial vectors, such as r = (x,y,z), represent the 1 | |2 spatial distance between a chosen point and the ori- ity transformation, such as the kinetic energy 2 m v (for a particle of mass m and velocity v); however, for gin, and of course the vector between two such points an evolving, nondissipative system such as a bouncing, r2 − r1 is independent of translations of this origin. perfectly elastic rubber ball, the total energy is constant Similarly, the scalar product r2 ·r1 is unchanged under in time—that is, energy is conserved. rotation of the coordinate system by an orthogonal The development of our understanding of fundamen- matrix R, under which r → Rr. tal laws of dynamics can be interpreted by progres- Continuous groups of transformations such as trans- sively more sophisticated and general representations lation and rotation, and their matrix representations, of space and time themselves: ancient Greek physi- are an important tool used in calculations of invari- cal science assumed absolute space with a privileged ants. For example, the set of two-dimensional matrices cos θ − sin θ spatial point (the center of the Earth), through static sin θ cos θ , representing rotations through angles θ, Euclidean space where all spatial points are equiv- may be considered as a continuous one-parameter Abe- matrix expo- alent, through classical mechanics [IV.19] where lian group of matrices generated by the nential θA 0 −1 all inertial frames, moving at uniform velocity with [II.14] e , where A is the generator 10 . respect to each other, are equivalent according to The generator itself is found as the derivative of the Newton’s first law, to the modern theories of spe- original matrix with respect to θ, evaluated at θ = 0. cial and general relativity. The theory of relativity Translations are less obviously represented by matri- (both general and special) is motivated by Einstein’s ces; one approach is to append an extra dimension to principle of covariance, which is described below. the position vector with unit entry, such as (1,x)spec- In this theory, space and time in different frames ifying one-dimensional position x; a translation by X is of reference are treated as coordinate systems on a thus represented by four-dimensional pseudo-Riemannian manifold (whose 10 00 tensors and = exp X . (1) mathematical background is described in X 1 10 manifolds [II.33]), which manifestly combines conser- When a physical system is invariant under a one- vation laws and continuous geometric symmetries of parameter group of transformations, the correspond- space and time. In special relativity (described in some ing generator plays a role in determining the associated detail in this article), this manifold is flat Minkowski conservation law. space-time, generalizing Euclidean space to include time in a physically natural way. In general relativity, 1 Mechanics in Euclidean Space described in detail in general relativity and cos- mology [IV.40], this manifold may be curved, depend- It is conventional in classical mechanics to define the ing in part on the distribution of matter and energy positions of a set of interacting particles in a vector according to einstein’s field equations [III.10]. space. However, we do not observe any unique origin In quantum physics, the description of a system in to the three-dimensional space we inhabit, which we terms of a complex vector in Hilbert space gives rise therefore take to be the Euclidean space E3; only rela- to new symmetries. An important example is the fact tive positions between different interacting subsystems For general queries, contact [email protected] © Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. 108 II. Concepts (i.e., positions relative to the common center of mass) (thereby capturing the forces as gradients of the poten- enter the equations of motion. The entire system may tial energy). Using the calculus of variations [IV.6], be translated in space without any effect on the phe- the functions qj (t) that satisfy the laws of mechanics nomena. are those that make the action stationary, and these Of course, external forces acting on the system may satisfy Lagrange’s equations of motion prevent this, such as a rubber ball in a linear gravity ∂L d ∂L − = 0,j= 1,...,n.
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]
  • Cen, Julia.Pdf
    City Research Online City, University of London Institutional Repository Citation: Cen, J. (2019). Nonlinear classical and quantum integrable systems with PT -symmetries. (Unpublished Doctoral thesis, City, University of London) This is the accepted version of the paper. This version of the publication may differ from the final published version. Permanent repository link: https://openaccess.city.ac.uk/id/eprint/23891/ Link to published version: Copyright: City Research Online aims to make research outputs of City, University of London available to a wider audience. Copyright and Moral Rights remain with the author(s) and/or copyright holders. URLs from City Research Online may be freely distributed and linked to. Reuse: Copies of full items can be used for personal research or study, educational, or not-for-profit purposes without prior permission or charge. Provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way. City Research Online: http://openaccess.city.ac.uk/ [email protected] Nonlinear Classical and Quantum Integrable Systems with PT -symmetries Julia Cen A Thesis Submitted for the Degree of Doctor of Philosophy School of Mathematics, Computer Science and Engineering Department of Mathematics Supervisor : Professor Andreas Fring September 2019 Thesis Commission Internal examiner Dr Olalla Castro-Alvaredo Department of Mathematics, City, University of London, UK External examiner Professor Andrew Hone School of Mathematics, Statistics and Actuarial Science University of Kent, UK Chair Professor Joseph Chuang Department of Mathematics, City, University of London, UK i To my parents.
    [Show full text]
  • Relativity with a Preferred Frame. Astrophysical and Cosmological Implications Monday, 21 August 2017 15:30 (30 Minutes)
    6th International Conference on New Frontiers in Physics (ICNFP2017) Contribution ID: 1095 Type: Talk Relativity with a preferred frame. Astrophysical and cosmological implications Monday, 21 August 2017 15:30 (30 minutes) The present analysis is motivated by the fact that, although the local Lorentz invariance is one of thecorner- stones of modern physics, cosmologically a preferred system of reference does exist. Modern cosmological models are based on the assumption that there exists a typical (privileged) Lorentz frame, in which the universe appear isotropic to “typical” freely falling observers. The discovery of the cosmic microwave background provided a stronger support to that assumption (it is tacitly assumed that the privileged frame, in which the universe appears isotropic, coincides with the CMB frame). The view, that there exists a preferred frame of reference, seems to unambiguously lead to the abolishmentof the basic principles of the special relativity theory: the principle of relativity and the principle of universality of the speed of light. Correspondingly, the modern versions of experimental tests of special relativity and the “test theories” of special relativity reject those principles and presume that a preferred inertial reference frame, identified with the CMB frame, is the only frame in which the two-way speed of light (the average speed from source to observer and back) is isotropic while it is anisotropic in relatively moving frames. In the present study, the existence of a preferred frame is incorporated into the framework of the special relativity, based on the relativity principle and universality of the (two-way) speed of light, at the expense of the freedom in assigning the one-way speeds of light that exists in special relativity.
    [Show full text]
  • A Negative Result for Hearing the Shape of a Triangle: a Computer-Assisted Proof
    AN ELECTRONIC JOURNAL OF THE SOCIETAT CATALANA DE MATEMATIQUES` A negative result for hearing the shape of a triangle: a computer-assisted proof ⇤Gerard Orriols Gim´enez Resum (CAT) ETH Z¨urich. En aquest article demostrem que existeixen dos triangles diferents pels quals [email protected] el primer, segon i quart valor propi del Laplaci`a amb condicions de Dirichlet coincideixen. Aix`o resol una conjectura proposada per Antunes i Freitas i Corresponding author ⇤ suggerida per la seva evid`encia num`erica. La prova ´esassistida per ordinador i utilitza una nova t`ecnica per tractar l’espectre d’un l’operador, que consisteix a combinar un M`etode d’Elements Finits per localitzar aproximadament els primers valors propis i controlar la seva posici´oa l’espectre, juntament amb el M`etode de Solucions Particulars per confinar aquests valors propis a un interval molt m´esprec´ıs. Abstract (ENG) We prove that there exist two distinct triangles for which the first, second and fourth eigenvalues of the Laplace operator with zero Dirichlet boundary conditions coincide. This solves a conjecture raised by Antunes and Freitas and suggested by their numerical evidence. We use a novel technique for a computer-assisted proof about the spectrum of an operator, which combines a Finite Element Method, to locate roughly the first eigenvalues keeping track of their position in the spectrum, and the Method of Particular Solutions, to get a much more precise bound of these eigenvalues. Keywords: computer-assisted proof, Laplace eigenvalues, spectral geome- Acknowledgement try, Finite Element Method, Method The author would like to thank Re- of Particular Solutions.
    [Show full text]
  • Frames of Reference
    Galilean Relativity 1 m/s 3 m/s Q. What is the women velocity? A. With respect to whom? Frames of Reference: A frame of reference is a set of coordinates (for example x, y & z axes) with respect to whom any physical quantity can be determined. Inertial Frames of Reference: - The inertia of a body is the resistance of changing its state of motion. - Uniformly moving reference frames (e.g. those considered at 'rest' or moving with constant velocity in a straight line) are called inertial reference frames. - Special relativity deals only with physics viewed from inertial reference frames. - If we can neglect the effect of the earth’s rotations, a frame of reference fixed in the earth is an inertial reference frame. Galilean Coordinate Transformations: For simplicity: - Let coordinates in both references equal at (t = 0 ). - Use Cartesian coordinate systems. t1 = t2 = 0 t1 = t2 At ( t1 = t2 ) Galilean Coordinate Transformations are: x2= x 1 − vt 1 x1= x 2+ vt 2 or y2= y 1 y1= y 2 z2= z 1 z1= z 2 Recall v is constant, differentiation of above equations gives Galilean velocity Transformations: dx dx dx dx 2 =1 − v 1 =2 − v dt 2 dt 1 dt 1 dt 2 dy dy dy dy 2 = 1 1 = 2 dt dt dt dt 2 1 1 2 dz dz dz dz 2 = 1 1 = 2 and dt2 dt 1 dt 1 dt 2 or v x1= v x 2 + v v x2 =v x1 − v and Similarly, Galilean acceleration Transformations: a2= a 1 Physics before Relativity Classical physics was developed between about 1650 and 1900 based on: * Idealized mechanical models that can be subjected to mathematical analysis and tested against observation.
    [Show full text]
  • Weak Galilean Invariance As a Selection Principle for Coarse
    Weak Galilean invariance as a selection principle for coarse-grained diffusive models Andrea Cairolia,b, Rainer Klagesb, and Adrian Bauleb,1 aDepartment of Bioengineering, Imperial College London, London SW7 2AZ, United Kingdom; and bSchool of Mathematical Sciences, Queen Mary University of London, London E1 4NS, United Kingdom Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved April 16, 2018 (received for review October 2, 2017) How does the mathematical description of a system change in dif- in the long-time limit, hX 2i ∼ t β with β = 1, for anomalous dif- ferent reference frames? Galilei first addressed this fundamental fusion it scales nonlinearly with β 6=1. Anomalous dynamics has question by formulating the famous principle of Galilean invari- been observed experimentally for a wide range of physical pro- ance. It prescribes that the equations of motion of closed systems cesses like particle transport in plasmas, molecular diffusion in remain the same in different inertial frames related by Galilean nanopores, and charge transport in amorphous semiconductors transformations, thus imposing strong constraints on the dynam- (5–7) that was first theoretically described in refs. 8 and 9 based ical rules. However, real world systems are often described by on the continuous-time random walk (CTRW) (10). Likewise, coarse-grained models integrating complex internal and exter- anomalous diffusion was later found for biological motion (11– nal interactions indistinguishably as friction and stochastic forces. 13) and even human movement (14). Recently, it was established Since Galilean invariance is then violated, there is seemingly no as a ubiquitous characteristic of cellular processes on a molec- alternative principle to assess a priori the physical consistency of a ular level (15).
    [Show full text]
  • From Relativistic Time Dilation to Psychological Time Perception
    From relativistic time dilation to psychological time perception: an approach and model, driven by the theory of relativity, to combine the physical time with the time perceived while experiencing different situations. Andrea Conte1,∗ Abstract An approach, supported by a physical model driven by the theory of relativity, is presented. This approach and model tend to conciliate the relativistic view on time dilation with the current models and conclusions on time perception. The model uses energy ratios instead of geometrical transformations to express time dilation. Brain mechanisms like the arousal mechanism and the attention mechanism are interpreted and combined using the model. Matrices of order two are generated to contain the time dilation between two observers, from the point of view of a third observer. The matrices are used to transform an observer time to another observer time. Correlations with the official time dilation equations are given in the appendix. Keywords: Time dilation, Time perception, Definition of time, Lorentz factor, Relativity, Physical time, Psychological time, Psychology of time, Internal clock, Arousal, Attention, Subjective time, Internal flux, External flux, Energy system ∗Corresponding author Email address: [email protected] (Andrea Conte) 1Declarations of interest: none Preprint submitted to PsyArXiv - version 2, revision 1 June 6, 2021 Contents 1 Introduction 3 1.1 The unit of time . 4 1.2 The Lorentz factor . 6 2 Physical model 7 2.1 Energy system . 7 2.2 Internal flux . 7 2.3 Internal flux ratio . 9 2.4 Non-isolated system interaction . 10 2.5 External flux . 11 2.6 External flux ratio . 12 2.7 Total flux .
    [Show full text]
  • 1 Introduction
    数理解析研究所講究録 1169 巻 2000 年 27-56 27 Numerical Verification Methods for Solutions of Ordinary and Partial Differential Equations Mitsuhiro T. Nakao 中尾 充宏 Graduate School of Mathematics, Kyushu University 33 Fukuoka 812-8581, Japan $\mathrm{e}$ -mail: [email protected] Abstract. In this article, we describe on a state of the art of validated numerical computations for solutions of differential equations. A brief overview of the main techniques in self-validating numerics for initial and boundary value problems in ordinary and partial differential equations including eigenvalue problems will be presented. A fairly detailed introductions are given for the author’s own method related to second-order elliptic boundary value problems. Many references which seem to be useful for readers are supplied at the end of the article. Key words. Numerical verification, nonlinear differential equation, computer assisted proof in analysis 1 Introduction If we denote an equation for unknown $u$ by $F(u)=0$ (1.1) then the problem of finding the solution $u$ generally implies an $n$ dimensional simulta- neous system of equations provided $u$ is in some finite dimensional ( $\mathrm{n}$ -dimension) space. Since very large scale, e.g. several thousand, linear system of equations can be easily solved on computers nowadays, it is not too surprising that, even if (1.1) is nonlinear, we can verify the existence and uniqueness of the solution as well as its domain by some numerical computations on the digital computer. However, in the case where (1.1) is a differential equation, it becomes a simultaneous equation which has infinitely many unknowns because the dimension of the potential function space containing $\mathrm{u}$ is infinite.
    [Show full text]
  • Coordinates and Proper Time
    Coordinates and Proper Time Edmund Bertschinger, [email protected] January 31, 2003 Now it came to me: . the independence of the gravitational acceleration from the na- ture of the falling substance, may be expressed as follows: In a gravitational ¯eld (of small spatial extension) things behave as they do in a space free of gravitation. This happened in 1908. Why were another seven years required for the construction of the general theory of relativity? The main reason lies in the fact that it is not so easy to free oneself from the idea that coordinates must have an immediate metrical meaning. | A. Einstein (quoted in Albert Einstein: Philosopher-Scientist, ed. P.A. Schilpp, 1949). 1. Introduction These notes supplement Chapter 1 of EBH (Exploring Black Holes by Taylor and Wheeler). They elaborate on the discussion of bookkeeper coordinates and how coordinates are related to actual physical distances and times. Also, a brief discussion of the classic Twin Paradox of special relativity is presented in order to illustrate the principal of maximal (or extremal) aging. Before going to details, let us review some jargon whose precise meaning will be important in what follows. You should be familiar with these words and their meaning. Spacetime is the four- dimensional playing ¯eld for motion. An event is a point in spacetime that is uniquely speci¯ed by giving its four coordinates (e.g. t; x; y; z). Sometimes we will ignore two of the spatial dimensions, reducing spacetime to two dimensions that can be graphed on a sheet of paper, resulting in a Minkowski diagram.
    [Show full text]
  • Higher Dimensional Conundra
    Higher Dimensional Conundra Steven G. Krantz1 Abstract: In recent years, especially in the subject of harmonic analysis, there has been interest in geometric phenomena of RN as N → +∞. In the present paper we examine several spe- cific geometric phenomena in Euclidean space and calculate the asymptotics as the dimension gets large. 0 Introduction Typically when we do geometry we concentrate on a specific venue in a particular space. Often the context is Euclidean space, and often the work is done in R2 or R3. But in modern work there are many aspects of analysis that are linked to concrete aspects of geometry. And there is often interest in rendering the ideas in Hilbert space or some other infinite dimensional setting. Thus we want to see how the finite-dimensional result in RN changes as N → +∞. In the present paper we study some particular aspects of the geometry of RN and their asymptotic behavior as N →∞. We choose these particular examples because the results are surprising or especially interesting. We may hope that they will lead to further studies. It is a pleasure to thank Richard W. Cottle for a careful reading of an early draft of this paper and for useful comments. 1 Volume in RN Let us begin by calculating the volume of the unit ball in RN and the surface area of its bounding unit sphere. We let ΩN denote the former and ωN−1 denote the latter. In addition, we let Γ(x) be the celebrated Gamma function of L. Euler. It is a helpful intuition (which is literally true when x is an 1We are happy to thank the American Institute of Mathematics for its hospitality and support during this work.
    [Show full text]
  • ON the GALILEAN COVARIANCE of CLASSICAL MECHANICS Abstract
    Report INP No 1556/PH ON THE GALILEAN COVARIANCE OF CLASSICAL MECHANICS Andrzej Horzela, Edward Kapuścik and Jarosław Kempczyński Department of Theoretical Physics, H. Niewodniczański Institute of Nuclear Physics, ul. Radzikowskiego 152, 31 342 Krakow. Poland and Laboratorj- of Theoretical Physics, Joint Institute for Nuclear Research, 101 000 Moscow, USSR, Head Post Office. P.O. Box 79 Abstract: A Galilean covariant approach to classical mechanics of a single interact¬ ing particle is described. In this scheme constitutive relations defining forces are rejected and acting forces are determined by some fundamental differential equa¬ tions. It is shown that the total energy of the interacting particle transforms under Galilean transformations differently from the kinetic energy. The statement is il¬ lustrated on the exactly solvable examples of the harmonic oscillator and the case of constant forces and also, in the suitable version of the perturbation theory, for the anharmonic oscillator. Kraków, August 1991 I. Introduction According to the principle of relativity the laws of physics do' not depend on the choice of the reference frame in which these laws are formulated. The change of the reference frame induces only a change in the language used for the formulation of the laws. In particular, changing the reference frame we change the space-time coordinates of events and functions describing physical quantities but all these changes have to follow strictly defined ways. When that is achieved we talk about a covariant formulation of a given theory and only in such formulation we may satisfy the requirements of the principle of relativity. The non-covariant formulations of physical theories are deprived from objectivity and it is difficult to judge about their real physical meaning.
    [Show full text]
  • The Experimental Verdict on Spacetime from Gravity Probe B
    The Experimental Verdict on Spacetime from Gravity Probe B James Overduin Abstract Concepts of space and time have been closely connected with matter since the time of the ancient Greeks. The history of these ideas is briefly reviewed, focusing on the debate between “absolute” and “relational” views of space and time and their influence on Einstein’s theory of general relativity, as formulated in the language of four-dimensional spacetime by Minkowski in 1908. After a brief detour through Minkowski’s modern-day legacy in higher dimensions, an overview is given of the current experimental status of general relativity. Gravity Probe B is the first test of this theory to focus on spin, and the first to produce direct and unambiguous detections of the geodetic effect (warped spacetime tugs on a spin- ning gyroscope) and the frame-dragging effect (the spinning earth pulls spacetime around with it). These effects have important implications for astrophysics, cosmol- ogy and the origin of inertia. Philosophically, they might also be viewed as tests of the propositions that spacetime acts on matter (geodetic effect) and that matter acts back on spacetime (frame-dragging effect). 1 Space and Time Before Minkowski The Stoic philosopher Zeno of Elea, author of Zeno’s paradoxes (c. 490-430 BCE), is said to have held that space and time were unreal since they could neither act nor be acted upon by matter [1]. This is perhaps the earliest version of the relational view of space and time, a view whose philosophical fortunes have waxed and waned with the centuries, but which has exercised enormous influence on physics.
    [Show full text]