DOCSLIB.ORG

Time Scales and Time Differences

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Time Scales and Time Differences SECTION 2 TIME SCALES AND TIME DIFFERENCES Contents 2.1 Introduction .....................................................................................2–3 2.2 Time Scales .......................................................................................2–3 2.2.1 Ephemeris Time (ET) .......................................................2–3 2.2.2 International Atomic Time (TAI) ...................................2–3 2.2.3 Universal Time (UT1 and UT1R)....................................2–4 2.2.4 Coordinated Universal Time (UTC)..............................2–5 2.2.5 GPS or TOPEX Master Time (GPS or TPX)...................2–5 2.2.6 Station Time (ST) ..............................................................2–5 2.3 Time Differences..............................................................................2–6 2.3.1 ET − TAI.............................................................................2–6 2.3.1.1 The Metric Tensor and the Metric...................2–6 2.3.1.2 Solar-System Barycentric Frame of Reference............................................................2–10 2.3.1.2.1 Tracking Station on Earth .................2–13 2.3.1.2.2 Earth Satellite ......................................2–16 2.3.1.2.3 Approximate Expression...................2–17 2.3.1.3 Geocentric Frame of Reference.......................2–18 2–1 SECTION 2 2.3.1.3.1 Tracking Station on Earth .................2–19 2.3.1.3.2 Earth Satellite ......................................2–20 2.3.2 TAI − UTC .........................................................................2–20 2.3.3 TAI − GPS and TAI − TPX ...............................................2–21 2.3.4 TAI − UT1 and TAI − UT1R.............................................2–21 2.3.5 Quadratic Offsets Between Station Time ST and UTC or (GPS or TPX) Master Time................................2–21 2.4 Input Files for Time Differences, Polar Motion, and Nutation Angle Corrections..........................................................2–21 2.5 Time Transformation Trees...........................................................2–23 2.5.1 Reception at DSN Tracking Station on Earth...............2–24 2.5.2 Reception at GPS Receiving Station on Earth ..............2–25 2.5.3 Reception at the TOPEX Satellite....................................2–26 2.5.4 Transmission at DSN Tracking Station on Earth.........2–27 2.5.5 Transmission at a GPS Satellite.......................................2–27 Figures 2–1 Time Transformations at a Tracking Station on Earth ..............2–25 2–2 Time Transformations at an Earth Satellite.................................2–26 2–2 TIME SCALES 2.1 INTRODUCTION This section is presented first because time is discussed in all of the other sections of this report. The various time scales used in programs PV and Regres of the ODP are described in Section 2.2. A time difference is the difference between values of an epoch recorded in two different time scales. Section 2.3 describes the time differences and gives the equations used for calculating them. Some of the time differences are obtained by interpolation of input files, which are described in Section 2.4. Section 2.5 presents time transformation trees. These figures indicate how to transform an epoch in one time scale to the corresponding epoch in any other time scale by adding and/or subtracting the intervening time differences. Time transformation trees are given for reception or transmission at a tracking station on Earth and at an Earth satellite. Time in any time scale is represented as seconds past January 1, 2000, 12h in that time scale. This epoch is J2000.0, which is the start of the Julian year 2000. The Julian Date for this epoch is JD 245,1545.0. 2.2 TIME SCALES 2.2.1 EPHEMERIS TIME (ET) Ephemeris time (ET) means coordinate time, which is the time coordinate of general relativity. It is either coordinate time of the Solar-System barycentric space-time frame of reference or coordinate time of the local geocentric space- time frame of reference, depending upon which reference frame the ODP user has selected. It is the independent variable for the motion of celestial bodies, spacecraft, and light rays. The scale of ET in each of these two reference frames is defined below in Section 2.3.1. 2.2.2 INTERNATIONAL ATOMIC TIME (TAI) International Atomic Time (TAI) is based upon the SI second (International System of Units). From p. 40–41 of the Explanatory Supplement to 2–3 SECTION 2 the Astronomical Almanac (1992), it is defined to be the duration of 9,192,631,770 periods of the radiation corresponding to the transition between two hyperfine levels of the ground state of the cesium-133 atom. It is further stated that this definition applies on the geoid (mean sea level). TAI is obtained from a worldwide system of synchronized atomic clocks. It is calculated as a weighted average of times obtained from the individual clocks, and corrections are applied for known effects. Time obtained from a clock on board an Earth satellite will be referenced to satellite International Atomic Time. Satellite TAI is an imaginary time scale obtained from an ideal atomic clock on the satellite. It agrees on average with TAI obtained from atomic clocks on Earth. 2.2.3 UNIVERSAL TIME (UT1 AND UT1R) Universal Time (UT) is the measure of time that is the basis for all civil time-keeping. It is an observed time scale, and the specific version used in the ODP is UT1. It is used to calculate mean sidereal time, which is the Greenwich hour angle of the mean equinox of date, measured in the true equator of date. Adding the equation of the equinoxes gives true sidereal time, which is used to calculate the position of the tracking station relative to the true equator and equinox of date. The equation for calculating mean sidereal time from observed UT1 is given in Section 5.3.6. From p. 51 of the Explanatory Supplement to the Astronomical Almanac (1992), the rate of UT1 is chosen so that a day of 86400 s of UT1 is close to the duration of the mean solar day. The phase of UT1 is chosen so that the Sun crosses the Greenwich meridian at approximately 12h UT1. Observed UT1 contains 41 short-period terms with periods between 5 and 35 days which are caused by long-period solid Earth tides. The algorithm for calculating the sum ∆UT1 of the 41 short-period terms of UT1 is given in Section 5.3.3. If ∆UT1 is subtracted from UT1, the result is called UT1R (where R means regularized). If UT1R is input to the ODP, the sum ∆UT1 must be calculated and added to UT1R to produce UT1, which is used to calculate mean sidereal time. 2–4 TIME SCALES 2.2.4 COORDINATED UNIVERSAL TIME (UTC) Coordinated Universal Time (UTC) is standard time for 0° longitude. Since January 1, 1972, UTC uses the SI second and has been behind International Atomic Time TAI by an integer number of seconds. UTC is maintained within 0.90 s of observed UT1 by adding a positive or negative leap second to UTC. A leap second is usually positive, which has the effect of retarding UTC by one second; it is usually added at the end of June or December. After a positive leap second was added at the end of December, 1998, TAI − UTC increased from 31 s to 32 s; at the beginning of 1972, it was 10 s. The history of TAI − UTC is given in International Earth Rotation Service (1998), Table II-3, p. II-7. 2.2.5 GPS OR TOPEX MASTER TIME (GPS OR TPX) GPS master time (GPS) is an atomic time scale, which is used instead of UTC as a reference time scale for GPS receiving stations on Earth and for GPS satellites. Similarly, TOPEX master time (TPX) is an atomic time scale used as a reference time scale on the TOPEX satellite. GPS time and TPX time are each an integer number of seconds behind TAI or satellite TAI. As opposed to UTC, these atomic time scales do not contain leap seconds. Therefore, the constant offsets from TAI or satellite TAI do not change. 2.2.6 STATION TIME (ST) Station time (ST) is atomic time at a Deep Space Network (DSN) tracking station on Earth, a GPS receiving station on Earth, a GPS satellite, or the TOPEX satellite. These atomic time scales depart by small amounts from the corresponding reference time scales. The reference time scale for a DSN tracking station on Earth is UTC. For a GPS receiving station on Earth or a GPS satellite, the reference time scale is GPS master time (GPS). For the TOPEX satellite, the reference time scale is TOPEX master time (TPX). Note, the TPX and GPS time scales can be used for any Earth-orbiting spacecraft. 2–5 SECTION 2 2.3 TIME DIFFERENCES 2.3.1 ET − TAI 2.3.1.1 The Metric Tensor and the Metric This section gives the equations for the n-body metric tensor and the corresponding expression for the interval ds. All of the relativistic equations in programs PV and Regres of the ODP can be derived from these equations or from simplifications of them. The components of the Parameterized Post– Newtonian (PPN) n-body point-mass metric tensor, which contains the PPN parameters β and γ of Will and Nordtvedt (1972), are given by the following equations, where the subscripts 1 through 4 refer to the four space-time coordinates. Subscripts 1, 2, and 3 refer to position coordinates, and 4 refers to coordinate time t multiplied by the speed of light c. µ 2γ j ggg===−+1 (2–1) 11 22 33 2 ∑ c ≠ rij ji gpqpq==≠0123( , , , ; ) (2–2) pq µ 22+ γ jjxú gg== (2–3) 14 41 3 ∑ c rij ji≠ µ 22+ γ jjyú gg== (2–4) 24 42 3 ∑ c rij ji≠ µ 22+ γ jjzú gg== (2–5) 34 43 3 ∑ c rij ji≠ 2–6 TIME SCALES 2 µ µ µ 2 2 j 2β j 12+ γ jjsú g =−1 + − 44 24∑∑ 4 ∑ c rij c rij c rij ji≠≠ ji ji ≠ (2–6) β − µ µ ∂ 2r + 22() 1 j k − 1 µ ij 44∑∑ ∑j 2 c rrij jk c ∂t ji≠≠kj ji ≠ where the indices j and k refer to the n bodies and k includes body i, whose motion is desired.
Load more

Recommended publications
  • Time Dilation of Accelerated Clocks
    Astrophysical Institute Neunhof Circular se07117, February 2016 1 Time Dilation of accelerated Clocks The definitions of proper time and proper length in General Rela- tivity Theory are presented. Time dilation and length contraction are explicitly computed for the example of clocks and rulers which are at rest in a rotating reference frame, and hence accelerated versus inertial reference frames. Experimental proofs of this time dilation, in particular the observation of the decay rates of acceler- ated muons, are discussed. As an illustration of the equivalence principle, we show that the general relativistic equation of motion of objects, which are at rest in a rotating reference frame, reduces to Newtons equation of motion of the same objects at rest in an inertial reference system and subject to a gravitational field. We close with some remarks on real versus ideal clocks, and the “clock hypothesis”. 1. Coordinate Diffeomorphisms In flat Minkowski space, we place a rectangular three-dimensional grid of fiducial marks in three-dimensional position space, and assign cartesian coordinate values x1, x2, x3 to each fiducial mark. The values of the xi are constants for each fiducial mark, i. e. the fiducial marks are at rest in the coordinate frame, which they define. Now we stretch and/or compress and/or skew and/or rotate the grid of fiducial marks locally and/or globally, and/or re-name the fiducial marks, e. g. change from cartesian coordinates to spherical coordinates or whatever other coordinates. All movements and renamings of the fiducial marks are subject to the constraint that the map from the initial grid to the deformed and/or renamed grid must be differentiable and invertible (i.
    [Show full text]
  • COORDINATE TIME in the VICINITY of the EARTH D. W. Allan' and N
    COORDINATE TIME IN THE VICINITY OF THE EARTH D. W. Allan’ and N. Ashby’ 1. Time and Frequency Division, National Bureau of Standards Boulder, Colorado 80303 2. Department of Physics, Campus Box 390 University of Colorado, Boulder, Colorado 80309 ABSTRACT. Atomic clock accuracies continue to improve rapidly, requir- ing the inclusion of general relativity for unambiguous time and fre- quency clock comparisons. Atomic clocks are now placed on space vehi- cles and there are many new applications of time and frequency metrology. This paper addresses theoretical and practical limitations in the accuracy of atomic clock comparisons arising from relativity, and demonstrates that accuracies of time and frequency comparison can approach a few picoseconds and a few parts in respectively. 1. INTRODUCTION Recent experience has shown that the accuracy of atomic clocks has improved by about an order of magnitude every seven years. It has therefore been necessary to include relativistic effects in the reali- zation of state-of-the-art time and frequency comparisons for at least the last decade. There is a growing need for agreement about proce- dures for incorporating relativistic effects in all disciplines which use modern time and frequency metrology techniques. The areas of need include sophisticated communication and navigation systems and funda- mental areas of research such as geodesy and radio astrometry. Significant progress has recently been made in arriving at defini- tions €or coordinate time that are practical, and in experimental veri- fication of the self-consistency of these procedures. International Atomic Time (TAI) and Universal Coordinated Time (UTC) have been defin- ed as coordinate time scales to assist in the unambiguous comparison of time and frequency in the vicinity of the Earth.
    [Show full text]
  • Observer with a Constant Proper Acceleration Cannot Be Treated Within the Theory of Special Relativity and That Theory of General Relativity Is Absolutely Necessary
    Observer with a constant proper acceleration Claude Semay∗ Groupe de Physique Nucl´eaire Th´eorique, Universit´ede Mons-Hainaut, Acad´emie universitaire Wallonie-Bruxelles, Place du Parc 20, BE-7000 Mons, Belgium (Dated: February 2, 2008) Abstract Relying on the equivalence principle, a first approach of the general theory of relativity is pre- sented using the spacetime metric of an observer with a constant proper acceleration. Within this non inertial frame, the equation of motion of a freely moving object is studied and the equation of motion of a second accelerated observer with the same proper acceleration is examined. A com- parison of the metric of the accelerated observer with the metric due to a gravitational field is also performed. PACS numbers: 03.30.+p,04.20.-q arXiv:physics/0601179v1 [physics.ed-ph] 23 Jan 2006 ∗FNRS Research Associate; E-mail: [email protected] Typeset by REVTEX 1 I. INTRODUCTION The study of a motion with a constant proper acceleration is a classical exercise of special relativity that can be found in many textbooks [1, 2, 3]. With its analytical solution, it is possible to show that the limit speed of light is asymptotically reached despite the constant proper acceleration. The very prominent notion of event horizon can be introduced in a simple context and the problem of the twin paradox can also be analysed. In many articles of popularisation, it is sometimes stated that the point of view of an observer with a constant proper acceleration cannot be treated within the theory of special relativity and that theory of general relativity is absolutely necessary.
    [Show full text]
  • Coordinate Systems in De Sitter Spacetime Bachelor Thesis A.C
    Coordinate systems in De Sitter spacetime Bachelor thesis A.C. Ripken July 19, 2013 Radboud University Nijmegen Radboud Honours Academy Abstract The De Sitter metric is a solution for the Einstein equation with positive cosmological constant, modelling an expanding universe. The De Sitter metric has a coordinate sin- gularity corresponding to an event horizon. The physical properties of this horizon are studied. The Klein-Gordon equation is generalized for curved spacetime, and solved in various coordinate systems in De Sitter space. Contact Name Chris Ripken Email [email protected] Student number 4049489 Study Physicsandmathematics Supervisor Name prof.dr.R.H.PKleiss Email [email protected] Department Theoretical High Energy Physics Adress Heyendaalseweg 135, Nijmegen Cover illustration Projection of two-dimensional De Sitter spacetime embedded in Minkowski space. Three coordinate systems are used: global coordinates, static coordinates, and static coordinates rotated in the (x1,x2)-plane. Contents Preface 3 1 Introduction 4 1.1 TheEinsteinfieldequations . 4 1.2 Thegeodesicequations. 6 1.3 DeSitterspace ................................. 7 1.4 TheKlein-Gordonequation . 10 2 Coordinate transformations 12 2.1 Transformations to non-singular coordinates . ......... 12 2.2 Transformationsofthestaticmetric . ..... 15 2.3 Atranslationoftheorigin . 22 3 Geodesics 25 4 The Klein-Gordon equation 28 4.1 Flatspace.................................... 28 4.2 Staticcoordinates ............................... 30 4.3 Flatslicingcoordinates. 32 5 Conclusion 39 Bibliography 40 A Maple code 41 A.1 Theprocedure‘riemann’. 41 A.2 Flatslicingcoordinates. 50 A.3 Transformationsofthestaticmetric . ..... 50 A.4 Geodesics .................................... 51 A.5 TheKleinGordonequation . 52 1 Preface For ages, people have studied the motion of objects. These objects could be close to home, like marbles on a table, or far away, like planets and stars.
    [Show full text]
  • The Flow of Time in the Theory of Relativity Mario Bacelar Valente
    The flow of time in the theory of relativity Mario Bacelar Valente Abstract Dennis Dieks advanced the view that the idea of flow of time is implemented in the theory of relativity. The ‘flow’ results from the successive happening/becoming of events along the time-like worldline of a material system. This leads to a view of now as local to each worldline. Each past event of the worldline has occurred once as a now- point, and we take there to be an ever-changing present now-point ‘marking’ the unfolding of a physical system. In Dieks’ approach there is no preferred worldline and only along each worldline is there a Newtonian-like linear order between successive now-points. We have a flow of time per worldline. Also there is no global temporal order of the now-points of different worldlines. There is, as much, what Dieks calls a partial order. However Dieks needs for a consistency reason to impose a limitation on the assignment of the now-points along different worldlines. In this work it is made the claim that Dieks’ consistency requirement is, in fact, inbuilt in the theory as a spatial relation between physical systems and processes. Furthermore, in this work we will consider (very) particular cases of assignments of now-points restricted by this spatial relation, in which the now-points taken to be simultaneous are not relative to the adopted inertial reference frame. 1 Introduction When we consider any experiment related to the theory of relativity,1 like the Michelson-Morley experiment (see, e.g., Møller 1955, 26-8), we can always describe it in terms of an intuitive notion of passage or flow of time: light is send through the two arms of the interferometer at a particular moment – the now of the experimenter –, and the process of light propagation takes time to occur, as can be measured by a clock calibrated to the adopted time scale.
    [Show full text]
  • Space-Time Reference Systems
    Space-Time reference systems Michael Soffel Dresden Technical University What are Space-Time Reference Systems (RS)? Science talks about events (e.g., observations): something that happens during a short interval of time in some small volume of space Both space and time are considered to be continous combined they are described by a ST manifold A Space-Time RS basically is a ST coordinate system (t,x) describing the ST position of events in a certain part of space-time In practice such a coordinate system has to be realized in nature with certain observations; the realization is then called the corresponding Reference Frame Newton‘s space and time In Newton‘s ST things are quite simple: Time is absolute as is Space -> there exists a class of preferred inertial coordinates (t, x) that have direct physical meaning, i.e., observables can be obtained directly from the coordinate picture of the physical system. Example: ∆τ = ∆t (proper) time coordinate time indicated by some clock Newton‘s absolute space Newtonian astrometry physically preferred global inertial coordinates observables are directly related to the inertial coordinates Example: observed angle θ between two incident light-rays 1 and 2 x (t) = x0 + c n (t – t ) ii i 0 cos θ = n. n 1 2 Is Newton‘s conception of Time and Space in accordance with nature? NO! The reason for that is related with properties of light propagation upon which time measurements are based Presently: The (SI) second is the duration of 9 192 631 770 periods of radiation that corresponds to a certain transition
    [Show full text]
  • 1 Temporal Arrows in Space-Time Temporality
    Temporal Arrows in Space-Time Temporality (…) has nothing to do with mechanics. It has to do with statistical mechanics, thermodynamics (…).C. Rovelli, in Dieks, 2006, 35 Abstract The prevailing current of thought in both physics and philosophy is that relativistic space-time provides no means for the objective measurement of the passage of time. Kurt Gödel, for instance, denied the possibility of an objective lapse of time, both in the Special and the General theory of relativity. From this failure many writers have inferred that a static block universe is the only acceptable conceptual consequence of a four-dimensional world. The aim of this paper is to investigate how arrows of time could be measured objectively in space-time. In order to carry out this investigation it is proposed to consider both local and global arrows of time. In particular the investigation will focus on a) invariant thermodynamic parameters in both the Special and the General theory for local regions of space-time (passage of time); b) the evolution of the universe under appropriate boundary conditions for the whole of space-time (arrow of time), as envisaged in modern quantum cosmology. The upshot of this investigation is that a number of invariant physical indicators in space-time can be found, which would allow observers to measure the lapse of time and to infer both the existence of an objective passage and an arrow of time. Keywords Arrows of time; entropy; four-dimensional world; invariance; space-time; thermodynamics 1 I. Introduction Philosophical debates about the nature of space-time often centre on questions of its ontology, i.e.
    [Show full text]
  • Electro-Gravity Via Geometric Chronon Field
    Physical Science International Journal 7(3): 152-185, 2015, Article no.PSIJ.2015.066 ISSN: 2348-0130 SCIENCEDOMAIN international www.sciencedomain.org Electro-Gravity via Geometric Chronon Field Eytan H. Suchard1* 1Geometry and Algorithms, Applied Neural Biometrics, Israel. Author’s contribution The sole author designed, analyzed and interpreted and prepared the manuscript. Article Information DOI: 10.9734/PSIJ/2015/18291 Editor(s): (1) Shi-Hai Dong, Department of Physics School of Physics and Mathematics National Polytechnic Institute, Mexico. (2) David F. Mota, Institute of Theoretical Astrophysics, University of Oslo, Norway. Reviewers: (1) Auffray Jean-Paul, New York University, New York, USA. (2) Anonymous, Sãopaulo State University, Brazil. (3) Anonymous, Neijiang Normal University, China. (4) Anonymous, Italy. (5) Anonymous, Hebei University, China. Complete Peer review History: http://sciencedomain.org/review-history/9858 Received 13th April 2015 th Original Research Article Accepted 9 June 2015 Published 19th June 2015 ABSTRACT Aim: To develop a model of matter that will account for electro-gravity. Interacting particles with non-gravitational fields can be seen as clocks whose trajectory is not Minkowsky geodesic. A field, in which each such small enough clock is not geodesic, can be described by a scalar field of time, whose gradient has non-zero curvature. This way the scalar field adds information to space-time, which is not anticipated by the metric tensor alone. The scalar field can’t be realized as a coordinate because it can be measured from a reference sub-manifold along different curves. In a “Big Bang” manifold, the field is simply an upper limit on measurable time by interacting clocks, backwards from each event to the big bang singularity as a limit only.
    [Show full text]
  • Time in the Theory of Relativity: on Natural Clocks, Proper Time, the Clock Hypothesis, and All That
    Time in the theory of relativity: on natural clocks, proper time, the clock hypothesis, and all that Mario Bacelar Valente Abstract When addressing the notion of proper time in the theory of relativity, it is usually taken for granted that the time read by an accelerated clock is given by the Minkowski proper time. However, there are authors like Harvey Brown that consider necessary an extra assumption to arrive at this result, the so-called clock hypothesis. In opposition to Brown, Richard TW Arthur takes the clock hypothesis to be already implicit in the theory. In this paper I will present a view different from these authors by taking into account Einstein’s notion of natural clock and showing its relevance to the debate. 1 Introduction: the notion of natural clock th Up until the mid 20 century the metrological definition of second was made in terms of astronomical motions. First in terms of the Earth’s rotation taken to be uniform (Barbour 2009, 2-3), i.e. the sidereal time; then in terms of the so-called ephemeris time, in which time was calculated, using Newton’s theory, from the motion of the Moon (Jespersen and Fitz-Randolph 1999, 104-6). The measurements of temporal durations relied on direct astronomical observation or on instruments (clocks) calibrated to the motions in the ‘heavens’. However soon after the adoption of a definition of second based on the ephemeris time, the improvements on atomic frequency standards led to a new definition of the second in terms of the resonance frequency of the cesium atom.
    [Show full text]
  • The First Physics Picture of Contractions from a Fundamental Quantum Relativity Symmetry Including All Known Relativity Symmetri
    NCU-HEP-k071 Feb. 2018 ed. Mar. 2019 The First Physics Picture of Contractions from a Fundamental Quantum Relativity Symmetry Including all Known Relativity Symmetries, Classical and Quantum Otto C. W. Kong, Jason Payne Department of Physics and Center for High Energy and High Field Physics, National Central University, Chung-li, Taiwan 32054 Abstract In this article, we utilize the insights gleaned from our recent formulation of space(-time), as well as dynamical picture of quantum mechanics and its classical approximation, from the relativity symmetry perspective in order to push further into the realm of the proposed fundamental relativity symmetry SO(2, 4). The latter has its origin arising from the perspectives of Planck scale deformations of relativity symmetries. We explicitly trace how the diverse actors in this story change through various contraction limits, paying careful attention to the relevant physical units, in order to place all known relativity theories – quantum and classical – within a single framework. More specifically, we explore both of the possible contractions of SO(2, 4) and its coset spaces in order to determine how best to recover the lower-level theories. These include both new models and all familiar theories, as well as quantum and classical dynamics with and without Einsteinian special arXiv:1802.02372v2 [hep-th] 12 Jan 2021 relativity. Along the way, we also find connections with covariant quantum mechanics. The emphasis of this article rests on the ability of this language to not only encompass all known physical theories, but to also provide a path for extensions. It will serve as the basic background for more detailed formulations of the dynamical theories at each level, as well as the exact connections amongst them.
    [Show full text]
  • Coordinate Time and Proper Time in the GPS
    TB, SD, UK, EJP/281959, 23/08/2008 IOP PUBLISHING EUROPEAN JOURNAL OF PHYSICS Eur. J. Phys. 29 (2008) 1–5 UNCORRECTED PROOF Coordinate time and proper time in the GPS T Matolcsi1,4 and M Matolcsi2,3,5 1 Department of Applied Analysis, Eotv¨ os¨ University, Pazm´ any´ P. set´ any´ 1C., H–1117 Budapest, Hungary 2 Alfred´ Renyi´ Institute of Mathematics, Hungarian Academy of Sciences, Realtanoda´ utca 13–15, H–1053, Budapest, Hungary 3 BME, Department of Analysis, Egry J. u. 1, Budapest, Hungary E-mail: [email protected] and [email protected] Received 22 May 2008, in final form 6 August 2008 Published DD MMM 2008 Online at stacks.iop.org/EJP/29/1 Abstract The global positioning system (GPS) provides an excellent educational example as to how the theory of general relativity is put into practice and becomes part of our everyday life. This paper gives a short and instructive derivation of an important formula used in the GPS, and is aimed at graduate students and general physicists. The theoretical background of the GPS (see [1]) uses the Schwarzschild spacetime to deduce the approximate formula, ds/dt ≈ − |v|2 1+V 2 , for the relation between the proper time rate s of a satellite clock and the coordinate time rate t.HereV is the gravitational potential at the position of the satellite and v is its velocity (with light-speed being normalized as c = 1). In this paper we give a different derivation√ of this formula, without = −| |2 − 2V · 2 using approximations,toarriveatds/dt 1+2V v 1+2V (n v) , where n is the normal vector pointing outwards from the centre of Earth to the satellite.
    [Show full text]
  • Physical Time and Physical Space in General Relativity Richard J
    Physical time and physical space in general relativity Richard J. Cook Department of Physics, U.S. Air Force Academy, Colorado Springs, Colorado 80840 ͑Received 13 February 2003; accepted 18 July 2003͒ This paper comments on the physical meaning of the line element in general relativity. We emphasize that, generally speaking, physical spatial and temporal coordinates ͑those with direct metrical significance͒ exist only in the immediate neighborhood of a given observer, and that the physical coordinates in different reference frames are related by Lorentz transformations ͑as in special relativity͒ even though those frames are accelerating or exist in strong gravitational fields. ͓DOI: 10.1119/1.1607338͔ ‘‘Now it came to me:...the independence of the this case, the distance between FOs changes with time even gravitational acceleration from the nature of the though each FO is ‘‘at rest’’ (r,␪,␾ϭconstant) in these co- falling substance, may be expressed as follows: ordinates. In a gravitational field (of small spatial exten- sion) things behave as they do in space free of III. PHYSICAL SPACE gravitation,... This happened in 1908. Why were another seven years required for the construction What is the distance dᐉ between neighboring FOs sepa- of the general theory of relativity? The main rea- rated by coordinate displacement dxi when the line element son lies in the fact that it is not so easy to free has the general form oneself from the idea that coordinates must have 2 ␣ ␤ an immediate metrical meaning.’’ 1 Albert Ein- ds ϭg␣␤ dx dx ? ͑1͒ stein Einstein’s answer is simple.3 Let a light ͑or radar͒ pulse be transmitted from the first FO ͑observer A) to the second FO I.
    [Show full text]