DOCSLIB.ORG

Symmetry of the Relativistic Doppler Effect

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Symmetry of the Relativistic Doppler Effect Appendix A Holmes explains relativity theory to Watson Holmes explains the dual nature of space and time The day was a dreary one. A thin rain was falling, and a dense wet fog lay low over London. Holmes looked out at the mud-colored clouds and muddy streets. He exclaimed, “Watson, I cannot live without brain work. What else is there for me? Stand by the window here. See how the yellow fog swirls down the street and drifts across the dun-colored houses. What is the use of having an energetic mind, Doctor, if one is unwilling to use it?” “My dear Holmes, the tremendous exertions in trying to understand Einstein which I have gone through during the last few days have quite worn me out — not bodily exertion, but the strain on the mind. You will appreciate that.” “Nonsense, old chap!” Holmes said, slapping me on the shoulder. “Come, Watson, we have a good day’s work before us.” “Perhaps you can bring light into something as dark and forbidding as relativity theory.” “Watson, I will bring light. The velocity of light in a vacuum is, as a round number, 300,000 kilometers per second. A light-second is defined as that distance which light travels in one second of time, and so it is equal to 300,000 kilometers. A more practical unit in astronomy is the light- year. A light-year is the distance that light travels in a year. A light-year is equal to 300,000 times 365 times 24 times 60 times 60 kilometers, or 9.5 trillion kilometers. The units light-second and light-year represent time of length. In this ‘time-of-length’ unit, the star Vega is 27 light-years away.” “Excellent! Admirable! But what use will you make of it?” “Originally, the foot and a day were considered natural units because one could measure a distance with his foot and could count the days. However, for beings who live elsewhere in the universe, those units 217 Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenA_v2--SCS.indd 217 8/15/2014 11:56:21 AM 218 Remote Sensing in Action would be quite arbitrary, so instead, natural units are now defined in terms of fundamental physical constants. One such fundamental physical quantity is the velocity of light in a vacuum, which is believed to be the same constant at all places, in all directions, and at all times. In all our discussions, unless otherwise stated, we will use natural units based on the velocity of light in a vacuum.” “They might be useful to scientists,” I said, “but beyond that, I can hardly see what use it is to me in understanding relativity theory.” “In answer, let me show you the relationship between a quantity mea- sured in natural units and the same quantity measured in conventional units. Therefore, for the current discussion, let capital letters denote quan- tities in conventional units and the corresponding small letters denote the same quantities in natural units.” Holmes continued, “The velocity of light in a vacuum is 300 million meters per second; in natural units, it is one unit. In conventional units, distance is given in meters, which we may denote as capital X. In conven- tional units, time is given in seconds, say capital T, and velocity, capital V, is given in meters per second. In the time-of-length method, we convert all lengths to times. It may be easier for you to follow if I just write down the equations used to convert our conventional units to natural units. Here they are: ” “Yes, I quite follow you,” I said. “I will take this opportunity to remark that the unit of x is light-second, the unit of t is second, and the unit of v is light-second per second. Even though we say light-second, the unit is actually a second, which means that both x and t have the same unit, the second, and thus, velocity v is a pure number. It is important to recognize that in natural units, velocity is dimensionless.” “Well done!” I said in an encouraging tone. “Let me mention another fact. The use of the velocity of light in defining the unit of distance has the virtue of simplifying the connec- tion between space and time. Any message sent by light or any other electromagnetic signal from Vega to earth will take 27 years. We see that the use of light-years implies that the communication time and the distance are the same number. For example, from earth to Vega, the time of communication by an electromagnetic signal is 27 years, and the distance is also 27 light-years. For a light signal, time and distance are Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenA_v2--SCS.indd 218 8/15/2014 11:56:22 AM Appendix A: Relativity theory 219 numerically equal; for example, 27 years and 27 light-years are numeri- cally the same. “However,” Holmes went on, “depending on the convention used for the directions of the t- and x-axes, the value of t or x might be either posi- tive or negative. Therefore, for a light signal, we write the equation this way: t2 – x2 = 0. The square root of the quantity on the left is called the time-space interval. We denote the time-space interval by the letter p; that is, time-space interval ==p tx22 − . We therefore conclude that for any electromagnetic signal, such as light, the time-space interval is equal to zero. This equality is a vital concept in the special theory of relativity.” I said, “I am glad in my heart that you have explained the concept of the time-space interval so well, for indeed, the mystery of relativity theory was becoming too much for my nerves.” “Watson, let us return to Sir Isaac Newton. He introduced the con- cepts of absolute time and absolute space. Absolute space and absolute time do not depend on physical events but act as a backdrop, or stage set- ting, within which all physical phenomena occur.” Sherlock Holmes took a blank sheet of paper and scrawled a drawing (Figure A-1). He continued, “As before, we will express both time in seconds and distance in light-seconds. In this way, the velocity of light is unity. We plot the slow train and the fast train on the same set of axes. Remember, Watson, that I am using natural units. Velocity is distance divided by time. In this diagram, the elapsed time t1 is the same for both trains. However, in the case of the slow train, distance x1 is less than t1, so the slow train has a velocity less than one. In the case of the fast train, distance x2 is greater than t1, so the fast train has a velocity greater than one. Light goes at the speed of one.” “Surely that is evident,” I said. “Newton puts time on one leg and dis- tance on the other leg of a right triangle. Depending on the length of each leg, velocity can be any value.” “However,” Holmes replied, “the theory of relativity postulates that the speed of light represents the maximal velocity that can be achieved in Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenA_v2--SCS.indd 219 8/15/2014 11:56:22 AM 220 Remote Sensing in Action x-axis (for distance) (in light-seconds) Velocity of fast train x2 = x2/t1 = 1.5 (unacceptable to Einstein). Velocity of light t1 = t1/t1 = 1.0. Light Fast train Velocity of slow train x1 = x1/t1 = 0.5. Slow train t-axis (for time) t1 = constant for three events Figure A-1. The slow train and the fast train on the same set of axes. nature. The magnitude of the velocity of any physical object must be less than one. As my drawing shows, Newton allows the fast train to have a velocity greater than one. Such a situation is impossible. As a result, Ein- stein demands that Newtonian absolute space and absolute time must be discarded. Yes, thrown out like an old shoe!” “Most outrageous!” I exclaimed. “This is beyond anything which I could have imagined. What can be done, Holmes?” Sherlock Holmes laughed heartily at my perplexity. “My dear Watson, the answer is simple. If one universal coordinate system is impossible, it follows that each and every inertial frame must have its own coordinate system. Einstein gave the solution.” “Please tell me how Einstein accomplished that feat,” I said. “How- ever, limit your answer to those things that are absolutely essential.” “The important concept is that there is the dual nature between space and time. Newton took time as absolute and space as absolute. Einstein rejects this categorization and instead takes the speed of light as an abso- lute constant. Einstein supersedes the ideas of absolute time and absolute space by the notion of space-time.” I was attempting to come to mental grips with this fantastic notion when Holmes, as was his practice, suddenly seemed to change the subject. Downloaded from http://pubs.geoscienceworld.org/books/book/chapter-pdf/3789046/9781560803140_backmatter.pdf by guest on 28 September 2021 0510_01_AppenA_v2--SCS.indd 220 8/15/2014 11:56:22 AM Appendix A: Relativity theory 221 “It is a dark and gloomy day, Watson,” he said.
Load more

Recommended publications
  • 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]
  • (12) Patent Application Publication (10) Pub. No.: US 2015/0260527 A1 Fink (43) Pub
    US 20150260527A1 (19) United States (12) Patent Application Publication (10) Pub. No.: US 2015/0260527 A1 Fink (43) Pub. Date: Sep. 17, 2015 (54) SYSTEMAND METHOD OF DETERMINING (52) U.S. Cl. A POSITION OF A REMOTE OBJECT VA CPC ...................................... G0IC 2 1/20 (2013.01) ONE ORMORE IMAGE SENSORS (57) ABSTRACT (71) Applicant: Ian Michael Fink, Austin, TX (US) In one or more embodiments, one or more systems, methods (72) Inventor: Ian Michael Fink, Austin, TX (US) and/or processes may determine a location of a remote object (e.g., a point and/or area of interest, landmark, structure that (21) Appl. No.: 14/716,817 “looks interesting, buoy, anchored boat, etc.). For example, the location of a remote object may be determined via a first (22) Filed: May 19, 2015 bearing, at a first location, and a second bearing, at a second O O location, to the remote object. For instance, the first and Related U.S. Application Data second locations can be determined via a position device, (63) Continuation of application No. 13/837,853, filed on Such as a global positioning system device. In one or more Mar. 15, 2013, now Pat. No. 9,074,892. embodiments, the location of the remote object may be based on the first location, the second location, the first bearing, and Publication Classification the second bearing. For example, the location of the remote object may be provided to a user via a map. For instance, (51) Int. Cl. turn-by-turn direction to the location of the remote object GOIC 2L/20 (2006.01) may be provided to the user.
    [Show full text]
  • Spherical Geometry
    Spherical Geometry Eric Lehman february-april 2012 2 Table of content 1 Spherical biangles and spherical triangles 3 § 1. The unit sphere . .3 § 2. Biangles . .6 § 3. Triangles . .7 2 Coordinates on a sphere 13 § 1. Cartesian versus spherical coordinates . 13 § 2. Basic astronomy . 16 § 3. Geodesics . 16 3 Spherical trigonometry 23 § 1. Pythagoras’ theorem . 23 § 2. The three laws of spherical trigonometry . 24 4 Stereographic projection 29 § 1. Definition . 29 § 2. The stereographic projection preserves the angles . 34 § 3. Images of circles . 36 5 Projective geometry 49 § 1. First description of the real projective plane . 49 § 2. The real projective plane . 52 § 3. Generalisations . 59 3 4 TABLE OF CONTENT TABLE OF CONTENT 1 Introduction The aim of this course is to show different aspects of spherical geometry for itself, in relation to applications and in relation to other geometries and other parts of mathematics. The chapters will be (mostly) independant from each other. To begin, we’l work on the sphere as Euclid did in the plane looking at triangles. Many things look alike, but there are some striking differences. The second viewpoint will be the introduction of coordinates and the application to basic astronomy. The theorem of Pytha- goras has a very nice and simple shape in spherical geometry. To contemplate spherical trigonometry will give us respect for our ancestors and navigators, but we shall skip the computations! and let the GPS do them. The stereographic projection is a marvellous tool to understand the pencils of coaxial circles and many aspects of the relation between the spherical geometry, the euclidean affine plane, the complex projective line, the real projec- tive plane, the Möbius strip and even the hyperbolic plane.
    [Show full text]
  • The Book of Common Prayer
    The Book of Common Prayer and Administration of the Sacraments and Other Rites and Ceremonies of the Church Together with The Psalter or Psalms of David According to the use of The Episcopal Church Church Publishing Incorporated, New York Certificate I certify that this edition of The Book of Common Prayer has been compared with a certified copy of the Standard Book, as the Canon directs, and that it conforms thereto. Gregory Michael Howe Custodian of the Standard Book of Common Prayer January, 2007 Table of Contents The Ratification of the Book of Common Prayer 8 The Preface 9 Concerning the Service of the Church 13 The Calendar of the Church Year 15 The Daily Office Daily Morning Prayer: Rite One 37 Daily Evening Prayer: Rite One 61 Daily Morning Prayer: Rite Two 75 Noonday Prayer 103 Order of Worship for the Evening 108 Daily Evening Prayer: Rite Two 115 Compline 127 Daily Devotions for Individuals and Families 137 Table of Suggested Canticles 144 The Great Litany 148 The Collects: Traditional Seasons of the Year 159 Holy Days 185 Common of Saints 195 Various Occasions 199 The Collects: Contemporary Seasons of the Year 211 Holy Days 237 Common of Saints 246 Various Occasions 251 Proper Liturgies for Special Days Ash Wednesday 264 Palm Sunday 270 Maundy Thursday 274 Good Friday 276 Holy Saturday 283 The Great Vigil of Easter 285 Holy Baptism 299 The Holy Eucharist An Exhortation 316 A Penitential Order: Rite One 319 The Holy Eucharist: Rite One 323 A Penitential Order: Rite Two 351 The Holy Eucharist: Rite Two 355 Prayers of the People
    [Show full text]
  • Lecture Notes 17: Proper Time, Proper Velocity, the Energy-Momentum 4-Vector, Relativistic Kinematics, Elastic/Inelastic
    UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 17 Prof. Steven Errede LECTURE NOTES 17 Proper Time and Proper Velocity As you progress along your world line {moving with “ordinary” velocity u in lab frame IRF(S)} on the ct vs. x Minkowski/space-time diagram, your watch runs slow {in your rest frame IRF(S')} in comparison to clocks on the wall in the lab frame IRF(S). The clocks on the wall in the lab frame IRF(S) tick off a time interval dt, whereas in your 2 rest frame IRF( S ) the time interval is: dt dtuu1 dt n.b. this is the exact same time dilation formula that we obtained earlier, with: 2 2 uu11uc 11 and: u uc We use uurelative speed of an object as observed in an inertial reference frame {here, u = speed of you, as observed in the lab IRF(S)}. We will henceforth use vvrelative speed between two inertial systems – e.g. IRF( S ) relative to IRF(S): Because the time interval dt occurs in your rest frame IRF( S ), we give it a special name: ddt = proper time interval (in your rest frame), and: t = proper time (in your rest frame). The name “proper” is due to a mis-translation of the French word “propre”, meaning “own”. Proper time is different than “ordinary” time, t. Proper time is a Lorentz-invariant quantity, whereas “ordinary” time t depends on the choice of IRF - i.e. “ordinary” time is not a Lorentz-invariant quantity. 222222 The Lorentz-invariant interval: dI dx dx dx dx ds c dt dx dy dz Proper time interval: d dI c2222222 ds c dt dx dy dz cdtdt22 = 0 in rest frame IRF(S) 22t Proper time: ddtttt 21 t 21 11 Because d and are Lorentz-invariant quantities: dd and: {i.e.
    [Show full text]
  • Class 3: Time Dilation, Lorentz Contraction, Relativistic Velocity Transformations
    Class 3: Time dilation, Lorentz contraction, relativistic velocity transformations Transformation of time intervals Suppose two events are recorded in a laboratory (frame S), e.g. by a cosmic ray detector. Event A is recorded at location xa and time ta, and event B is recorded at location xb and time tb. We want to know the time interval between these events in an inertial frame, S ′, moving with relative to the laboratory at speed u. Using the Lorentz transform, ux x′=−γ() xut,,, yyzzt ′′′ = = =− γ t , (3.1) c2 we see, from the last equation, that the time interval in S ′ is u ttba′−= ′ γ() tt ba −− γ () xx ba − , (3.2) c2 i.e., the time interval in S ′ depends not only on the time interval in the laboratory but also in the separation. If the two events are at the same location in S, the time interval (tb− t a ) measured by a clock located at the events is called the proper time interval . We also see that, because γ >1 for all frames moving relative to S, the proper time interval is the shortest time interval that can be measured between the two events. The events in the laboratory frame are not simultaneous. Is there a frame, S ″ in which the separated events are simultaneous? Since tb′′− t a′′ must be zero, we see that velocity of S ″ relative to S must be such that u c( tb− t a ) β = = , (3.3) c xb− x a i.e., the speed of S ″ relative to S is the fraction of the speed of light equal to the time interval between the events divided by the light travel time between the events.
    [Show full text]
  • Debasishdas SUNDIALS to TELL the TIMES of PRAYERS in the MOSQUES of INDIA January 1, 2018 About
    Authior : DebasishDas SUNDIALS TO TELL THE TIMES OF PRAYERS IN THE MOSQUES OF INDIA January 1, 2018 About It is said that Delhi has almost 1400 historical monuments.. scattered remnants of layers of history, some refer it as a city of 7 cities, some 11 cities, some even more. So, even one is to explore one monument every single day, it will take almost 4 years to cover them. Narratives on Delhi’s historical monuments are aplenty: from amateur writers penning down their experiences, to experts and archaeologists deliberating on historic structures. Similarly, such books in the English language have started appearing from as early as the late 18th century by the British that were the earliest translation of Persian texts. Period wise, we have books on all of Delhi’s seven cities (some say the city has 15 or more such cities buried in its bosom) between their covers, some focus on one of the cities, some are coffee-table books, some attempt to create easy-to-follow guide-books for the monuments, etc. While going through the vast collection of these valuable works, I found the need to tell the city’s forgotten stories, and weave them around the lesser-known monuments and structures lying scattered around the city. After all, Delhi is not a mere necropolis, as may be perceived by the un-initiated. Each of these broken and dilapidated monuments speak of untold stories, and without that context, they can hardly make a connection, however beautifully their architectural style and building plan is explained. My blog is, therefore, to combine actual on-site inspection of these sites, with interesting and insightful anecdotes of the historical personalities involved, and prepare essays with photographs and words that will attempt to offer a fresh angle to look at the city’s history.
    [Show full text]
  • Minkowski's Proper Time and the Status of the Clock Hypothesis
    Minkowski’s Proper Time and the Status of the Clock Hypothesis1 1 I am grateful to Stephen Lyle, Vesselin Petkov, and Graham Nerlich for their comments on the penultimate draft; any remaining infelicities or confusions are mine alone. Minkowski’s Proper Time and the Clock Hypothesis Discussion of Hermann Minkowski’s mathematical reformulation of Einstein’s Special Theory of Relativity as a four-dimensional theory usually centres on the ontology of spacetime as a whole, on whether his hypothesis of the absolute world is an original contribution showing that spacetime is the fundamental entity, or whether his whole reformulation is a mere mathematical compendium loquendi. I shall not be adding to that debate here. Instead what I wish to contend is that Minkowski’s most profound and original contribution in his classic paper of 100 years ago lies in his introduction or discovery of the notion of proper time.2 This, I argue, is a physical quantity that neither Einstein nor anyone else before him had anticipated, and whose significance and novelty, extending beyond the confines of the special theory, has become appreciated only gradually and incompletely. A sign of this is the persistence of several confusions surrounding the concept, especially in matters relating to acceleration. In this paper I attempt to untangle these confusions and clarify the importance of Minkowski’s profound contribution to the ontology of modern physics. I shall be looking at three such matters in this paper: 1. The conflation of proper time with the time co-ordinate as measured in a system’s own rest frame (proper frame), and the analogy with proper length.
    [Show full text]
  • Primer of Celestial Navigation
    LIFEBOAT MANUAL By Lt. Comdr. A. E. Redifer, U.S.M.S. Lifeboat Training Officer, Avalon, Calif., Training Station *Indispensable to ordinary seamen preparing for the Lifeboat Examination. *A complete guide for anyone who may have to use a lifeboat. Construction, equipment, operation of lifeboats and other buoyant apparatus are here, along with the latest Government regulations which you need to know. 163 Pages More than 150 illustrations Indexed $2.00 METEOROLOGY WORKBOOK with Problems By Peter E. Kraght Senior Meteorologist, American Airlines, Inc. *Embodies author's experience in teaching hundreds of aircraft flight crews. *For self-examination and for classroom use. *For student navigator or meteorologist or weather hobbyist. *A good companion book to Mr. Kraght's Meteorology for Ship and Aircraft Operation, 0r any other standard reference work. 141 Illustrations Cloud form photographs Weather maps 148 Pages W$T x tl" Format $2.25 MERCHANT MARBMI OFFICERS' HANDBOOK By E. A. Turpin and W. A. MacEwen, Master Mariners "A valuable guide for both new candidates for . officer's license and the older, more experienced officers." Marine Engi- neering and Shipping Re$etu. An unequalled guide to ship's officers' duties, navigation, meteorology, shiphandling and cargo, ship construction and sta- bility, rules of the road, first aid and ship sanitation, and enlist- ments in the Naval Reserve all in one volume. 03 086 3rd Edition "This book will be a great help to not only beginners but to practical navigating officers." Harold Kildall, Instructor in Charge, Navigation School of the Washing- ton Technical Institute, "Lays bare the mysteries of navigation with a firm and sure touch .
    [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]
  • Great Circle If the Plane Passes Through the Centre of the Sphere, and a Small Circle If the Plane Does Not Pass Through the Centre of the Sphere
    1 Assoc. Prof. Dr. Bahattin ERDOĞAN Assoc. Prof. Dr. Nursu TUNALIOĞLU Yildiz Technical University Faculty of Civil Engineering Department of Geomatic Engineering Lecture Notes 2 1. A sphere is a solid bounded by a surface every point of which is equally distant from a fixed point which is called the centre of the sphere. The straight line which joins any point of the surface with the centre is called a radius. A straight line drawn through the centre and terminated both ways by the surface is called a diameter. 3 2. The section of the surface of a sphere made by any plane is a circle. 4 5 » The section of the surface of a sphere by a plane is called a great circle if the plane passes through the centre of the sphere, and a small circle if the plane does not pass through the centre of the sphere. Thus the radius of a great circle is equal to the radius of the sphere. » On the globe, equator circle and circles of longitudes are the examples of great circle and, circles of latitudes are the example of small circle. 6 7 » Latitude: latitude lines are also known as parallels (since they are parallel to one another). The lines are actually full circles that extend around the earth and vary in length depending on where we are located. The biggest circle is at the equator and represents the earth's circumference. This line is also called a Great Circle. There can be infinitely many great circles, but only one that is a line of latitude (the equator).
    [Show full text]