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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 ................. -
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. -
Kaluza-Klein Gravity, Concentrating on the General Rel- Ativity, Rather Than Particle Physics Side of the Subject
Kaluza-Klein Gravity J. M. Overduin Department of Physics and Astronomy, University of Victoria, P.O. Box 3055, Victoria, British Columbia, Canada, V8W 3P6 and P. S. Wesson Department of Physics, University of Waterloo, Ontario, Canada N2L 3G1 and Gravity Probe-B, Hansen Physics Laboratories, Stanford University, Stanford, California, U.S.A. 94305 Abstract We review higher-dimensional unified theories from the general relativity, rather than the particle physics side. Three distinct approaches to the subject are identi- fied and contrasted: compactified, projective and noncompactified. We discuss the cosmological and astrophysical implications of extra dimensions, and conclude that none of the three approaches can be ruled out on observational grounds at the present time. arXiv:gr-qc/9805018v1 7 May 1998 Preprint submitted to Elsevier Preprint 3 February 2008 1 Introduction Kaluza’s [1] achievement was to show that five-dimensional general relativity contains both Einstein’s four-dimensional theory of gravity and Maxwell’s the- ory of electromagnetism. He however imposed a somewhat artificial restriction (the cylinder condition) on the coordinates, essentially barring the fifth one a priori from making a direct appearance in the laws of physics. Klein’s [2] con- tribution was to make this restriction less artificial by suggesting a plausible physical basis for it in compactification of the fifth dimension. This idea was enthusiastically received by unified-field theorists, and when the time came to include the strong and weak forces by extending Kaluza’s mechanism to higher dimensions, it was assumed that these too would be compact. This line of thinking has led through eleven-dimensional supergravity theories in the 1980s to the current favorite contenders for a possible “theory of everything,” ten-dimensional superstrings. -
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. -
Philosophy and the Physicists Preview
L. Susan Stebbing Philosophy and the Physicists Preview il glifo ebooks ISBN: 9788897527466 First Edition: December 2018 (A) Copyright © il glifo, December 2018 www.ilglifo.it 1 Contents FOREWORD FROM THE EDITOR Note to the 2018 electronic edition PHILOSOPHY AND THE PHYSICISTS Original Title Page PREFACE NOTE PART I - THE ALARMING ASTRONOMERS Chapter I - The Common Reader and the Popularizing Scientist Chapter II - THE ESCAPE OF SIR JAMES JEANS PART II - THE PHYSICIST AND THE WORLD Chapter III - ‘FURNITURE OF THE EARTH’ Chapter IV - ‘THE SYMBOLIC WORLD OF PHYSICS’ Chapter V - THE DESCENT TO THE INSCRUTABLE Chapter VI - CONSEQUENCES OF SCRUTINIZING THE INSCRUTABLE PART III - CAUSALITY AND HUMAN FREEDOM Chapter VII - THE NINETEENTH-CENTURY NIGHTMARE Chapter VIII - THE REJECTION OF PHYSICAL DETERMINISM Chapter IX - REACTIONS AND CONSEQUENCES Chapter X - HUMAN FREEDOM AND RESPONSIBILITY PART IV - THE CHANGED OUTLOOK Chapter XI - ENTROPY AND BECOMING Chapter XII - INTERPRETATIONS BIBLIOGRAPHY INDEX BACK COVER Susan Stebbing 2 Foreword from the Editor In 1937 Susan Stebbing published Philosophy and the Physicists , an intense and difficult essay, in reaction to reading the works written for the general public by two physicists then at the center of attention in England and the world, James Jeans (1877-1946) and Arthur Eddington (1882- 1944). The latter, as is known, in 1919 had announced to the Royal Society the astronomical observations that were then considered experimental confirmations of the general relativity of Einstein, and who by that episode had managed to trigger the transformation of general relativity into a component of the mass and non-mass imaginary of the twentieth century. -
Hypercomplex Algebras and Their Application to the Mathematical
Hypercomplex Algebras and their application to the mathematical formulation of Quantum Theory Torsten Hertig I1, Philip H¨ohmann II2, Ralf Otte I3 I tecData AG Bahnhofsstrasse 114, CH-9240 Uzwil, Schweiz 1 [email protected] 3 [email protected] II info-key GmbH & Co. KG Heinz-Fangman-Straße 2, DE-42287 Wuppertal, Deutschland 2 [email protected] March 31, 2014 Abstract Quantum theory (QT) which is one of the basic theories of physics, namely in terms of ERWIN SCHRODINGER¨ ’s 1926 wave functions in general requires the field C of the complex numbers to be formulated. However, even the complex-valued description soon turned out to be insufficient. Incorporating EINSTEIN’s theory of Special Relativity (SR) (SCHRODINGER¨ , OSKAR KLEIN, WALTER GORDON, 1926, PAUL DIRAC 1928) leads to an equation which requires some coefficients which can neither be real nor complex but rather must be hypercomplex. It is conventional to write down the DIRAC equation using pairwise anti-commuting matrices. However, a unitary ring of square matrices is a hypercomplex algebra by definition, namely an associative one. However, it is the algebraic properties of the elements and their relations to one another, rather than their precise form as matrices which is important. This encourages us to replace the matrix formulation by a more symbolic one of the single elements as linear combinations of some basis elements. In the case of the DIRAC equation, these elements are called biquaternions, also known as quaternions over the complex numbers. As an algebra over R, the biquaternions are eight-dimensional; as subalgebras, this algebra contains the division ring H of the quaternions at one hand and the algebra C ⊗ C of the bicomplex numbers at the other, the latter being commutative in contrast to H. -
What Is Gravity Probe B? a Quest for Experimental Truth the GP-B Flight
What is Gravity Probe B? than Gravity Probe B. It’s just a star, a telescope, and a spinning sphere.” However, it took the exceptional collaboration of Stanford, Gravity Probe B (GP-B) is a NASA physics mission to experimentally MSFC, Lockheed Martin and a host of others more than four decades investigate Einstein’s 1916 general theory of relativity—his theory of gravity. to develop the ultra-precise gyroscopes and the other cutting- GP-B uses four spherical gyroscopes and a telescope, housed in a satellite edge technologies necessary to carry out this “simple” experiment. orbiting 642 km (400 mi) above the Earth, to measure, with unprecedented accuracy, two extraordinary effects predicted by the general theory of rela- The GP-B Flight Mission & Data Analysis tivity: 1) the geodetic effect—the amount by which the Earth warps the local spacetime in which it resides; and 2) the frame-dragging effect—the amount On April 20, 2004 at 9:57:24 AM PDT, a crowd of over 2,000 current and by which the rotating Earth drags its local spacetime around with it. GP-B former GP-B team members and supporters watched and cheered as the tests these two effects by precisely measuring the precession (displacement) GP-B spacecraft lifted off from Vandenberg Air Force Base. That emotionally angles of the spin axes of the four gyros over the course of a year and com- overwhelming day, culminating with the extraordinary live video of paring these experimental results with predictions from Einstein’s theory. the spacecraft separating from the second stage booster meant, as GP-B Program Manager Gaylord Green put it, “that 10,000 things went right.” A Quest for Experimental Truth Once in orbit, the spacecraft first underwent a four-month Initialization The idea of testing general relativity with orbiting gyroscopes was sug- and Orbit Checkout (IOC), in which all systems and instruments were gested independently by two physicists, George Pugh and Leonard Schiff, initialized, tested, and optimized for the data collection to follow. -
SEER for Hardware's Cost Model for Future Orbital Concepts (“FAR OUT”)
Presented at the 2008 SCEA-ISPA Joint Annual Conference and Training Workshop - www.iceaaonline.com SEER for Hardware’s Cost Model for Future Orbital Concepts (“FAR OUT”) Lee Fischman ISPA/SCEA Industry Hills 2008 Presented at the 2008 SCEA-ISPA Joint Annual Conference and Training Workshop - www.iceaaonline.com Introduction A model for predicting the cost of long term unmanned orbital spacecraft (Far Out) has been developed at the request of AFRL. Far Out has been integrated into SEER for Hardware. This presentation discusses the Far Out project and resulting model. © 2008 Galorath Incorporated Presented at the 2008 SCEA-ISPA Joint Annual Conference and Training Workshop - www.iceaaonline.com Goals • Estimate space satellites in any earth orbit. Deep space exploration missions may be considered as data is available, or may be an area for further research in Phase 3. • Estimate concepts to be launched 10-20 years into the future. • Cost missions ranging from exploratory to strategic, with a specific range decided based on estimating reliability. The most reliable estimates are likely to be in the middle of this range. • Estimates will include hardware, software, systems engineering, and production. • Handle either government or commercial missions, either “one-of-a-kind” or constellations. • Be used in a “top-down” manner using relatively less specific mission resumes, similar to those available from sources such as the Earth Observation Portal or Janes Space Directory. © 2008 Galorath Incorporated Presented at the 2008 SCEA-ISPA Joint Annual Conference and Training Workshop - www.iceaaonline.com Challenge: Technology Change Over Time Evolution in bus technologies Evolution in payload technologies and performance 1. -
Moment of Inertia of Superconductors
Moment of inertia of superconductors J. E. Hirsch Department of Physics, University of California, San Diego La Jolla, CA 92093-0319 We find that the bulk moment of inertia per unit volume of a metal becoming superconducting 2 2 increases by the amount me/(πrc), with me the bare electron mass and rc = e /mec the classical 2 electron radius. This is because superfluid electrons acquire an intrinsic moment of inertia me(2λL) , with λL the London penetration depth. As a consequence, we predict that when a rotating long cylinder becomes superconducting its angular velocity does not change, contrary to the prediction of conventional BCS-London theory that it will rotate faster. We explain the dynamics of magnetic field generation when a rotating normal metal becomes superconducting. PACS numbers: I. INTRODUCTION and is called the “London moment”. In this paper we propose that this effect reveals fundamental physics of A superconducting body rotating with angular veloc- superconductors not predicted by conventional BCS the- ity ~ω develops a uniform magnetic field throughout its ory [3, 4]. For a preliminary treatment where we argued interior, given by that BCS theory is inconsistent with the London moment we refer the reader to our earlier work [5]. Other non- 2m c B~ = − e ~ω ≡ B(ω)ˆω (1) conventional explanations of the London moment have e also been proposed [6, 7]. with e (< 0) the electron charge, and me the bare electron Eq. (1) has been verified experimentally for a variety mass. Numerically, B =1.137 × 10−7ω, with B in Gauss of superconductors [8–15]. -
Tests of General Relativity - Wikipedia
12/2/2018 Tests of general relativity - Wikipedia Tests of general relativity T ests of general relativity serve to establish observational evidence for the theory of general relativity. The first three tests, proposed by Einstein in 1915, concerned the "anomalous" precession of the perihelion of Mercury, the bending of light in gravitational fields, and the gravitational redshift. The precession of Mercury was already known; experiments showing light bending in line with the predictions of general relativity was found in 1919, with increasing precision measurements done in subsequent tests, and astrophysical measurement of the gravitational redshift was claimed to be measured in 1925, although measurements sensitive enough to actually confirm the theory were not done until 1954. A program of more accurate tests starting in 1959 tested the various predictions of general relativity with a further degree of accuracy in the weak gravitational field limit, severely limiting possible deviations from the theory. In the 197 0s, additional tests began to be made, starting with Irwin Shapiro's measurement of the relativistic time delay in radar signal travel time near the sun. Beginning in 197 4, Hulse, Taylor and others have studied the behaviour of binary pulsars experiencing much stronger gravitational fields than those found in the Solar System. Both in the weak field limit (as in the Solar System) and with the stronger fields present in systems of binary pulsars the predictions of general relativity have been extremely well tested locally. In February 2016, the Advanced LIGO team announced that they had directly detected gravitational waves from a black hole merger.[1] This discovery, along with additional detections announced in June 2016 and June 2017 ,[2] tested general relativity in the very strong field limit, observing to date no deviations from theory. -
VLBI for Gravity Probe B. VII. the Evolution of the Radio Structure Of
Accepted to the Astrophysical Journal Supplement Series VLBI for Gravity Probe B. VII. The Evolution of the Radio Structure of IM Pegasi M. F. Bietenholz1,2, N. Bartel1, D. E. Lebach3, R. R. Ransom1,4, M. I. Ratner3, and I. I. Shapiro3 ABSTRACT We present measurements of the total radio flux density as well as very-long-baseline interferometry (VLBI) images of the star, IM Pegasi, which was used as the guide star for the NASA/Stanford relativity mission Gravity Probe B. We obtained flux densities and images from 35 sessions of observations at 8.4 GHz (λ = 3.6 cm) between 1997 January and 2005 July. The observations were accurately phase-referenced to several extragalactic reference sources, and we present the images in a star-centered frame, aligned by the position of the star as derived from our fits to its orbital motion, parallax, and proper motion. Both the flux density and the morphology of IM Peg are variable. For most sessions, the emission region has a single-peaked structure, but 25% of the time, we observed a two-peaked (and on one occasion perhaps a three-peaked) structure. On average, the emission region is elongated by 1.4 ± 0.4 mas (FWHM), with the average direction of elongation being close to that of the sky projection of the orbit normal. The average length of the emission region is approximately equal to the diameter of the primary star. No significant correlation with the orbital phase is found for either the flux density or the direction of elongation, and no preference for any particular longitude on the star is shown by the emission region. -
The 29 of May Or Sir Arthur and The
International Journal of Management and Applied Science, ISSN: 2394-7926 Volume-4, Issue-4, Apr.-2018 http://iraj.in THE 29TH OF MAY OR SIR ARTHUR AND THE BENDING LIGHT (TEACHING SCIENCE AND LITERATURE JOINTLY) MICHAEL KATZ Faculty of Education, Haifa University, Mount Carmel, Haifa, Israel E-mail: [email protected] Abstract - In this paper I present an instance illustrating a program of joint teaching of science and literature. The instance sketched here rests on recognition that at the age when children read books like Kästner's "The 35th of May" they can already taste notions of Relativity Theory through the story of Eddington's expedition to Principe Island. Eddington's story is briefly recounted here with references to Kästner's book. The relationships thus exhibited between science and fiction, fantasy and reality, theory and actuality, humor and earnestness, should help evoke school age children's interest in science and literature at early stages in their studies' endeavor. Keywords - Bending Light, Fantasy, Gravitation, Relativity, Science and Literature. I. INTRODUCTION South Seas" – a story of adventure and humor by Erich Kästner, the well known German author of The idea that teaching science and humanities in more than a few dearly loved novels, mostly for primary and secondary schools needn't necessarily be children and adolescents. The book was first seen as disjoint objectives is prevalent in the thought published in German in 1931 and in English shortly and practice of more than a few scholars and teachers after. in recent years. Carole Cox [1], for example, unfolds forty strategies of literature based teaching of various Twelve years earlier, in 1919, another journey topics in arts and sciences.