Time Travel, Clock Puzzles and Their Experimental Tests
Total Page:16
File Type:pdf, Size:1020Kb
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 ................. -
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. -
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. -
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. -
Highlights in Space 2010
International Astronautical Federation Committee on Space Research International Institute of Space Law 94 bis, Avenue de Suffren c/o CNES 94 bis, Avenue de Suffren UNITED NATIONS 75015 Paris, France 2 place Maurice Quentin 75015 Paris, France Tel: +33 1 45 67 42 60 Fax: +33 1 42 73 21 20 Tel. + 33 1 44 76 75 10 E-mail: : [email protected] E-mail: [email protected] Fax. + 33 1 44 76 74 37 URL: www.iislweb.com OFFICE FOR OUTER SPACE AFFAIRS URL: www.iafastro.com E-mail: [email protected] URL : http://cosparhq.cnes.fr Highlights in Space 2010 Prepared in cooperation with the International Astronautical Federation, the Committee on Space Research and the International Institute of Space Law The United Nations Office for Outer Space Affairs is responsible for promoting international cooperation in the peaceful uses of outer space and assisting developing countries in using space science and technology. United Nations Office for Outer Space Affairs P. O. Box 500, 1400 Vienna, Austria Tel: (+43-1) 26060-4950 Fax: (+43-1) 26060-5830 E-mail: [email protected] URL: www.unoosa.org United Nations publication Printed in Austria USD 15 Sales No. E.11.I.3 ISBN 978-92-1-101236-1 ST/SPACE/57 *1180239* V.11-80239—January 2011—775 UNITED NATIONS OFFICE FOR OUTER SPACE AFFAIRS UNITED NATIONS OFFICE AT VIENNA Highlights in Space 2010 Prepared in cooperation with the International Astronautical Federation, the Committee on Space Research and the International Institute of Space Law Progress in space science, technology and applications, international cooperation and space law UNITED NATIONS New York, 2011 UniTEd NationS PUblication Sales no. -
JOHN EARMAN* and CLARK GL YMUURT the GRAVITATIONAL RED SHIFT AS a TEST of GENERAL RELATIVITY: HISTORY and ANALYSIS
JOHN EARMAN* and CLARK GL YMUURT THE GRAVITATIONAL RED SHIFT AS A TEST OF GENERAL RELATIVITY: HISTORY AND ANALYSIS CHARLES St. John, who was in 1921 the most widely respected student of the Fraunhofer lines in the solar spectra, began his contribution to a symposium in Nncure on Einstein’s theories of relativity with the following statement: The agreement of the observed advance of Mercury’s perihelion and of the eclipse results of the British expeditions of 1919 with the deductions from the Einstein law of gravitation gives an increased importance to the observations on the displacements of the absorption lines in the solar spectrum relative to terrestrial sources, as the evidence on this deduction from the Einstein theory is at present contradictory. Particular interest, moreover, attaches to such observations, inasmuch as the mathematical physicists are not in agreement as to the validity of this deduction, and solar observations must eventually furnish the criterion.’ St. John’s statement touches on some of the reasons why the history of the red shift provides such a fascinating case study for those interested in the scientific reception of Einstein’s general theory of relativity. In contrast to the other two ‘classical tests’, the weight of the early observations was not in favor of Einstein’s red shift formula, and the reaction of the scientific community to the threat of disconfirmation reveals much more about the contemporary scientific views of Einstein’s theory. The last sentence of St. John’s statement points to another factor that both complicates and heightens the interest of the situation: in contrast to Einstein’s deductions of the advance of Mercury’s perihelion and of the bending of light, considerable doubt existed as to whether or not the general theory did entail a red shift for the solar spectrum. -
The Theory of Relativity and Applications: a Simple Introduction
The Downtown Review Volume 5 Issue 1 Article 3 December 2018 The Theory of Relativity and Applications: A Simple Introduction Ellen Rea Cleveland State University Follow this and additional works at: https://engagedscholarship.csuohio.edu/tdr Part of the Engineering Commons, and the Physical Sciences and Mathematics Commons How does access to this work benefit ou?y Let us know! Recommended Citation Rea, Ellen. "The Theory of Relativity and Applications: A Simple Introduction." The Downtown Review. Vol. 5. Iss. 1 (2018) . Available at: https://engagedscholarship.csuohio.edu/tdr/vol5/iss1/3 This Article is brought to you for free and open access by the Student Scholarship at EngagedScholarship@CSU. It has been accepted for inclusion in The Downtown Review by an authorized editor of EngagedScholarship@CSU. For more information, please contact [email protected]. Rea: The Theory of Relativity and Applications What if I told you that time can speed up and slow down? What if I told you that everything you think you know about gravity is a lie? When Albert Einstein presented his theory of relativity to the world in the early 20th century, he was proposing just that. And what’s more? He’s been proven correct. Einstein’s theory has two parts: special relativity, which deals with inertial reference frames and general relativity, which deals with the curvature of space- time. A surface level study of the theory and its consequences followed by a look at some of its applications will provide an introduction to one of the most influential scientific discoveries of the last century. -
A Test of General Relativity Using the LARES and LAGEOS Satellites And
A Test of General Relativity Using the LARES and LAGEOS Satellites and a GRACE Earth’s Gravity Model Measurement of Earth’s Dragging of Inertial Frames Ignazio Ciufolini∗1,2, Antonio Paolozzi2,3, Erricos C. Pavlis4, Rolf Koenig5, John Ries6, Vahe Gurzadyan7, Richard Matzner8, Roger Penrose9, Giampiero Sindoni10, Claudio Paris2,3, Harutyun Khachatryan7 and Sergey Mirzoyan7 1 Dip. Ingegneria dell’Innovazione, Universit`adel Salento, Lecce, Italy 2Museo della fisica e Centro studi e ricerche Enrico Fermi, Rome, Italy 3Scuola di Ingegneria Aerospaziale, Sapienza Universit`adi Roma, Italy 4Joint Center for Earth Systems Technology (JCET), University of Maryland, Baltimore County, USA 5Helmholtz Centre Potsdam, GFZ German Research Centre for Geosciences, Potsdam, Germany 6Center for Space Research, University of Texas at Austin, Austin, USA 7Center for Cosmology and Astrophysics, Alikhanian National Laboratory and Yerevan State University, Yerevan, Armenia 8Theory Center, University of Texas at Austin, Austin, USA 9Mathematical Institute, University of Oxford, Oxford, UK 10DIAEE, Sapienza Universit`adi Roma, Rome, Italy April 1, 2016 We present a test of General Relativity, the measurement of the Earth’s dragging of inertial frames. Our result is obtained using about 3.5 years of laser-ranged observations of the LARES, LAGEOS and LAGEOS 2 laser- ranged satellites together with the Earth’s gravity field model GGM05S pro- arXiv:1603.09674v1 [gr-qc] 29 Mar 2016 duced by the space geodesy mission GRACE. We measure µ = (0.994 ± 0.002) ± 0.05, where µ is the Earth’s dragging of inertial frames normalized to its General Relativity value, 0.002 is the 1-sigma formal error and 0.05 is the estimated systematic error mainly due to the uncertainties in the Earth’s gravity model GGM05S. -
Albert Einstein and Relativity
The Himalayan Physics, Vol.1, No.1, May 2010 Albert Einstein and Relativity Kamal B Khatri Department of Physics, PN Campus, Pokhara, Email: [email protected] Albert Einstein was born in Germany in 1879.In his the follower of Mach and his nascent concept helped life, Einstein spent his most time in Germany, Italy, him to enter the world of relativity. Switzerland and USA.He is also a Nobel laureate and worked mostly in theoretical physics. Einstein In 1905, Einstein propounded the “Theory of is best known for his theories of special and general Special Relativity”. This theory shows the observed relativity. I will not be wrong if I say Einstein a independence of the speed of light on the observer’s deep thinker, a philosopher and a real physicist. The state of motion. Einstein deduced from his concept philosophies of Henri Poincare, Ernst Mach and of special relativity the twentieth century’s best David Hume infl uenced Einstein’s scientifi c and known equation, E = m c2.This equation suggests philosophical outlook. that tiny amounts of mass can be converted into huge amounts of energy which in deed, became the Einstein at the age of 4, his father showed him a boon for the development of nuclear power. pocket compass, and Einstein realized that there must be something causing the needle to move, despite Einstein realized that the principle of special relativity the apparent ‘empty space’. This shows Einstein’s could be extended to gravitational fi elds. Since curiosity to the space from his childhood. The space Einstein believed that the laws of physics were local, of our intuitive understanding is the 3-dimensional described by local fi elds, he concluded from this that Euclidean space. -
Derivation of Generalized Einstein's Equations of Gravitation in Some
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 5 February 2021 doi:10.20944/preprints202102.0157.v1 Derivation of generalized Einstein's equations of gravitation in some non-inertial reference frames based on the theory of vacuum mechanics Xiao-Song Wang Institute of Mechanical and Power Engineering, Henan Polytechnic University, Jiaozuo, Henan Province, 454000, China (Dated: Dec. 15, 2020) When solving the Einstein's equations for an isolated system of masses, V. Fock introduces har- monic reference frame and obtains an unambiguous solution. Further, he concludes that there exists a harmonic reference frame which is determined uniquely apart from a Lorentz transformation if suitable supplementary conditions are imposed. It is known that wave equations keep the same form under Lorentz transformations. Thus, we speculate that Fock's special harmonic reference frames may have provided us a clue to derive the Einstein's equations in some special class of non-inertial reference frames. Following this clue, generalized Einstein's equations in some special non-inertial reference frames are derived based on the theory of vacuum mechanics. If the field is weak and the reference frame is quasi-inertial, these generalized Einstein's equations reduce to Einstein's equa- tions. Thus, this theory may also explain all the experiments which support the theory of general relativity. There exist some differences between this theory and the theory of general relativity. Keywords: Einstein's equations; gravitation; general relativity; principle of equivalence; gravitational aether; vacuum mechanics. I. INTRODUCTION p. 411). Theoretical interpretation of the small value of Λ is still open [6]. The Einstein's field equations of gravitation are valid 3.