Gravitational Waves in Special and General Relativity
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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 ................. -
Measurement of the Speed of Gravity
Measurement of the Speed of Gravity Yin Zhu Agriculture Department of Hubei Province, Wuhan, China Abstract From the Liénard-Wiechert potential in both the gravitational field and the electromagnetic field, it is shown that the speed of propagation of the gravitational field (waves) can be tested by comparing the measured speed of gravitational force with the measured speed of Coulomb force. PACS: 04.20.Cv; 04.30.Nk; 04.80.Cc Fomalont and Kopeikin [1] in 2002 claimed that to 20% accuracy they confirmed that the speed of gravity is equal to the speed of light in vacuum. Their work was immediately contradicted by Will [2] and other several physicists. [3-7] Fomalont and Kopeikin [1] accepted that their measurement is not sufficiently accurate to detect terms of order , which can experimentally distinguish Kopeikin interpretation from Will interpretation. Fomalont et al [8] reported their measurements in 2009 and claimed that these measurements are more accurate than the 2002 VLBA experiment [1], but did not point out whether the terms of order have been detected. Within the post-Newtonian framework, several metric theories have studied the radiation and propagation of gravitational waves. [9] For example, in the Rosen bi-metric theory, [10] the difference between the speed of gravity and the speed of light could be tested by comparing the arrival times of a gravitational wave and an electromagnetic wave from the same event: a supernova. Hulse and Taylor [11] showed the indirect evidence for gravitational radiation. However, the gravitational waves themselves have not yet been detected directly. [12] In electrodynamics the speed of electromagnetic waves appears in Maxwell equations as c = √휇0휀0, no such constant appears in any theory of gravity. -
Mass Spectrometry: Quadrupole Mass Filter
Advanced Lab, Jan. 2008 Mass Spectrometry: Quadrupole Mass Filter The mass spectrometer is essentially an instrument which can be used to measure the mass, or more correctly the mass/charge ratio, of ionized atoms or other electrically charged particles. Mass spectrometers are now used in physics, geology, chemistry, biology and medicine to determine compositions, to measure isotopic ratios, for detecting leaks in vacuum systems, and in homeland security. Mass Spectrometer Designs The first mass spectrographs were invented almost 100 years ago, by A.J. Dempster, F.W. Aston and others, and have therefore been in continuous development over a very long period. However the principle of using electric and magnetic fields to accelerate and establish the trajectories of ions inside the spectrometer according to their mass/charge ratio is common to all the different designs. The following description of Dempster’s original mass spectrograph is a simple illustration of these physical principles: The magnetic sector spectrograph PUMP F DD S S3 1 r S2 Fig. 1: Dempster’s Mass Spectrograph (1918). Atoms/molecules are first ionized by electrons emitted from the hot filament (F) and then accelerated towards the entrance slit (S1). The ions then follow a semicircular trajectory established by the Lorentz force in a uniform magnetic field. The radius of the trajectory, r, is defined by three slits (S1, S2, and S3). Ions with this selected trajectory are then detected by the detector D. How the magnetic sector mass spectrograph works: Equating the Lorentz force with the centripetal force gives: qvB = mv2/r (1) where q is the charge on the ion (usually +e), B the magnetic field, m is the mass of the ion and r the radius of the ion trajectory. -
Ph501 Electrodynamics Problem Set 8 Kirk T
Princeton University Ph501 Electrodynamics Problem Set 8 Kirk T. McDonald (2001) [email protected] http://physics.princeton.edu/~mcdonald/examples/ Princeton University 2001 Ph501 Set 8, Problem 1 1 1. Wire with a Linearly Rising Current A neutral wire along the z-axis carries current I that varies with time t according to ⎧ ⎪ ⎨ 0(t ≤ 0), I t ( )=⎪ (1) ⎩ αt (t>0),αis a constant. Deduce the time-dependence of the electric and magnetic fields, E and B,observedat apoint(r, θ =0,z = 0) in a cylindrical coordinate system about the wire. Use your expressions to discuss the fields in the two limiting cases that ct r and ct = r + , where c is the speed of light and r. The related, but more intricate case of a solenoid with a linearly rising current is considered in http://physics.princeton.edu/~mcdonald/examples/solenoid.pdf Princeton University 2001 Ph501 Set 8, Problem 2 2 2. Harmonic Multipole Expansion A common alternative to the multipole expansion of electromagnetic radiation given in the Notes assumes from the beginning that the motion of the charges is oscillatory with angular frequency ω. However, we still use the essence of the Hertz method wherein the current density is related to the time derivative of a polarization:1 J = p˙ . (2) The radiation fields will be deduced from the retarded vector potential, 1 [J] d 1 [p˙ ] d , A = c r Vol = c r Vol (3) which is a solution of the (Lorenz gauge) wave equation 1 ∂2A 4π ∇2A − = − J. (4) c2 ∂t2 c Suppose that the Hertz polarization vector p has oscillatory time dependence, −iωt p(x,t)=pω(x)e . -
Einstein's Mistakes
Einstein’s Mistakes Einstein was the greatest genius of the Twentieth Century, but his discoveries were blighted with mistakes. The Human Failing of Genius. 1 PART 1 An evaluation of the man Here, Einstein grows up, his thinking evolves, and many quotations from him are listed. Albert Einstein (1879-1955) Einstein at 14 Einstein at 26 Einstein at 42 3 Albert Einstein (1879-1955) Einstein at age 61 (1940) 4 Albert Einstein (1879-1955) Born in Ulm, Swabian region of Southern Germany. From a Jewish merchant family. Had a sister Maja. Family rejected Jewish customs. Did not inherit any mathematical talent. Inherited stubbornness, Inherited a roguish sense of humor, An inclination to mysticism, And a habit of grüblen or protracted, agonizing “brooding” over whatever was on its mind. Leading to the thought experiment. 5 Portrait in 1947 – age 68, and his habit of agonizing brooding over whatever was on its mind. He was in Princeton, NJ, USA. 6 Einstein the mystic •“Everyone who is seriously involved in pursuit of science becomes convinced that a spirit is manifest in the laws of the universe, one that is vastly superior to that of man..” •“When I assess a theory, I ask myself, if I was God, would I have arranged the universe that way?” •His roguish sense of humor was always there. •When asked what will be his reactions to observational evidence against the bending of light predicted by his general theory of relativity, he said: •”Then I would feel sorry for the Good Lord. The theory is correct anyway.” 7 Einstein: Mathematics •More quotations from Einstein: •“How it is possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of physical reality?” •Questions asked by many people and Einstein: •“Is God a mathematician?” •His conclusion: •“ The Lord is cunning, but not malicious.” 8 Einstein the Stubborn Mystic “What interests me is whether God had any choice in the creation of the world” Some broadcasters expunged the comment from the soundtrack because they thought it was blasphemous. -
Linearized Einstein Field Equations
General Relativity Fall 2019 Lecture 15: Linearized Einstein field equations Yacine Ali-Ha¨ımoud October 17th 2019 SUMMARY FROM PREVIOUS LECTURE We are considering nearly flat spacetimes with nearly globally Minkowski coordinates: gµν = ηµν + hµν , with jhµν j 1. Such coordinates are not unique. First, we can make Lorentz transformations and keep a µ ν globally-Minkowski coordinate system, with hµ0ν0 = Λ µ0 Λ ν0 hµν , so that hµν can be seen as a Lorentz tensor µ µ µ ν field on flat spacetime. Second, if we make small changes of coordinates, x ! x − ξ , with j@µξ j 1, the metric perturbation remains small and changes as hµν ! hµν + 2ξ(µ,ν). By analogy with electromagnetism, we can see these small coordinate changes as gauge transformations, leaving the Riemann tensor unchanged at linear order. Since we will linearize the relevant equations, we may work in Fourier space: each Fourier mode satisfies an independent equation. We denote by ~k the wavenumber and by k^ its direction and k its norm. We have decomposed the 10 independent components of the metric perturbation according to their transformation properties under spatial rotations: there are 4 independent \scalar" components, which can be taken, for instance, ^i ^i^j to be h00; k h0i; hii, and k k hij { or any 4 linearly independent combinations thereof. There are 2 independent ilm^ ilm^ ^j transverse \vector" components, each with 2 independent components: klh0m and klhmjk { these are proportional to the curl of h0i and to the curl of the divergence of hij, and are divergenceless (transverse to the ~ TT Fourier wavenumber k). -
A Brief History of Gravitational Waves
universe Review A Brief History of Gravitational Waves Jorge L. Cervantes-Cota 1, Salvador Galindo-Uribarri 1 and George F. Smoot 2,3,4,* 1 Department of Physics, National Institute for Nuclear Research, Km 36.5 Carretera Mexico-Toluca, Ocoyoacac, C.P. 52750 Mexico, Mexico; [email protected] (J.L.C.-C.); [email protected] (S.G.-U.) 2 Helmut and Ana Pao Sohmen Professor at Large, Institute for Advanced Study, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, 999077 Hong Kong, China 3 Université Sorbonne Paris Cité, Laboratoire APC-PCCP, Université Paris Diderot, 10 rue Alice Domon et Leonie Duquet, 75205 Paris Cedex 13, France 4 Department of Physics and LBNL, University of California; MS Bldg 50-5505 LBNL, 1 Cyclotron Road Berkeley, 94720 CA, USA * Correspondence: [email protected]; Tel.:+1-510-486-5505 Academic Editors: Lorenzo Iorio and Elias C. Vagenas Received: 21 July 2016; Accepted: 2 September 2016; Published: 13 September 2016 Abstract: This review describes the discovery of gravitational waves. We recount the journey of predicting and finding those waves, since its beginning in the early twentieth century, their prediction by Einstein in 1916, theoretical and experimental blunders, efforts towards their detection, and finally the subsequent successful discovery. Keywords: gravitational waves; General Relativity; LIGO; Einstein; strong-field gravity; binary black holes 1. Introduction Einstein’s General Theory of Relativity, published in November 1915, led to the prediction of the existence of gravitational waves that would be so faint and their interaction with matter so weak that Einstein himself wondered if they could ever be discovered. -
Optimal Design of the Magnet Profile for a Compact Quadrupole on Storage Ring at the Canadian Synchrotron Facility
OPTIMAL DESIGN OF THE MAGNET PROFILE FOR A COMPACT QUADRUPOLE ON STORAGE RING AT THE CANADIAN SYNCHROTRON FACILITY A Thesis Submitted to the College of Graduate and Postdoctoral Studies in Partial Fulfillment of the Requirements for the Degree of Master of Science in the Department of Mechanical Engineering, University of Saskatchewan, Saskatoon By MD Armin Islam ©MD Armin Islam, December 2019. All rights reserved. Permission to Use In presenting this thesis in partial fulfillment of the requirements for a Postgraduate degree from the University of Saskatchewan, I agree that the Libraries of this University may make it freely available for inspection. I further agree that permission for copying of this thesis in any manner, in whole or in part, for scholarly purposes may be granted by the professors who supervised my thesis work or, in their absence, by the Head of the Department or the Dean of the College in which my thesis work was done. It is understood that any copying or publication or use of this thesis or parts thereof for financial gain shall not be allowed without my written permission. It is also understood that due recognition shall be given to me and to the University of Saskatchewan in any scholarly use which may be made of any material in my thesis. Requests for permission to copy or to make other use of the material in this thesis in whole or part should be addressed to: Dean College of Graduate and Postdoctoral Studies University of Saskatchewan 116 Thorvaldson Building, 110 Science Place Saskatoon, Saskatchewan, S7N 5C9, Canada Or Head of the Department of Mechanical Engineering College of Engineering University of Saskatchewan 57 Campus Drive, Saskatoon, SK S7N 5A9, Canada i Abstract This thesis concerns design of a quadrupole magnet for the next generation Canadian Light Source storage ring. -
Equivalence Principle (WEP) of General Relativity Using a New Quantum Gravity Theory Proposed by the Authors Called Electro-Magnetic Quantum Gravity Or EMQG (Ref
WHAT ARE THE HIDDEN QUANTUM PROCESSES IN EINSTEIN’S WEAK PRINCIPLE OF EQUIVALENCE? Tom Ostoma and Mike Trushyk 48 O’HARA PLACE, Brampton, Ontario, L6Y 3R8 [email protected] Monday April 12, 2000 ACKNOWLEDGMENTS We wish to thank R. Mongrain (P.Eng) for our lengthy conversations on the nature of space, time, light, matter, and CA theory. ABSTRACT We provide a quantum derivation of Einstein’s Weak Equivalence Principle (WEP) of general relativity using a new quantum gravity theory proposed by the authors called Electro-Magnetic Quantum Gravity or EMQG (ref. 1). EMQG is manifestly compatible with Cellular Automata (CA) theory (ref. 2 and 4), and is also based on a new theory of inertia (ref. 5) proposed by R. Haisch, A. Rueda, and H. Puthoff (which we modified and called Quantum Inertia, QI). QI states that classical Newtonian Inertia is a property of matter due to the strictly local electrical force interactions contributed by each of the (electrically charged) elementary particles of the mass with the surrounding (electrically charged) virtual particles (virtual masseons) of the quantum vacuum. The sum of all the tiny electrical forces (photon exchanges with the vacuum particles) originating in each charged elementary particle of the accelerated mass is the source of the total inertial force of a mass which opposes accelerated motion in Newton’s law ‘F = MA’. The well known paradoxes that arise from considerations of accelerated motion (Mach’s principle) are resolved, and Newton’s laws of motion are now understood at the deeper quantum level. We found that gravity also involves the same ‘inertial’ electromagnetic force component that exists in inertial mass. -
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. -
Formulation of Einstein Field Equation Through Curved Newtonian Space-Time
Formulation of Einstein Field Equation Through Curved Newtonian Space-Time Austen Berlet Lord Dorchester Secondary School Dorchester, Ontario, Canada Abstract This paper discusses a possible derivation of Einstein’s field equations of general relativity through Newtonian mechanics. It shows that taking the proper perspective on Newton’s equations will start to lead to a curved space time which is basis of the general theory of relativity. It is important to note that this approach is dependent upon a knowledge of general relativity, with out that, the vital assumptions would not be realized. Note: A number inside of a double square bracket, for example [[1]], denotes an endnote found on the last page. 1. Introduction The purpose of this paper is to show a way to rediscover Einstein’s General Relativity. It is done through analyzing Newton’s equations and making the conclusion that space-time must not only be realized, but also that it must have curvature in the presence of matter and energy. 2. Principal of Least Action We want to show here the Lagrangian action of limiting motion of Newton’s second law (F=ma). We start with a function q mapping to n space of n dimensions and we equip it with a standard inner product. q : → (n ,(⋅,⋅)) (1) We take a function (q) between q0 and q1 and look at the ds of a section of the curve. We then look at some properties of this function (q). We see that the classical action of the functional (L) of q is equal to ∫ds, L denotes the systems Lagrangian. -
Einstein's 1916 Derivation of the Field Equations
1 Einstein's 1916 derivation of the Field Equations Galina Weinstein 24/10/2013 Abstract: In his first November 4, 1915 paper Einstein wrote the Lagrangian form of his field equations. In the fourth November 25, 1915 paper, Einstein added a trace term of the energy- momentum tensor on the right-hand side of the generally covariant field equations. The main purpose of the present work is to show that in November 4, 1915, Einstein had already explored much of the main ingredients that were required for the formulation of the final form of the field equations of November 25, 1915. The present work suggests that the idea of adding the second-term on the right-hand side of the field equation might have originated in reconsideration of the November 4, 1915 field equations. In this regard, the final form of Einstein's field equations with the trace term may be linked with his work of November 4, 1915. The interesting history of the derivation of the final form of the field equations is inspired by the exchange of letters between Einstein and Paul Ehrenfest in winter 1916 and by Einstein's 1916 derivation of the November 25, 1915 field equations. In 1915, Einstein wrote the vacuum (matter-free) field equations in the form:1 for all systems of coordinates for which It is sufficient to note that the left-hand side represents the gravitational field, with g the metric tensor field. Einstein wrote the field equations in Lagrangian form. The action,2 and the Lagrangian, 2 Using the components of the gravitational field: Einstein wrote the variation: which gives:3 We now come back to (2), and we have, Inserting (6) into (7) gives the field equations (1).