Space, Time, and Spacetime
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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 ................. -
The Son of Lamoraal Ulbo De Sitter, a Judge, and Catharine Theodore Wilhelmine Bertling
558 BIOGRAPHIES v.i WiLLEM DE SITTER viT 1872-1934 De Sitter was bom on 6 May 1872 in Sneek (province of Friesland), the son of Lamoraal Ulbo de Sitter, a judge, and Catharine Theodore Wilhelmine Bertling. His father became presiding judge of the court in Arnhem, and that is where De Sitter attended gymna sium. At the University of Groniiigen he first studied mathematics and physics and then switched to astronomy under Jacobus Kapteyn. De Sitter spent two years observing and studying under David Gill at the Cape Obsen'atory, the obseivatory with which Kapteyn was co operating on the Cape Photographic Durchmusterung. De Sitter participated in the program to make precise measurements of the positions of the Galilean moons of Jupiter, using a heliometer. In 1901 he received his doctorate under Kapteyn on a dissertation on Jupiter's satellites: Discussion of Heliometer Observations of Jupiter's Satel lites. De Sitter remained at Groningen as an assistant to Kapteyn in the astronomical laboratory, until 1909, when he was appointed to the chair of astronomy at the University of Leiden. In 1919 he be came director of the Leiden Observatory. He remained in these posts until his death in 1934. De Sitter's work was highly mathematical. With his work on Jupi ter's satellites, De Sitter pursued the new methods of celestial me chanics of Poincare and Tisserand. His earlier heliometer meas urements were later supplemented by photographic measurements made at the Cape, Johannesburg, Pulkowa, Greenwich, and Leiden. De Sitter's final results on this subject were published as 'New Math ematical Theory of Jupiter's Satellites' in 1925. -
The Influence of Copernicus on Physics
radiosources, obJects believed to be extremely distant. Counts of these The influence of Copernicus sources indicate that their number density is different at large distances on physics from what it is here. Or we may put it differently : the number density of B. Paczynski, Institute of Astronomy, Warsaw radiosources was different a long time ago from what it is now. This is in This month sees the 500th Anniver tific one, and there are some perils contradiction with the strong cosmo sary of the birth of Copernicus. To associated with it. logical principle. Also, the Universe mark the occasion, Europhysics News Why should we not make the cos seems to be filled with a black body has invited B. Paczynski, Polish Board mological principle more perfect ? microwave radiation which has, at member of the EPS Division on Phy Once we say that the Universe is the present, the temperature of 2.7 K. It is sics in Astronomy, to write this article. same everywhere, should we not also almost certain that this radiation was say that it is the same at all times ? produced about 1010 years ago when Once we have decided our place is the Universe was very dense and hot, Copernicus shifted the centre of and matter was in thermal equilibrium the Universe from Earth to the Sun, not unique, why should the time we live in be unique ? In fact such a with radiation. Such conditions are and by doing so he deprived man vastly different from the present kind of its unique position in the Uni strong cosmological principle has been adopted by some scientists. -
Cosmology Timeline
Timeline Cosmology • 2nd Millennium BCEBC Mesopotamian cosmology has a flat,circular Earth enclosed in a cosmic Ocean • 12th century BCEC Rigveda has some cosmological hymns, most notably the Nasadiya Sukta • 6th century BCE Anaximander, the first (true) cosmologist - pre-Socratic philosopher from Miletus, Ionia - Nature ruled by natural laws - Apeiron (boundless, infinite, indefinite), that out of which the universe originates • 5th century BCE Plato - Timaeus - dialogue describing the creation of the Universe, - demiurg created the world on the basis of geometric forms (Platonic solids) • 4th century BCE Aristotle - proposes an Earth-centered universe in which the Earth is stationary and the cosmos, is finite in extent but infinite in time • 3rd century BCE Aristarchus of Samos - proposes a heliocentric (sun-centered) Universe, based on his conclusion/determination that the Sun is much larger than Earth - further support in 2nd century BCE by Seleucus of Seleucia • 3rd century BCE Archimedes - book The Sand Reckoner: diameter of cosmos � 2 lightyears - heliocentric Universe not possible • 3rd century BCE Apollonius of Perga - epicycle theory for lunar and planetary motions • 2nd century CE Ptolemaeus - Almagest/Syntaxis: culmination of ancient Graeco-Roman astronomy - Earth-centered Universe, with Sun, Moon and planets revolving on epicyclic orbits around Earth • 5th-13th century CE Aryabhata (India) and Al-Sijzi (Iran) propose that the Earth rotates around its axis. First empirical evidence for Earth’s rotation by Nasir al-Din al-Tusi. • 8th century CE Puranic Hindu cosmology, in which the Universe goes through repeated cycles of creation, destruction and rebirth, with each cycle lasting 4.32 billion years. • • 1543 Nicolaus Copernicus - publishes heliocentric universe in De Revolutionibus Orbium Coelestium - implicit introduction Copernican principle: Earth/Sun is not special • 1609-1632 Galileo Galilei - by means of (telescopic) observations, proves the validity of the heliocentric Universe. -
Chapter 5 the Relativistic Point Particle
Chapter 5 The Relativistic Point Particle To formulate the dynamics of a system we can write either the equations of motion, or alternatively, an action. In the case of the relativistic point par- ticle, it is rather easy to write the equations of motion. But the action is so physical and geometrical that it is worth pursuing in its own right. More importantly, while it is difficult to guess the equations of motion for the rela- tivistic string, the action is a natural generalization of the relativistic particle action that we will study in this chapter. We conclude with a discussion of the charged relativistic particle. 5.1 Action for a relativistic point particle How can we find the action S that governs the dynamics of a free relativis- tic particle? To get started we first think about units. The action is the Lagrangian integrated over time, so the units of action are just the units of the Lagrangian multiplied by the units of time. The Lagrangian has units of energy, so the units of action are L2 ML2 [S]=M T = . (5.1.1) T 2 T Recall that the action Snr for a free non-relativistic particle is given by the time integral of the kinetic energy: 1 dx S = mv2(t) dt , v2 ≡ v · v, v = . (5.1.2) nr 2 dt 105 106 CHAPTER 5. THE RELATIVISTIC POINT PARTICLE The equation of motion following by Hamilton’s principle is dv =0. (5.1.3) dt The free particle moves with constant velocity and that is the end of the story. -
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. -
Supplemental Lecture 5 Precession in Special and General Relativity
Supplemental Lecture 5 Thomas Precession and Fermi-Walker Transport in Special Relativity and Geodesic Precession in General Relativity Abstract This lecture considers two topics in the motion of tops, spins and gyroscopes: 1. Thomas Precession and Fermi-Walker Transport in Special Relativity, and 2. Geodesic Precession in General Relativity. A gyroscope (“spin”) attached to an accelerating particle traveling in Minkowski space-time is seen to precess even in a torque-free environment. The angular rate of the precession is given by the famous Thomas precession frequency. We introduce the notion of Fermi-Walker transport to discuss this phenomenon in a systematic fashion and apply it to a particle propagating on a circular orbit. In a separate discussion we consider the geodesic motion of a particle with spin in a curved space-time. The spin necessarily precesses as a consequence of the space-time curvature. We illustrate the phenomenon for a circular orbit in a Schwarzschild metric. Accurate experimental measurements of this effect have been accomplished using earth satellites carrying gyroscopes. This lecture supplements material in the textbook: Special Relativity, Electrodynamics and General Relativity: From Newton to Einstein (ISBN: 978-0-12-813720-8). The term “textbook” in these Supplemental Lectures will refer to that work. Keywords: Fermi-Walker transport, Thomas precession, Geodesic precession, Schwarzschild metric, Gravity Probe B (GP-B). Introduction In this supplementary lecture we discuss two instances of precession: 1. Thomas precession in special relativity and 2. Geodesic precession in general relativity. To set the stage, let’s recall a few things about precession in classical mechanics, Newton’s world, as we say in the textbook. -
What Is Gravity Probe B?
What is Gravity Probe B? Gravity Probe B (GP-B) is a NASA physics mission to experimentally in• William Fairbank once remarked: “No mission could be sim• vestigate Einstein’s 1916 general theory of relativity—his theory of gravity. pler than Gravity Probe B. It’s just a star, a telescope, and a spin• GB-B uses four spherical gyroscopes and a telescope, housed in a satellite ning sphere.” However, it took the exceptional collaboration of Stan• orbiting 642 km (400 mi) above the Earth, to measure, with unprecedented ford, MSFC, Lockheed Martin and a host of others more than four accuracy, two extraordinary effects predicted by the general theory of rela• decades to develop the ultra-precise gyroscopes and the other cut• tivity: 1) the geodetic effect—the amount by which the Earth warps the local ting-edge technologies necessary to carry out this “simple” experiment. spacetime in which it resides; and 2) the frame-dragging effect—the amount by which the rotating Earth drags its local spacetime around with it. GP-B tests these two effects by precisely measuring the precession (displacement) The GP-B Flight Mission angles of the spin axes of the four gyros over the course of a year and com• paring these experimental results with predictions from Einstein’s theory. On April 20, 2004 at 9:57:24 AM PDT, a crowd of over 2,000 current and former GP-B team members and supporters watched and cheered as the A Quest for Experimental Truth GP-B spacecraftlift ed offf rom Vandenberg Air Force Base. -
Chapter 1 Introduction
Chapter 1 Introduction 1.1 Introduction Einsteins theory of general relativity (GR) [Einstein, 1915b,a, de Sitter, 1917, Silberstein, 1917] together with quantum field theory are considered to be the pillars of modern physics. GR deals with gravitational phenomena treating space-time as a four-dimensional manifold. The theory has a well understood and mathematically very elegant structure and is internally consistent. GR describes gracefully all the experimental tests of gravity [Weinberg, 1972, Berti et al., 2015] conducted till now without a single adjustable parameter. 1.2 General Relativity The inverse-square gravitational force law introduced by Newton predicted cor- rectly a variety of gravitational phenomena including planetary motion. In fact the 10 Chapter 1: Introduction 11 Newtons theory was very successful in describing all the known gravitational phe- nomena at that time. The first deviation from the Newtons theory was noticed in the precession of Mercurys orbit; a 43 arc-seconds per century excess precession over that given by Newtons theory was observed. However, not because of such small deviation in just one experimental results, Einstein developed General relativity to make the theory of gravitation consistent with Special Theory of Relativity (STR). Newtonian mechanics is based on the existence of absolute space and thereby absolute motion whereas STR rests on the idea that all (un-accelerated) motions are relative in nature. A consequent feature emerges in STR that no information can be transmitted with speed greater than the speed of light whereas the Newtons law of gravity implies action at a distance i.e. in Newtonian paradigm the gravita- tional information moves with infinite speed. -
Partículas De Prueba Con Espín En Experimentos Tipo Michelson Y Morley
Partículas de prueba con espín en experimentos tipo Michelson y Morley A Dissertation Presented to the Faculty of Science of Universidad Nacional de Colombia in Candidacy for the Degree of Doctor of Philosophy by Nelson Velandia-Heredia, S.J. Dissertation Director: Juan Manuel Tejeiro, Dr. Rer. Nat. November 2017 Abstract Partículas de prueba con espín en experimentos tipo Michelson y Morley Nelson Velandia-Heredia, S.J. 2018 Resumen En esta tesis, nosotros caracterizamos, con ayuda de la relatividad numérica, los efectos gravitomagnéticos para partículas de prueba con espín cuando se mueven en un campo rotante. Dado este propósito, nosotros resolveremos numéricamente las ecuaciones de Mathisson-Papapetrou- Dixon en una métrica de Kerr. Además, estu- diaremos la influencia del valor y la orientación de espín en el efecto reloj. Palabras clave: Relatividad general, relatividad numérica, ecuaciones de Mathisson-Papapetrou-Dixon, métrica de Kerr, partículas de prueba con es- pín. Abstract In this thesis, we characterize, with help of the numerical relativity, the gravito- magnetic effects for spinning test particles when are moving in a rotating field. Since this aim, we numerically solve the Mathisson-Papapetrou-Dixon equations in a Kerr metric. Also, we study the influence of value and orientation of the spin in the clock effect. Keywords: General relativity, numerical relativity, Mathisson-Papapetrou-Dixon equations, Kerr metric, the spinning test particles ii Copyright c 2018 by Nelson Velandia-Heredia, S.J. All rights reserved. i Contents 1 Introduction 1 2 Formulation for the equations of motion 8 2.1 Introduction................................8 2.2 Equation of linear field . 10 2.2.1 Newtonian mechanics . -
Variable Planck's Constant
Variable Planck’s Constant: Treated As A Dynamical Field And Path Integral Rand Dannenberg Ventura College, Physics and Astronomy Department, Ventura CA [email protected] [email protected] January 28, 2021 Abstract. The constant ħ is elevated to a dynamical field, coupling to other fields, and itself, through the Lagrangian density derivative terms. The spatial and temporal dependence of ħ falls directly out of the field equations themselves. Three solutions are found: a free field with a tadpole term; a standing-wave non-propagating mode; a non-oscillating non-propagating mode. The first two could be quantized. The third corresponds to a zero-momentum classical field that naturally decays spatially to a constant with no ad-hoc terms added to the Lagrangian. An attempt is made to calibrate the constants in the third solution based on experimental data. The three fields are referred to as actons. It is tentatively concluded that the acton origin coincides with a massive body, or point of infinite density, though is not mass dependent. An expression for the positional dependence of Planck’s constant is derived from a field theory in this work that matches in functional form that of one derived from considerations of Local Position Invariance violation in GR in another paper by this author. Astrophysical and Cosmological interpretations are provided. A derivation is shown for how the integrand in the path integral exponent becomes Lc/ħ(r), where Lc is the classical action. The path that makes stationary the integral in the exponent is termed the “dominant” path, and deviates from the classical path systematically due to the position dependence of ħ. -
The Big-Bang Theory AST-101, Ast-117, AST-602
AST-101, Ast-117, AST-602 The Big-Bang theory Luis Anchordoqui Thursday, November 21, 19 1 17.1 The Expanding Universe! Last class.... Thursday, November 21, 19 2 Hubbles Law v = Ho × d Velocity of Hubbles Recession Distance Constant (Mpc) (Doppler Shift) (km/sec/Mpc) (km/sec) velocity Implies the Expansion of the Universe! distance Thursday, November 21, 19 3 The redshift of a Galaxy is: A. The rate at which a Galaxy is expanding in size B. How much reader the galaxy appears when observed at large distances C. the speed at which a galaxy is orbiting around the Milky Way D. the relative speed of the redder stars in the galaxy with respect to the blues stars E. The recessional velocity of a galaxy, expressed as a fraction of the speed of light Thursday, November 21, 19 4 The redshift of a Galaxy is: A. The rate at which a Galaxy is expanding in size B. How much reader the galaxy appears when observed at large distances C. the speed at which a galaxy is orbiting around the Milky Way D. the relative speed of the redder stars in the galaxy with respect to the blues stars E. The recessional velocity of a galaxy, expressed as a fraction of the speed of light Thursday, November 21, 19 5 To a first approximation, a rough maximum age of the Universe can be estimated using which of the following? A. the age of the oldest open clusters B. 1/H0 the Hubble time C. the age of the Sun D.