From Relativistic Time Dilation to Psychological Time Perception
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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 ................. -
Time Perception 1
TIME PERCEPTION 1 Time Perception: A Review on Psychological, Computational and Robotic Models Hamit Basgol, Inci Ayhan, and Emre Ugur Abstract—Animals exploit time to survive in the world. Tem- Learning to Time Prospective Behavioral Timing poral information is required for higher-level cognitive abilities Algorithmic Level Encoding Type Theory of Timing Retrospective such as planning, decision making, communication, and effective Internal Timing cooperation. Since time is an inseparable part of cognition, Clock Theory there is a growing interest in the artificial intelligence approach Models Abilities to subjective time, which has a possibility of advancing the field. The current survey study aims to provide researchers Dedicated Models with an interdisciplinary perspective on time perception. Firstly, Implementational Time Perception Sensory Timing Level Modality Type we introduce a brief background from the psychology and Intrinsic Models in Natural Motor Timing Cognitive neuroscience literature, covering the characteristics and models Systems of time perception and related abilities. Secondly, we summa- Language rize the emergent computational and robotic models of time Multi-modality Action perception. A general overview to the literature reveals that a Relationships Characteristics Multiple Timescales Decision Making substantial amount of timing models are based on a dedicated Scalar Property time processing like the emergence of a clock-like mechanism Magnitudes from the neural network dynamics and reveal a relationship between the embodiment and time perception. We also notice Fig. 1. Mind map for natural cognitive systems of time perception that most models of timing are developed for either sensory timing (i.e. ability to assess an interval) or motor timing (i.e. -
Using Time As a Sustainable Tool to Control Daylight in Space
USING TIME AS A SUSTAINABLE TOOL TO CONTROL DAYLIGHT IN SPACE Creating the Chronoform SAMER EL SAYARY Beirut Arab University and Alexandria University, Egypt [email protected] Abstract. Just as Einstein's own Relativity Theory led Einstein to reject time, Feynman’s Sum over histories theory led him to describe time simply as a direction in space. Many artists tried to visualize time as Étienne-Jules Marey when he invented the chronophotography. While the wheel of development of chronophotography in the Victorian era had ended in inventing the motion picture, a lot of research is still in its earlier stages regarding the time as a shaping media for the architectural form. Using computer animation now enables us to use time as a flexible tool to be stretched and contracted to visualize the time dilation of the Einstein's special relativity. The presented work suggests using time as a sustainable tool to shape the generated form in response to the sun movement to control the amount of daylighting entering the space by stretching the time duration and contracting time frames at certain locations of the sun trajectory along a summer day to control the amount of daylighting in the morning and afternoon versus the noon time. INTRODUCTION According to most dictionaries (Oxford, 2011; Collins, 2011; Merriam- Webster, 2015) Time is defined as a nonspatial continued progress of unlimited duration of existence and events that succeed one another ordered in the past, present, and future regarded as a whole measured in units such as minutes, hours, days, months, or years. -
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. -
The Time Wave in Time Space: a Visual Exploration Environment for Spatio
THE TIME WAVE IN TIME SPACE A VISUAL EXPLORATION ENVIRONMENT FOR SPATIO-TEMPORAL DATA Xia Li Examining Committee: prof.dr.ir. M. Molenaar University of Twente prof.dr.ir. A.Stein University of Twente prof.dr. F.J. Ormeling Utrecht University prof.dr. S.I. Fabrikant University of Zurich ITC dissertation number 175 ITC, P.O. Box 217, 7500 AE Enschede, The Netherlands ISBN 978-90-6164-295-4 Cover designed by Xia Li Printed by ITC Printing Department Copyright © 2010 by Xia Li THE TIME WAVE IN TIME SPACE A VISUAL EXPLORATION ENVIRONMENT FOR SPATIO-TEMPORAL DATA DISSERTATION to obtain the degree of doctor at the University of Twente, on the authority of the rector magnificus, prof.dr. H. Brinksma, on account of the decision of the graduation committee, to be publicly defended on Friday, October 29, 2010 at 13:15 hrs by Xia Li born in Shaanxi Province, China on May 28, 1977 This thesis is approved by Prof. Dr. M.J. Kraak promotor Prof. Z. Ma assistant promoter For my parents Qingjun Li and Ruixian Wang Acknowledgements I have a thousand words wandering in my mind the moment I finished this work. However, when I am trying to write them down, I lose almost all of them. The only word that remains is THANKS. I sincerely thank all the people who have been supporting, guiding, and encouraging me throughout my study and research period at ITC. First, I would like to express my gratitude to ITC for giving me the opportunity to carry out my PhD research. -
Physics 200 Problem Set 7 Solution Quick Overview: Although Relativity Can Be a Little Bewildering, This Problem Set Uses Just A
Physics 200 Problem Set 7 Solution Quick overview: Although relativity can be a little bewildering, this problem set uses just a few ideas over and over again, namely 1. Coordinates (x; t) in one frame are related to coordinates (x0; t0) in another frame by the Lorentz transformation formulas. 2. Similarly, space and time intervals (¢x; ¢t) in one frame are related to inter- vals (¢x0; ¢t0) in another frame by the same Lorentz transformation formu- las. Note that time dilation and length contraction are just special cases: it is time-dilation if ¢x = 0 and length contraction if ¢t = 0. 3. The spacetime interval (¢s)2 = (c¢t)2 ¡ (¢x)2 between two events is the same in every frame. 4. Energy and momentum are always conserved, and we can make e±cient use of this fact by writing them together in an energy-momentum vector P = (E=c; p) with the property P 2 = m2c2. In particular, if the mass is zero then P 2 = 0. 1. The earth and sun are 8.3 light-minutes apart. Ignore their relative motion for this problem and assume they live in a single inertial frame, the Earth-Sun frame. Events A and B occur at t = 0 on the earth and at 2 minutes on the sun respectively. Find the time di®erence between the events according to an observer moving at u = 0:8c from Earth to Sun. Repeat if observer is moving in the opposite direction at u = 0:8c. Answer: According to the formula for a Lorentz transformation, ³ u ´ 1 ¢tobserver = γ ¢tEarth-Sun ¡ ¢xEarth-Sun ; γ = p : c2 1 ¡ (u=c)2 Plugging in the numbers gives (notice that the c implicit in \light-minute" cancels the extra factor of c, which is why it's nice to measure distances in terms of the speed of light) 2 min ¡ 0:8(8:3 min) ¢tobserver = p = ¡7:7 min; 1 ¡ 0:82 which means that according to the observer, event B happened before event A! If we reverse the sign of u then 2 min + 0:8(8:3 min) ¢tobserver 2 = p = 14 min: 1 ¡ 0:82 2. -
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. -
Simulating Gamma-Ray Binaries with a Relativistic Extension of RAMSES? A
A&A 560, A79 (2013) Astronomy DOI: 10.1051/0004-6361/201322266 & c ESO 2013 Astrophysics Simulating gamma-ray binaries with a relativistic extension of RAMSES? A. Lamberts1;2, S. Fromang3, G. Dubus1, and R. Teyssier3;4 1 UJF-Grenoble 1 / CNRS-INSU, Institut de Planétologie et d’Astrophysique de Grenoble (IPAG) UMR 5274, 38041 Grenoble, France e-mail: [email protected] 2 Physics Department, University of Wisconsin-Milwaukee, Milwaukee WI 53211, USA 3 Laboratoire AIM, CEA/DSM - CNRS - Université Paris 7, Irfu/Service d’Astrophysique, CEA-Saclay, 91191 Gif-sur-Yvette, France 4 Institute for Theoretical Physics, University of Zürich, Winterthurestrasse 190, 8057 Zürich, Switzerland Received 11 July 2013 / Accepted 29 September 2013 ABSTRACT Context. Gamma-ray binaries are composed of a massive star and a rotation-powered pulsar with a highly relativistic wind. The collision between the winds from both objects creates a shock structure where particles are accelerated, which results in the observed high-energy emission. Aims. We want to understand the impact of the relativistic nature of the pulsar wind on the structure and stability of the colliding wind region and highlight the differences with colliding winds from massive stars. We focus on how the structure evolves with increasing values of the Lorentz factor of the pulsar wind, keeping in mind that current simulations are unable to reach the expected values of pulsar wind Lorentz factors by orders of magnitude. Methods. We use high-resolution numerical simulations with a relativistic extension to the hydrodynamics code RAMSES we have developed. We perform two-dimensional simulations, and focus on the region close to the binary, where orbital motion can be ne- glected. -
Doppler Boosting and Cosmological Redshift on Relativistic Jets
Doppler Boosting and cosmological redshift on relativistic jets Author: Jordi Fernández Vilana Advisor: Valentí Bosch i Ramon Facultat de Física, Universitat de Barcelona, Diagonal 645, 08028 Barcelona, Spain. Abstract: In this study our goal was to identify the possible effects that influence with the Spectral Energy Distribution of a blazar, which we deducted that are Doppler Boosting effect and cosmological redshift. Then we investigated how the spectral energy distribution varies by changing intrinsic parameters of the jet like its Lorentz factor, the angle of the line of sight respect to the jet orientation or the blazar redshift. The obtained results showed a strong enhancing of the Spectral Energy Distribution for nearly 0º angles and high Lorentz factors, while the increase in redshift produced the inverse effect, reducing the normalization of the distribution and moving the peak to lower energies. These two effects compete in the observations of all known blazars and our simple model confirmed the experimental results, that only those blazars with optimal characteristics, if at very high redshift, are suitable to be measured by the actual instruments, making the study of blazars with a high redshift an issue that needs very deep observations at different wavelengths to collect enough data to properly characterize the source population. jet those particles are moving at relativistic speeds, producing I. INTRODUCTION high energy interactions in the process. This scenario produces a wide spectrum of radiation that mainly comes, as we can see in more detail in [2], from Inverse Compton We know that active galactic nuclei or AGN are galaxies scattering. Low energy photons inside the jet are scattered by with an accreting supermassive blackhole in the centre. -
(Special) Relativity
(Special) Relativity With very strong emphasis on electrodynamics and accelerators Better: How can we deal with moving charged particles ? Werner Herr, CERN Reading Material [1 ]R.P. Feynman, Feynman lectures on Physics, Vol. 1 + 2, (Basic Books, 2011). [2 ]A. Einstein, Zur Elektrodynamik bewegter K¨orper, Ann. Phys. 17, (1905). [3 ]L. Landau, E. Lifschitz, The Classical Theory of Fields, Vol2. (Butterworth-Heinemann, 1975) [4 ]J. Freund, Special Relativity, (World Scientific, 2008). [5 ]J.D. Jackson, Classical Electrodynamics (Wiley, 1998 ..) [6 ]J. Hafele and R. Keating, Science 177, (1972) 166. Why Special Relativity ? We have to deal with moving charges in accelerators Electromagnetism and fundamental laws of classical mechanics show inconsistencies Ad hoc introduction of Lorentz force Applied to moving bodies Maxwell’s equations lead to asymmetries [2] not shown in observations of electromagnetic phenomena Classical EM-theory not consistent with Quantum theory Important for beam dynamics and machine design: Longitudinal dynamics (e.g. transition, ...) Collective effects (e.g. space charge, beam-beam, ...) Dynamics and luminosity in colliders Particle lifetime and decay (e.g. µ, π, Z0, Higgs, ...) Synchrotron radiation and light sources ... We need a formalism to get all that ! OUTLINE Principle of Relativity (Newton, Galilei) - Motivation, Ideas and Terminology - Formalism, Examples Principle of Special Relativity (Einstein) - Postulates, Formalism and Consequences - Four-vectors and applications (Electromagnetism and accelerators) § ¤ some slides are for your private study and pleasure and I shall go fast there ¦ ¥ Enjoy yourself .. Setting the scene (terminology) .. To describe an observation and physics laws we use: - Space coordinates: ~x = (x, y, z) (not necessarily Cartesian) - Time: t What is a ”Frame”: - Where we observe physical phenomena and properties as function of their position ~x and time t. -
Durham E-Theses
Durham E-Theses Factors Underlying Students' Conceptions of Deep Time: An Exploratory Study CHEEK, KIM How to cite: CHEEK, KIM (2010) Factors Underlying Students' Conceptions of Deep Time: An Exploratory Study, Durham theses, Durham University. Available at Durham E-Theses Online: http://etheses.dur.ac.uk/277/ Use policy The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in Durham E-Theses • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. Please consult the full Durham E-Theses policy for further details. Academic Support Oce, Durham University, University Oce, Old Elvet, Durham DH1 3HP e-mail: [email protected] Tel: +44 0191 334 6107 http://etheses.dur.ac.uk FACTORS UNDERLYING STUDENTS’ CONCEPTIONS OF DEEP TIME: AN EXPLORATORY STUDY By Kim A. Cheek ABSTRACT Geologic or “deep time” is important for understanding many geologic processes. There are two aspects to deep time. First, events in Earth’s history can be placed in temporal order on an immense time scale (succession). Second, rates of geologic processes vary significantly. Thus, some events and processes require time periods (durations) that are outside a human lifetime by many orders of magnitude. Previous research has demonstrated that learners of all ages and many teachers have poor conceptions of succession and duration in deep time. -
Reconsidering Time Dilation of Special Theory of Relativity
International Journal of Advanced Research in Physical Science (IJARPS) Volume 5, Issue 8, 2018, PP 7-11 ISSN No. (Online) 2349-7882 www.arcjournals.org Reconsidering Time Dilation of Special Theory of Relativity Salman Mahmud Student of BIAM Model School and College, Bogra, Bangladesh *Corresponding Author: Salman Mahmud, Student of BIAM Model School and College, Bogra, Bangladesh Abstract : In classical Newtonian physics, the concept of time is absolute. With his theory of relativity, Einstein proved that time is not absolute, through time dilation. According to the special theory of relativity, time dilation is a difference in the elapsed time measured by two observers, either due to a velocity difference relative to each other. As a result a clock that is moving relative to an observer will be measured to tick slower than a clock that is at rest in the observer's own frame of reference. Through some thought experiments, Einstein proved the time dilation. Here in this article I will use my thought experiments to demonstrate that time dilation is incorrect and it was a great mistake of Einstein. 1. INTRODUCTION A central tenet of special relativity is the idea that the experience of time is a local phenomenon. Each object at a point in space can have it’s own version of time which may run faster or slower than at other objects elsewhere. Relative changes in the experience of time are said to happen when one object moves toward or away from other object. This is called time dilation. But this is wrong. Einstein’s relativity was based on two postulates: 1.