Spacetime diagrams and Bondi s k-calculus
|
|
- Annabel Byrd
- 6 years ago
- Views:
Transcription
1 Spacetime diagrams and Bondi s k-calculus Two important consequences of Einstein s 1905 paper on SR: 1. It was immediately apparent that physicists had been taking the assignment and measurement of coordinates for granted when, in fact, it was important to do it correctly, in a way that could be "operationally" verified by experiment. For instance, the assumption that time was universal, and that the time measured between any two events was independent of the observer, turned out to be an unexamined and incorrect idea. Another was that a rod of length 1 meter in some inertial frame would naturally be found to have the same length in any other such frame. 2. At the same time, it became clear that in thinking (and drawing pictures!) of objects, we should pay careful attention to the object s extension in time as well as space. This leads to the subject of spacetime diagrams, to which we now turn. We first have to set the mathematical scene: Units: diagram. The essential features of SR can be understood using a 2-dimensional spacetime Geometrically, this is a plane, but not, as we ll see, the Euclidean plane. One of the two dimensions corresponds to time and one to space. It makes for simplicity if both time and space have the same units. Our convention is to measure everything in seconds. This means that we replace x, measured in centimeters, by x = x cm c cm/sec = x c sec, 1
2 which means: x is the time it takes to travel a distance x moving at the speed of light. Having made this definition, we now remove the accent and just write x. So for example, an object moving at the speed of light will have an equation x = ±t, and will have velocity v = ±1. Notice that in these units, velocity is dimensionless. The conventional units can always be recovered by replacing x and v with x/c and v/c. We make the following assumptions: (1) there exists an inertial observer (one for whom Newton s first law holds), (2) any other observer moving with constant velocity relative to this one is likewise an inertial observer. In what follows, the word observer will be used synonymously with inertial observer. We also assume that all our observers are equipped with identical standard clocks and with signalling devices which can emit, detect, and reflect light rays. More about this shortly. Points in a spacetime diagram are called events. An observer A will label an event by two coordinates (t a, x a ). The observer has a world line which we label A, and which, in his coordinates, is just the time axis. Points on A s world line have the coordinates 2
3 (t a, 0). The light rays, the world lines of photons or quanta of radiation, are drawn with slopes of ±1. Time increases as we move from the bottom to the top of the diagram. Figure 1 shows the world lines of three observers, together with a number of light rays. The k-factor Suppose A and B are two observers in relative motion, and that A emits two photons T seconds apart. If B is motion relative to A, then B will receive the two photons T seconds apart, where T < T if their motion is toward one another and T > T if they re moving apart. In either case, there s a number k such that T = kt. If B reflects these two photons back to A, then A will receive them k 2 T seconds apart. The k-factor between A and B is the same as that between B and A; k is also known as the relativistic Doppler shift. See figure 2. 3
4 Exercises: (a) If 2 1-dimensional observers are in relative motion, then their paths cross. Prior to that event, they are approaching each other, and afterwards, receding from each other. The picture above illustrates the case when they re receding. By extending the spacetime diagram into the past, show that the k-factor as they approach each other is 1/k. (b) If if there are 3 observers A, B, and C, with k-factors k ab, k ac, and k bc, then k ac = k ab k bc. Coordinates We are ready to introduce coordinates in a well-defined way. Fix an observer A and an event E. A measures the distance to E by using radar ranging : he sends a photon to E at time t 1 on his clock; the photon is reflected at E and received back by A at time t 2. The round-trip distance travelled by the photon, moving at the speed of light, is just c(t 2 t 1 ). This is twice the distance to E. Since c = 1 in our system of units, 4
5 A infers that E is located at a distance of x a (E) = (1/2)(t 2 t 1 ). The event E must also be assigned a time, and sensible choice is the midpoint of the interval [t 1, t 2 ]: t a (E) = (1/2)(t 2 + t 1 ). Note that, given the coordinates (t a, x a ), we have immediately t 2 = t a + x a, t 1 = t a x a. (We ll need this in the derivation of the Lorentz transformation later.) Velocity and the k-factor Not surprisingly, if A and B are in relative motion, their velocity and their k-factor are related. 5
6 If "we" are observer A and wish to find the velocity of observer B, we need to take two events on B s worldline, assign coordinates to them, and then compute x/ t. It doesn t matter which two events we take - we re really just computing the slope of a line. So we make an easy choice: since the two are in relative motion, their worldlines cross at a point we label O in the figure above, and which we take as the first event. If we set A s clock to t = 0 at O, then we know the coordinates of O; they are (0, 0). Reading things off from figure 4, we see that the event E has the coordinates x a (E) = (1/2)(k 2 T T ), t a (E) = (1/2)(k 2 T + T ). Given the coordinates of O, this means that v = x a (E)/t a (E) = (k 2 T T )/(k 2 T + T ) = (k 2 1)/(k 2 + 1) Exercises: Show that k 2 = 1 + v 1 v. What was the velocity of B relative to A at a time before the event O? Isn t that cute? that x a We can get all the signs right if we re tricky like this, but remember has really been defined as a distance, and so sometimes the correct signs have to be put in by hand. Use a spacetime diagram, similar triangles, etc. to draw the lines t a = constant for some observer A. On the same diagram, draw some of the lines t b = constant for an observer B in relative motion to A. Two events E and F are simultaneous to observer A if they take place at the same time t a. Will they be simultaneous for observer B? This is called the relativity of simultaneity. It is REAL! 6
7 By drawing some more pictures convince yourself that if the events E and F are simultaneous for A there exist observers B and C such that E precedes F according to B and F precedes E according to C. The composition of velocities We are about to encounter our first strange result (or second, if you think the relativity of simultaneity is strange). Remember that in Newtonian mechanics, relative velocities simply add: if v ab is the velocity of B relative to A, and v bc the velocity of C relative to B, then v ac = v ab + v bc. If I m walking at a speed of 5 mph toward the front of a train moving at 70 mph, then my velocity as measured by someone standing by the tracks is 75 mph. However, we already know that if I m on a rocket ship moving relative to observer A at v = c/2 and shine my flashlight in the direction of motion, then the speed of the light rays as measured by A is not c+c/2; it s just c. So the Newtonian addition of velocities won t hold in SR. What does happen can be computed from the k-factors: We start with the fact (exercise above) that for 3 observers, k ac = k ab k bc. Using the 7
8 expression for v in terms of k, this means that v ac = k2 ab k2 bc 1 k 2 ab k2 bc + 1 after some algebra... = (1 + v ab)(1 + v bc ) (1 v ab )(1 v bc ) (1 + v ab )(1 + v bc ) + (1 v ab )(1 v bc ) or Examples: v ac = v ab + v bc 1 + v ab v bc Suppose v ab = 0.9c (or 0.9 in our units), and v bc = 0.9 too. Then v ac = = = Exercise: Show that if v ab < 1 and v bc < 1, then v ac < 1. If you re moving at less than c, no matter how much acceleration you can muster, you ll always be moving at less thant c. On the other hand, suppose v ab = v bc = 3 km/sec, (a fairly hefty speed which would take you more than 1/4 of the way around the earth at the equator in an hour). This is "just" 10 5 c, however, and we compute v ac = , which is to many decimal places. This is why NASA doesn t use special relativity when it computes trajectories to the moon. 8
Introduction to Special Relativity
1 Introduction to Special Relativity PHYS 1301 F99 Prof. T.E. Coan version: 20 Oct 98 Introduction This lab introduces you to special relativity and, hopefully, gives you some intuitive understanding of
More informationSimultaneity, Time Dilation, and Length Contraction Using Minkowski Diagrams and Lorentz Transformations
Simultaneity, Time Dilation, and Length Contraction Using Minkowski Diagrams and Lorentz Transformations Dr. Russell L. Herman January 25, 2008 (modified: January 17, 2018) Abstract In these notes we present
More informationKinematics of special relativity
Chapter 2 Kinematics of special relativity 2.1 Special Relativity 2.1.1 Principles of Relativity Einstein postulated that there was still Galilean invariance, i. e. all uniformly moving observers had the
More informationThe Philosophy of Physics. Special Relativity and Minkowski Spacetime
The Philosophy of Physics Lecture Five Special Relativity and Minkowski Spacetime Rob Trueman rob.trueman@york.ac.uk University of York Special Relativity a quick refresher Special Relativity and Minkowski
More informationThe spacetime of special relativity
1 The spacetime of special relativity We begin our discussion of the relativistic theory of gravity by reviewing some basic notions underlying the Newtonian and special-relativistic viewpoints of space
More informationPHYSICS - CLUTCH CH 34: SPECIAL RELATIVITY.
!! www.clutchprep.com CONCEPT: INERTIAL REFERENCE FRAMES A reference frame is a coordinate system that you make measurements in, and there are two types: - Inertial reference frames, which move at velocity
More informationMassachusetts Institute of Technology Physics Department
Massachusetts Institute of Technology Physics Department Physics 8.20 IAP 2005 Introduction to Special Relativity Problem Set 2 1. A Lorentz transformation The origins of two inertial frames, Σ and Σ,
More informationPhysics 8.20 Special Relativity IAP 2008
Physics 8.20 Special Relativity IAP 2008 Problem Set # 2 Solutions Problems 1. A Moving Clock (7 points) A clock moves along the x-axis in your reference frame at a speed of 0.80c and reads zero as it
More informationYour (primed) frame frame
6 Special Relativity 6.1 Galiean Relativity 6.1.1 Spacetime Diagrams We keep seeing the word relativity appear in our discussion. To the person on the street, relativity (normally associated with Einstein)
More informationSpecial Relativity - Math Circle
Special Relativity - Math Circle Jared Claypoole Julio Parra Andrew Yuan January 24, 2016 Introduction: The Axioms of Special Relativity The principle of relativity existed long before Einstein. It states:
More information2.6 Invariance of the Interval
2.6 Invariance of the Interval Note. In this section, we define a quantity called the interval between two events which is invariant under a change of spacetime coordinates from one inertial frame to another
More informationwhere = observed wavelength when v = 0
where = observed wavelength when v = 0 called the k-factor and c = speed of light. NOTE: if v > 0, then the source and observer are moving away from each other This is the famous galactic red shift observed
More informationCorrect Resolution of the Twin Paradox
Correct Resolution of the Twin Paradox Michael Huemer In the following, I explain the Twin Paradox, which is supposed to be a paradoxical consequence of the Special Theory of Relativity (STR). I give the
More informationChanging coordinates - the Lorentz transformation
Changing coordinates - the Lorentz transformation In figure 5, we have two observers, A and B, and the two photons passing through the event E are shown. Each observer assigns coordinates to the event
More informationLorentz Transformations and Special Relativity
Lorentz Transformations and Special Relativity Required reading: Zwiebach 2.,2,6 Suggested reading: Units: French 3.7-0, 4.-5, 5. (a little less technical) Schwarz & Schwarz.2-6, 3.-4 (more mathematical)
More informationChapter 1. Relativity 1
Chapter 1 Relativity 1 Classical Relativity inertial vs noninertial reference frames Inertial Reference Frames Galilean transformation: x = x vt; y = y; z = z; t = t u x = u x v; u y = u y ; u z = u z
More information2.1 Einstein s postulates of Special Relativity. (i) There is no ether (there is no absolute system of reference).
Chapter 2 Special Relativity The contradiction brought about by the development of Electromagnetism gave rise to a crisis in the 19th century that Special Relativity resolved. 2.1 Einstein s postulates
More informationThe result is; distances are contracted in the direction of motion.
The result is; distances are contracted in the direction of motion. t = t/(1 v 2 /c 2 ) 0.5 d = d(1- v 2 /c 2 ) 0.5 These are the Lorentz equations. The Twin-Paradox. A woman astronaut is going to fly
More informationPhysics 225 Relativity and Math Applications. Fall Unit 7 The 4-vectors of Dynamics
Physics 225 Relativity and Math Applications Fall 2011 Unit 7 The 4-vectors of Dynamics N.C.R. Makins University of Illinois at Urbana-Champaign 2010 Physics 225 7.2 7.2 Physics 225 7.3 Unit 7: The 4-vectors
More informationMore Relativity: The Train and The Twins
previous index next More Relativity: The Train and The Twins Michael F owler, UVa Physics, 11/28/07 Einstein s Definition of Common Sense As you can see from the lectures so far, although Einstein s Theory
More informationRELATIVITY. Special Relativity
RELATIVITY Special Relativity FROM WARMUP It was all interesting! How important is it for us to know the Galilean transformation equations and the math of the Michelson-Morley experiment? Know the Galilean
More informationWhat is allowed? relativity: physics is the same for all observers so light travels at the same speed for everyone. so what? THE UNIVERSITY OF ALABAMA
Relativity, part 2 What is allowed? relativity: physics is the same for all observers so light travels at the same speed for everyone so what? THE UNIVERSITY OF ALABAMA CENTER FOR MATERIALS FOR INFORMATION
More informationRelativity. Overview & Postulates Events Relativity of Simultaneity. Relativity of Time. Relativity of Length Relativistic momentum and energy
Relativity Overview & Postulates Events Relativity of Simultaneity Simultaneity is not absolute Relativity of Time Time is not absolute Relativity of Length Relativistic momentum and energy Relativity
More informationRelative Motion (a little more than what s in your text, so pay attention)
Lab Activity Relative Motion (a little more than what s in your tet, so pay attention) Relative motion is something we use everyday, but we don t really think about it. For eample, passing a truck on the
More informationConsequences of special relativity.
PHYS419 Lecture 12 Consequences of special relativity 1 Consequences of special relativity. The length of moving objects. Recall that in special relativity, simultaneity depends on the frame of reference
More informationRelativity. An explanation of Brownian motion in terms of atoms. An explanation of the photoelectric effect ==> Quantum Theory
Relativity Relativity In 1905 Albert Einstein published five articles in Annalen Der Physik that had a major effect upon our understanding of physics. They included:- An explanation of Brownian motion
More informationThe Lorentz Transformation
The Lorentz Transformation During the fourth week of the course, we spent some time discussing how the coordinates of two different reference frames were related to each other. Now that we know about the
More informationIntroduction to General Relativity
Introduction to General Relativity 1 Recall Newtonian gravitation: Clearly not Lorentz invariant, since Laplacian appears rather than d'alembertian. No attempt to find Lorentz invariant equations that
More informationNotes - Special Relativity
Notes - Special Relativity 1.) The problem that needs to be solved. - Special relativity is an interesting branch of physics. It often deals with looking at how the laws of physics pan out with regards
More informationSPECIAL RELATIVITY. Introduction:
SPECIAL RELATIVITY Introduction: When Mr. La Rosa started talking about relativity in class, I first was lost and didn t get the idea. At That time I was thinking of a subject for my project. I thought
More informationBell s spaceship paradox
Bell s spaceship paradox If the two ships start accelerating at the same time, I always see them travelling at the same velocity, and keeping a constant distance... But I said the objects get shorter when
More informationConsequences of special relativity.
PHYS419 Lecture 12 Consequences of special relativity 1 Consequences of special relativity. The length of moving objects. Recall that in special relativity, simultaneity depends on the frame of reference
More informationPhysics E-1ax, Fall 2014 Experiment 3. Experiment 3: Force. 2. Find your center of mass by balancing yourself on two force plates.
Learning Goals Experiment 3: Force After you finish this lab, you will be able to: 1. Use Logger Pro to analyze video and calculate position, velocity, and acceleration. 2. Find your center of mass by
More informationMichael Fowler, UVa Physics, 12/1/07. Momentum has Direction
Michael Fowler, UVa Physics, //07 Momentum has Direction As we discussed in the last lecture, even before Newton formulated his laws, Descartes, with a little help from Huygens, had discovered a deep dynamical
More informationCHAPTER 2 Special Theory of Relativity-part 1
CHAPTER 2 Special Theory of Relativity-part 1 2.1 The Apparent Need for Ether 2.2 The Michelson-Morley Experiment 2.3 Einstein s Postulates 2.4 The Lorentz Transformation 2.5 Time Dilation and Length Contraction
More informationModern Physics. Third Edition RAYMOND A. SERWAY CLEMENT J. MOSES CURT A. MOYER
Modern Physics Third Edition RAYMOND A. SERWAY CLEMENT J. MOSES CURT A. MOYER 1 RELATIVITY 1.1 Special Relativity 1.2 The Principle of Relativity, The Speed of Light 1.3 The Michelson Morley Experiment,
More informationEPGY Special and General Relativity. Lecture 4B
Lecture 4B In the previous lecture we found that the proper description of the universe is one consisting of a four-dimensional manifold (space) endowed with a Lorentzian metric, (of course we are restricting
More informationNewtonian or Galilean Relativity
Relativity Eamples 1. What is the velocity of an electron in a 400 kv transmission electron microscope? What is the velocity in the 6 GeV CESR particle accelerator?. If one million muons enter the atmosphere
More informationThe Foundations of Special Relativity
The Foundations of Special Relativity 1 Einstein's postulates of SR: 1. The laws of physics are identical in all inertial reference frames (IFs). 2. The speed of light in vacuum, c, is the same in all
More informationSpecial relativity, 3. How big is gamma? The Lorentz transformations depend on the factor γ =
Special relativity, 3 A few kinematic consequences of the Lorentz transformations How big is gamma? The Lorentz transformations depend on the factor γ = 1 1 β 2, where β = V c. For macroscopic objects,
More informationLight and Relativity
PHY1033C Fall 2017 Lecture W11 Light and Relativity 1. Light, a Special Wave For more than 200 years, Newton s theory of mechanics, condensed into the three laws of motion, have been accepted as the correct
More informationRelativistic Boats: an explanation of special relativity. Brianna Thorpe, Dr. Michael Dugger
height Relativistic Boats: an explanation of special relativity Brianna Thorpe, Dr. Michael Dugger Time Dilation Relativity is all about your point of view. We are working with the speed of light and some
More informationReview Special Relativity. February 3, Absolutes of Relativity. Key Ideas of Special Relativity. Path of Ball in a Moving Train
February 3, 2009 Review Special Relativity General Relativity Key Ideas of Special Relativity No material object can travel faster than light If you observe something moving near light speed: Its time
More informationGeneral Relativity. on the frame of reference!
General Relativity Problems with special relativity What makes inertial frames special? How do you determine whether a frame is inertial? Inertial to what? Problems with gravity: In equation F = GM 1M
More informationSpecial and General Relativity (PHZ 4601/5606) Fall 2018 Classwork and Homework. Every exercise counts 10 points unless stated differently.
1 Special and General Relativity (PHZ 4601/5606) Fall 2018 Classwork and Homework Every exercise counts 10 points unless stated differently. Set 1: (1) Homework, due ( F ) 8/31/2018 before ( ) class. Consider
More informationSpecial Relativity 05/09/2008. Lecture 14 1
How Fast Are You Moving Right Now? Special Relativity Einstein messes with space and time 0 m/s relative to your chair 400 m/s relative to earth center (rotation) 30,000 m/s relative to the sun (orbit)
More informationObservations versus Measurements
Observations versus Measurements Pre-Collegiate Institutes Special and General Relativity 2015 In all discussions pertaining to relativity one needs to be careful to distinguish the reference frame from
More informationLecture 2 - Length Contraction
Lecture 2 - Length Contraction A Puzzle We are all aware that if you jump to the right, your reflection in the mirror will jump left. But if you raise your hand up, your reflection will also raise its
More informationModern Physics notes Paul Fendley Lecture 34. Born, chapter III (most of which should be review for you), chapter VII
Modern Physics notes Paul Fendley fendley@virginia.edu Lecture 34 General Relativity Born, chapter III (most of which should be review for you), chapter VII Fowler, Remarks on General Relativity Ashby
More informationRELATIVITY. Special Relativity
RELATIVITY Special Relativity FROM WARMUP How does special relativity differ from general? Special relativity deals with inertial reference frames. General relativity deals with gravity and acceleration
More informationRelativity. Physics April 2002 Lecture 8. Einstein at 112 Mercer St. 11 Apr 02 Physics 102 Lecture 8 1
Relativity Physics 102 11 April 2002 Lecture 8 Einstein at 112 Mercer St. 11 Apr 02 Physics 102 Lecture 8 1 Physics around 1900 Newtonian Mechanics Kinetic theory and thermodynamics Maxwell s equations
More informationMATH 12 CLASS 2 NOTES, SEP Contents. 2. Dot product: determining the angle between two vectors 2
MATH 12 CLASS 2 NOTES, SEP 23 2011 Contents 1. Dot product: definition, basic properties 1 2. Dot product: determining the angle between two vectors 2 Quick links to definitions/theorems Dot product definition
More informationAST2000 Lecture Notes
AST2000 Lecture Notes Part 2A The special theory of relativity: Basic principles Questions to ponder before the lecture 1. You have already used the Lorentz transformations. Do you know where they come
More informationSpecial Theory of Relativity. The Newtonian Electron. Newton vs. Einstein. So if Newtonian Physics is wrong. It is all Relative.
Special Theory of Relativity Chapter 26 The Newtonian Electron Newtonian Theory (everything we have done so far in class) can be tested at high speeds by accelerating electrons or other charged particles
More informationGravity and Spacetime: Why do things fall?
Gravity and Spacetime: Why do things fall? A painless introduction to Einstein s theory of space, time and gravity David Blair University of WA Abstract I present a simple description of Einstein s theory
More informationMassachusetts Institute of Technology Physics Department. Midterm
Massachusetts Institute of Technology Physics Department Physics 8.20 IAP 2005 Special Relativity January 18, 2005 7:30 9:30 pm Midterm Instructions This exam contains SIX problems pace yourself accordingly!
More informationMINKOWSKIAN SPACE, GRAPHS, EVENTS, WORLD LINES and MEASURING TIME
MINKOWSKIAN SPACE, GRAPHS, EVENTS, WORLD LINES and MEASURING TIME One second after we started our stop watch, little Johnny pukes in the corner of his room. Relative to the corner, the happening occurs
More informationTherefore F = ma = ma = F So both observers will not only agree on Newton s Laws, but will agree on the value of F.
Classical Physics Inertial Reference Frame (Section 5.2): a reference frame in which an object obeys Newton s Laws, i.e. F = ma and if F = 0 (object does not interact with other objects), its velocity
More informationModern Physics Part 2: Special Relativity
Modern Physics Part 2: Special Relativity Last modified: 23/08/2018 Links Relative Velocity Fluffy and the Tennis Ball Fluffy and the Car Headlights Special Relativity Relative Velocity Example 1 Example
More informationMassachusetts Institute of Technology Physics Department
Massachusetts Institute of Technology Physics Department Physics 8.20 IAP 2003 Introduction to Special Relativity January 10, 2003 Assignment 2 Due January 17, 2003 Announcements Please remember to put
More informationChapter 37. Relativity. PowerPoint Lectures for University Physics, 14th Edition Hugh D. Young and Roger A. Freedman Lectures by Jason Harlow
Chapter 37 Relativity PowerPoint Lectures for University Physics, 14th Edition Hugh D. Young and Roger A. Freedman Lectures by Jason Harlow Learning Goals for Chapter 37 Looking forward at why different
More informationExtra notes on rela,vity. Wade Naylor
Extra notes on rela,vity Wade Naylor Over 105 years since Einstein s Special theory of relativity A. Einstein, 1879-1955 The postulates of special relativity 1. The principle of relativity (Galileo) states
More informationWhy do we need a new theory?
Lecture 7: General Relativity We now begin our foray into the beautiful, yet sophisticated theory of General Relativity. At first we will explain the need for a new theory and then outline the ideas. We
More informationModern Physics notes Spring 2005 Paul Fendley Lecture 35
Modern Physics notes Spring 2005 Paul Fendley fendley@virginia.edu Lecture 35 Gravity and clocks Curved spacetime Born, chapter III (most of which should be review for you), chapter VII Fowler, Remarks
More informationSpacetime Diagrams Lab Exercise
Spacetime Diagrams Lab Exercise The spacetime diagram (also known as a Minkowski diagram) is a tool that can used to graphically describe complex problems in special relativity. In many cases, with a properly
More informationPhysics 2D Lecture Slides Lecture 4. April 3, 2009
Physics 2D Lecture Slides Lecture 4 April 3, 2009 Synchronizing Clocks Sam v Sally After coincidence of their origins at t=0, t = 0 Sam and Sally agree to send light signals to each other after time t
More informationSpecial Relativity: Basics
Special Relativity: Basics High-energy astrophysics involves not only light, which is intrinsically relativistic, but also particles that are accelerated to nearly the speed of light. Newtonian mechanics
More informationVectors in Special Relativity
Chapter 2 Vectors in Special Relativity 2.1 Four - vectors A four - vector is a quantity with four components which changes like spacetime coordinates under a coordinate transformation. We will write the
More information1.1 Units and unit conversions
Fundamentals This chapter reviews four important mathematical concepts and techniques that will be helpful in many quantitative problems you re likely to encounter in a college-level introductory astronomy
More information1 Review of the dot product
Any typographical or other corrections about these notes are welcome. Review of the dot product The dot product on R n is an operation that takes two vectors and returns a number. It is defined by n u
More informationSpecial Relativity: What Time is it?
Special Relativity: What Time is it? Michael Fowler, Physics Department, UVa. Special Relativity in a Nutshell Einstein s Theory of Special Relativity, discussed in the last lecture, may be summarized
More informationInflation, vacua and the end of the Universe
Inflation, 10 500 vacua and the end of the Universe Homework Problems: 1-7 (15 points) 1-10 (25 points) 2-9 (20 points) 2-13 (20 points) from Spacetime Physics Physics 311 Special Relativity Lecture 4:
More informationLecture Notes on Relativity. Last updated 10/1/02 Pages 1 65 Lectures 1 10
Lecture Notes on Relativity Last updated 10/1/02 Pages 1 65 Lectures 1 10 Special Relativity: Introduction Describes physics of fast motion i.e. when objects move relative to each other at very high speeds,
More information2.1 The Ether and the Michelson-Morley Experiment
Chapter. Special Relativity Notes: Some material presented in this chapter is taken The Feynman Lectures on Physics, Vol. I by R. P. Feynman, R. B. Leighton, and M. Sands, Chap. 15 (1963, Addison-Wesley)..1
More informationChapter 2: The Special Theory of Relativity. A reference fram is inertial if Newton s laws are valid in that frame.
Chapter 2: The Special Theory of Relativity What is a reference frame? A reference fram is inertial if Newton s laws are valid in that frame. If Newton s laws are valid in one reference frame, they are
More informationThe Nature of Space-Time
Chapter 9 The Nature of Space-Time 9.1 The Problem of Coordinates The basic problem of physics is to track in space and time the development of elements of a system. This requires that we have some method
More informationMassachusetts Institute of Technology Physics Department. Physics 8.20 IAP 2005 Special Relativity January 28, 2005 FINAL EXAM
Massachusetts Institute of Technology Physics Department Physics 8.20 IAP 2005 Special Relativity January 28, 2005 FINAL EXAM Instructions You have 2.5 hours for this test. Papers will be picked up promptly
More informationModern Physics notes Spring 2006 Paul Fendley Lecture 35
Modern Physics notes Spring 2006 Paul Fendley fendley@virginia.edu Lecture 35 Gravity and clocks Curved spacetime Born, chapter III (most of which should be review for you), chapter VII Fowler, Remarks
More informationLecture 2. When we studied dimensional analysis in the last lecture, I defined speed. The average speed for a traveling object is quite simply
Lecture 2 Speed Displacement Average velocity Instantaneous velocity Cutnell+Johnson: chapter 2.1-2.2 Most physics classes start by studying the laws describing how things move around. This study goes
More informationMassachusetts Institute of Technology Physics Department Physics 8.20 IAP 2005 Introduction to Special Relativity
Massachusetts Institute of Technology Physics Department Physics 8.20 IAP 2005 Introduction to Special Relativity Problem Set 1 1. Speeds What fraction of the speed of light does each of the following
More informationLecture 13 Notes: 07 / 20. Invariance of the speed of light
Lecture 13 Notes: 07 / 20 Invariance of the speed of light The Michelson-Morley experiment, among other experiments, showed that the speed of light in vacuum is a universal constant, as predicted by Maxwell's
More informationGeneral Relativity and Black Holes
General Relativity and Black Holes Lecture 19 1 Lecture Topics General Relativity The Principal of Equivalence Consequences of General Relativity slowing of clocks curvature of space-time Tests of GR Escape
More informationBlack Holes -Chapter 21
Black Holes -Chapter 21 The most massive stellar cores If the core is massive enough (~3 M ; total initial mass of star > 25 M or so), even neutron degeneracy pressure can be overwhelmed by gravity. A
More informationMoon Project Handout. I: A Mental Model of the Sun, Moon, and Earth (Do in class.)
Moon Project Handout Summary: You will recreate and interpret the geometric and timing measurements performed by the Ancient Greeks in order to determine the sizes of the Sun, Moon, and Earth and the distances
More informationAstronomical Distances
Astronomical Distances 13 April 2012 Astronomical Distances 13 April 2012 1/27 Last Time We ve been discussing methods to measure lengths and objects such as mountains, trees, and rivers. Astronomical
More informationCollege Physics B - PHY2054C. Special Relativity 11/10/2014. My Office Hours: Tuesday 10:00 AM - Noon 206 Keen Building.
College - PHY2054C 11/10/2014 My Office Hours: Tuesday 10:00 AM - Noon 206 Keen Building Outline 1 2 3 1 The speed of light is the maximum possible speed, and it is always measured to have the same value
More informationParadoxes in Special Relativity Paradoxes in Special Relativity. Dr. Naylor
October 2006 Paradoxes in Special Relativity Paradoxes in Special Relativity Dr. Naylor 1 102 years after Einstein s A. Einstein, 1879-1955 Special theory of relativity 2 Paradoxes? Twin Paradox Time dilation
More informationGeneral Relativity. Einstein s Theory of Gravitation. March R. H. Gowdy (VCU) General Relativity 03/06 1 / 26
General Relativity Einstein s Theory of Gravitation Robert H. Gowdy Virginia Commonwealth University March 2007 R. H. Gowdy (VCU) General Relativity 03/06 1 / 26 What is General Relativity? General Relativity
More informationParabolas and lines
Parabolas and lines Study the diagram at the right. I have drawn the graph y = x. The vertical line x = 1 is drawn and a number of secants to the parabola are drawn, all centred at x=1. By this I mean
More informationEverything should be made as simple as possible, but not simpler -A. Einstein
r1 Eerything should be made as simple as possible, but not simpler -A. Einstein r2 SR1... -3-2 -1 0 1 2 3... Synchronizing clocks At the origin, at three o clock, the clock sends out a light signal to
More informationChapter 26. Relativity
Chapter 26 Relativity Time Dilation The vehicle is moving to the right with speed v A mirror is fixed to the ceiling of the vehicle An observer, O, at rest in this system holds a laser a distance d below
More informationRotational Mechanics and Relativity --- Summary sheet 1
Rotational Mechanics and Relativity --- Summary sheet 1 Centre of Mass 1 1 For discrete masses: R m r For continuous bodies: R dm i i M M r body i Static equilibrium: the two conditions for a body in static
More informationPhysics H7C Midterm 2 Solutions
Physics H7C Midterm 2 Solutions Eric Dodds 21 November, 2013 1 Qualitative questions a) The angular resolution of a space based telescope is limited by the wave properties of light, that is, by diffraction.
More information! Exam 2 in this classroom on Friday! 35 Multiple choice questions! Will cover material from Lecture 12 to 22.!
This Class (Lecture 24): Black Holes Are Fun Next Class: Death by Black Hole: Spaghettification Night Obs/Computer labs due in class on Nov 9 th. HW 2 due on the 7 th. Exam 2 on Friday!! Exam 2 in this
More informationChapter S3 Spacetime and Gravity Pearson Education, Inc.
Chapter S3 Spacetime and Gravity What are the major ideas of general relativity? Spacetime Special relativity showed that space and time are not absolute. Instead, they are inextricably linked in a four-dimensional
More informationIntroduction to Physics Physics 114 Eyres
What is Physics? Introduction to Physics Collecting and analyzing experimental data Making explanations and experimentally testing them Creating different representations of physical processes Finding
More informationVectors and their uses
Vectors and their uses Sharon Goldwater Institute for Language, Cognition and Computation School of Informatics, University of Edinburgh DRAFT Version 0.95: 3 Sep 2015. Do not redistribute without permission.
More informationSpecial Relativity. The principle of relativity. Invariance of the speed of light
Special Relativity Einstein's special theory of relativity has two fundamental postulates: the principle of relativity and the principle of the invariance of the speed of light. The principle of relativity
More informationRelating measurements in one reference frame to those in a different reference frame moving relative to the first
What is Relativity? Relating measurements in one reference frame to those in a different reference frame moving relative to the first 1905 - Einstein s first paper on relativity, dealt with inertial reference
More informationAnnouncement. Einstein s Postulates of Relativity: PHYS-3301 Lecture 3. Chapter 2. Sep. 5, Special Relativity
Announcement PHYS-3301 Lecture 3 Sep. 5, 2017 2 Einstein s Postulates of Relativity: Chapter 2 Special Relativity 1. Basic Ideas 6. Velocity Transformation 2. Consequences of Einstein s Postulates 7. Momentum
More information