hl physics notes - part 3 (astrophysics & relativity)

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  • 8/9/2019 HL Physics Notes - Part 3 (Astrophysics & Relativity)

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    Sy Hai DinhKungsholmens Gymnasium

    IB Physics HL Notes (May 2014 Session)

    Table of Contents

    INTRODUCTION.............................................................................................................................................. i

    SYMBOLS & ABBREVIATIONS .................................................................................................................. ii

    ASTROPHYSICS ............................................................................................................................................. 1

    Solar System ........................................................................................................................................................ 1

    1. Solar System Objects ................................................................................................................................. 1

    2. Motions of Stars ......................................................................................................................................... 1

    Basic Definitions in Astrophysics ....................................................................................................................... 2

    Observing & Analyzing Methods ........................................................................................................................ 3

    1. Information Collected on Earth ................................................................................................................. 3

    2. Classification of Stars ................................................................................................................................ 4

    3.

    Parallax Method ......................................................................................................................................... 4

    4. Absolute & Apparent Magnitudes ............................................................................................................. 4

    5. Spectroscopic Parallax & Cepheid Star Method ........................................................................................ 4

    Stellar Evolution (Nucleosynthesis) .................................................................................................................... 5

    1. Formations of Stars .................................................................................................................................... 5

    2. Fates of Stars .............................................................................................................................................. 5

    3. Hertzsprung-Russell (HR) Diagram & Evolutionary Paths ....................................................................... 6

    Cosmology........................................................................................................................................................... 7

    1. Olbers Paradox......................................................................................................................................... 7

    2. Evidence of The Big Bang ......................................................................................................................... 7

    3. Big Bang Model & Evolution of Universe ................................................................................................ 8

    4. Type of Galaxies ........................................................................................................................................ 9

    RELATIVITY ................................................................................................................................................. 10

    Background & Definitions in SR....................................................................................................................... 10

    The Basic of Special Relativity ......................................................................................................................... 10

    Lorentzs Transformation................................................................................................................................. 11

    Simultaneity & Addition of Velocity ................................................................................................................. 11

    Time Dilation & Length Contraction ................................................................................................................ 13

    Relativistic Mechanics ...................................................................................................................................... 14

    General Relativity ............................................................................................................................................. 14

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    INTRODUCTION

    This is part of my series of detailed IB Physics HL notes that I took during my physics course at Kungsholmens

    Gymnasium. This part consists of HL options (Astrophysics and Relativity) and contains lots of derivations andexplanations for different laws and concepts that students need to master at the end of their IB Physics HL course. In

    order to make the study guide short, concise but still understandable, the language of mathematics (i.e. symbols) is

    used intensively throughout the study guide; thus people who have a stronger background in mathematics will clearly

    be more advantaged when using this guide.

    Note that therere things within this guide that are totally not part of the syllabus, but are still put here for the ones who

    are interested. These supplementary materials are put within the information boxes such as the one presented below:

    The texts that are emphasized throughout the guide can be divided into 4 types:

    -

    Bold: means that the phrase is either a headline or a key terminology

    - Bold & Italic: means important phrases

    - Bold, Italic, & Underlined: means vitally important key words

    - Mathematical formulas that are marked bold are the ones that appear within the IB physics formula booklet

    It is necessary to understand that images/figures that are used here in the study guide are as equally important as the

    texts themselves; and the text that is placed nextto a figure always contains some contents that refer to the picture.

    There are many common symbols and abbreviations used repeatedly throughout the study guide, and for that reason,

    they are listed in the next section down below for easy referencing; however, note that all symbols that are not

    commonly used are defined within the guide itself, so pay attention to these definitions when using the guide!

    This guide is not for commercial use; hence, the information presented here (in which most are taken from the internet,

    and from the Physics for the IB Diploma 5th Edition textbook, written by K. A. Tsokos) will not be formally

    referenced. In addition, I also used Professor Leonard Susskinds Special-Relativity course (in 2012) Lecture (https://www.youtube.com/watch?v=toGH5BdgRZ4 )& Lecture (https://www.youtube.com/watch?v=qfTJP7Soto4 ),to extend my understanding in the Relativity Option.

    Please feel free to contact me through email (my email address [email protected])if you have any questions

    regarding this study guide. Thank you!

    https://www.youtube.com/watch?v=toGH5BdgRZ4https://www.youtube.com/watch?v=toGH5BdgRZ4https://www.youtube.com/watch?v=toGH5BdgRZ4https://www.youtube.com/watch?v=qfTJP7Soto4https://www.youtube.com/watch?v=qfTJP7Soto4https://www.youtube.com/watch?v=qfTJP7Soto4mailto:[email protected]:[email protected]:[email protected]:[email protected]://www.youtube.com/watch?v=qfTJP7Soto4https://www.youtube.com/watch?v=toGH5BdgRZ4
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    SYMBOLS & ABBREVIATIONS

    General Symbols & Abbreviations:

    Electrons Photons Delta/Change

    Protons

    Speed of light in vacuum

    Kinetic energy

    Neutrons Plancks constant Potential energy Electron neutrino Wavelength Schwarzchild radius

    Symbols & Abbreviations used in Astrophysics

    Astronomical Unit Arrow used in nuclear reaction Emissivity Light year Hydrogen Surface area Parsec Hydrogen gas Temperature Solar masses Helium Power

    Arcsecond Carbon Electromagnetic Shnberg-Chandrasekhar Limit Iron Magnetic field Chandrasekhar Limit Main-sequence star Density Oppen-heimer Volkoff Limit Mass of a stars core Critical density of universe Luminosity A stars initial mass Hubbles constant Apparent brightness Centre of mass Planks time Apparent magnitude Stefan-Boltzmann constant Absolute magnitude Charged couple device

    Symbols & Abbreviations used in Relativity

    Special relativity Gamma factor Relativistic energy General relativity Position in space Relativistic mass Permittivity of free space Time Relativistic momentum Permeability of free space Proper time Rest energy Height Length Rest mass Gravitational field strength Proper length

    Gravitational constant

    ,

    ,

    Velocities of different objects

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    ASTROPHYSICS

    Solar System

    1. Solar System Objects

    - Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus & Neptune; whilePluto is:

    o Classified as dwarfplanet as it hasnt cleared its orbi tal regionof other objects (its not a dominant object in

    its orbit in terms of mass as it shares orbits with many Kui per Belt objects of equal size)

    o Pluto has orbits crossing Neptunes; howevertheir orbital planesare different

    o Orders of magnitude of radius of planets are (Mercury Mars) & (Jupiter Neptune)o Orders of magnitude of planets average distance to Sunrange from

    - Cometscontain dust & frozen gases which evaporate when its near Sun, forming its tail

    - Planets & comets follow elliptical orbits (with Sun at 1 foci) with the same directions orbital planesdiffer only

    slightly; comets orbits actually extend beyondsolar systemlong periods

    - Large asteroids belt(minor planets) between Mars & Jupiter collisions between themselves knock them out of

    orbits & head towards Earth with most being burnt in atmosphere the belt exists due to either:

    o Disruption of a planet into many pieces

    o Jupiter (with large mass) doesnt allow these materials to assemble into a planet

    2. Motions of Stars

    - Relativepositions of star constellations remain unchanged motions of stars in the sky are a consequence of

    Earths rotation about its axis & around Sun stars follow circularpatterns on Celestial Sphere

    o

    At Earths poles, stars circle around celestial poles (CP) which appear to not move at all (which points at

    Polaris in NCPhowever as Earths axis of rotation precesseswith period of 26000 years, different stars will

    be at CP)anti-clockwise at Northern Hemisphere & clockwise at Southern Hemisphere

    o At equator, stars move across the sky from West East over night as Earth rotates around itselfo

    At equator, different stars will appear directly overheadat midnightat different times of year as Earth rotates

    around Sun with change of direction from one night to the next of

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    These figures (Figure is the work ofLCGS Russ,publishedhereon Wikimedia

    Commons, under CC BY-SA 3.0 license)

    show the imaginary celestial sphere &

    the night sky at North Poles where the

    aths of the stars on the sky are tracked

    throughout the night

    Basic Definitions in Astrophysics

    -

    Conversion of units of distances:

    o o light yearo parsec average distance between stars in a galaxy is , while

    average distance between galaxies varies from 100 for galaxies within the same cluster to a fewforgalaxies belonging to different clusters

    -

    I nterstell ar mediumis made of gases (mainly & ) & dust grains (silicates, , & ), which fills the spacebetween starsdensity of interstellar mass is very low ( )temp is

    -

    Constellationis a collection of stars that are far from each other but make a recognizable pattern; stell ar clusteris

    a group of nearby stars, created by the collapse of the same cloud during stars formation

    - Dwarf is a star with no nuclear reactions taking placecome with different coloursBrown, White, Black

    - Variableis star with variable due to changes in internal structurelight curve(graphs of vs time) is either:o Non-periodice.g. when stars undergo planetary nebula or supernova, or in binary star system, matters can

    be transferred between the 2 stars, thus raising temperature & radiating energy

    o Periodice.g. cepheids(the greater peak , the longer the period; this is due to interaction of EM waves withmatter in cepheids atmospheres which causes outer layers to undergo periodic expansions &contractions, thus

    varying the surface areas which changes periodically)- Pulsaris a rapidly( period) rotatingneutron star emitting radiowaves due to its strong magnetic fields

    (), as it rotates, EM waves are produced in-continuouslywhich head to Earth if direction of coincides withrotation axis, no EM waves are produced

    -

    Quasars are quasi-stellar radio-emitting objects with spectrum that has been greatly red-shiftedvery powerful

    which contains active cores of young galaxiesby Hubbles law, quasars are far awaylooking back in time

    http://commons.wikimedia.org/wiki/User:LCGS_Russhttp://commons.wikimedia.org/wiki/User:LCGS_Russhttp://commons.wikimedia.org/wiki/File:Circumpolar_AZ81.jpghttp://commons.wikimedia.org/wiki/File:Circumpolar_AZ81.jpghttp://commons.wikimedia.org/wiki/File:Circumpolar_AZ81.jpghttp://creativecommons.org/licenses/by-sa/3.0/deed.enhttp://creativecommons.org/licenses/by-sa/3.0/deed.enhttp://creativecommons.org/licenses/by-sa/3.0/deed.enhttp://commons.wikimedia.org/wiki/File:Circumpolar_AZ81.jpghttp://commons.wikimedia.org/wiki/User:LCGS_Russ
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    - Novais when a whi te dwarfattracts material from nearby star, which heats up & leads to a suddenincrease of - 2 binary starsrotate around a centre of mass (CM) CM formula deduces that &

    by

    replacing &

    into

    , we get

    (where is period ofbinary-system &

    is distance between them)by finding

    (period) &

    , we can find mass of binary system

    o Visualbinary stars can be viewed by a telescope as 2 separate stars to calculate massof each star, either

    measure radius to CM using parallax, or measure velocities of each star by analyzing red- & blue-shift

    o Earth is contained within planeof rotation of eclipsingbinary stars when dimmer star is behind brighter

    one, theres a smalldipin apparent brightness, & vice versa(period) is equal the period of the dipo Spectroscopicbinary stars can onlybe observed by noticing Dopplers shift in their spectrum spectral lines

    regularly split into 2 lines & then recombine as one star approaches while other recedes

    Observing Analyzing Methods

    Summary of methods of measur ing distances:

    - Parallaxup to hundreds of - Spectroscopic parallaxup to a few -

    Standard Candles (e.g. Cepheid variablesup to a few ) are stars of known

    1. Information Collected on Earth

    - As

    (

    for a black-body), apparent bri ghtnessis

    (unit of

    ), which is

    measured by CCD or by a bolometer (supersensitive thermometer)

    -

    Peakon a black-body radiation curve is used to determine the stars colour & surface temp (using Wiens law)- Stellar spectrum can be used to measure Doppler Shifts to determine stars velocities (with respect to Earth) &

    rotational velocities(by measuring shifts on different parts of the star)

    - By comparing stellar spectrum with absorption lines of an element (measured on Earth) shows stars chemical

    compositionit is however unreliableas stars have different temp but essentially the same composition:

    o Hot star has ionizedhydrogen (without ), thus cant absorb any from EM waves before they reach Eartho

    Cold star () can onlyabsorb UVlight due to high electrical energy between & o Only stars that are neithertoo hot or too cool can give characteristic spectrum for hydrogen

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    2. Classification of Stars

    - Spectral classesof stars are based on temp& colour(which are determined by stellar spectrum):

    o Oh Be A Fine Girl Kiss Me! (blue, ), (), (white, ), (), (yellow, Sun), (), (red, )

    o

    Each class is divided into subclassesto indicate temp (the lower the number, the higher the temp)o To indicate size, scale of RomanNumeralis added(for supergiant) (for dwarf)

    Sun is

    3.

    Parallax Method

    - As Earth moves around Sun, position of a close (up to hundreds of ) star with respect to a background of distantstars would be different at different times of a year (6 months in between for simplicity)

    This figure illustrates the parallax method to measure the distance to a

    star; it shows the example of where the distance is - as is typically small, (in which is the parallax angle, measuring the difference

    between 2 different angular position of the star, & is distance between Earth & Sun)- Definition of a parsec

    4. Absolute & Apparent Magnitudes

    Apparent magnitude is apparent brightness in logarithmic scale, i.e.

    ( is the reference value) the larger the magnitude, the dimmer the starhuman eyes cant detect , while the largest telescopes can record up to Absolu te magnitudeis the apparent magnitude that the star wouldhave at from Earth- Let be the apparent brightness at ; as

    &

    , thus the

    distance in parseccan be calculated using

    5.

    Spectroscopic Parallax & Cepheid Star Method

    These methods find distance to a star by in which is measured on Earth, or for a Cepheid starit may be graphed against time, while can be either found by:-

    Relationship between & on HR diagram (can be deduced from studying stars spectrum)- Relationship between & Cepheid stars periods

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    Stellar Evolution (Nucleosynthesis)

    1. Formations of Stars

    Gases in interstellar space (nebulae) become unstable & collapse under gravity when gravitational energy exceeds

    thermal kinetic energythis is called Jeans Cri terion, which is satisfied when:

    -

    (is no. of molecules, is Boltzmans constant, & typically )- Substituting (is mass of 1 gas molecule) into 1stformula - As density

    As nebulae collapse under Jeans Criterion, protostars& then MSS are formed because gravitational potential energy

    has been transformed into kinetic energy, thus raising enough temp & pressure to initiate fusion

    - Proton-protoncycle: this issummarized into a single equation:

    - Energetic & then transfer to the whole star which stabilize the star against gravity

    2.

    Fates of Stars

    Cores Mass (in ) Initial Star Mass (in ) Outcome

    Usually White Dwarf with core

    White Dwarf with core White Dwarf with core

    Usually Neutron Star Usually Black Hole

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    Red Giants & Super-giants:

    - After (SCL) of of a star has been used, it becomes a giantor supergiantgravitational potential energyis transformed into heat which forces outer layers (envelope) to expandcore is heated while envelope is cooled

    -

    & are formed in the core through tr ipl e alpha process (with atoms)

    - Fusion cant produce elements heavier than , since binding energy per nucleon peaks with , thus furtherfusion is not possiblemassive stars contain different layers made of different elements with at the core

    Planetary Nebulas & White Dwarfs:

    - Stablewhite dwarfs (e.g. Sirius B, binary-system companion of Sirius A star) are formed due to electron pressur e

    (Paulis exclusion principle forces to have large to not have same quantum states when compressed in smallvolumes)white dwarfs then become black dwarfs as it cools down to absolute zero

    Supernova, Neutron Stars, & Black Holes:

    -

    Stars with -core are full of energetic , which can rip apart into , , & due to high densities &temp, (a temporary neutron stars) as a result of Paulis exclusion principle, neutronpressurecreates a force outward, causing a supernova

    - Black holesSchwarzschild radius is derived from escape velocity formulao Effects of a black holes to nearby star (e.g. its binary-system companion star) are observable gases of

    companion star are accelerated into black hole, thus become hot & radiate X-rays

    o Much heavier black holes exist in a galactic centrewhich affect motions of stars in the galaxy

    3.

    Hertzsprung-Russell (HR) Diagram & Evolutionary Paths

    This image (taken from NASA/CXC/SAO at

    http://chandra.harvard.edu/edu/formal/variable_st

    ars/bg_info.html)shows the HR-Diagram Pay

    attention to the axes & their scale!!!- Hertzsprung plotted vs temp (), while Russell plotted absolute magnitude vs spectral class 4 axes &

    absolute magnitude are similar concepts, while & spectral class are also similar concepts- Main SequenceStars (MSS):

    o Relationship between & helps make good prediction for spectroscopic paral lax methodo Contains stars which produce enough fusion energy to counterbalance gravity

    http://chandra.harvard.edu/edu/formal/variable_stars/bg_info.htmlhttp://chandra.harvard.edu/edu/formal/variable_stars/bg_info.htmlhttp://chandra.harvard.edu/edu/formal/variable_stars/bg_info.htmlhttp://chandra.harvard.edu/edu/formal/variable_stars/bg_info.html
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    - Evolutionary pathof a star can be drawn right on the HR graph

    - (where , & is mass of a MSS) this comes from nuclear physics & uncertainty of is dueto unknowncomposition of stars (unknown equation of state) the bigger , the bigger , the bigger rate of

    burning , thus the less time the star spends on the main sequence

    Cosmology

    Cosmological Principle:

    -

    Homogeneityon a large scale, universe looks uniform

    - Isotropythere is no direction that is special in comparison with another

    Implies that universe has no edge and no centre

    Newton used an extreme version of cosmological principlesuggested that universe is infinite & static, thus had

    no beginningsince there is no centre, gravity also spreads out evenly, keeping the universe static all the time

    1.

    Olbers Paradox

    This spherewhere stars are filled on the surface areawill aid the

    calculations down below

    Let the sphere have density

    stars per unit areatotal no. of stars is

    , & observer at the center measures

    for each startotal light received is (a constant)since theres an infiniteno.of such a sphere with each contributing infiniteenergy receivedsky must be infinitelybright!paradox!Big Bang model solves Olbers Paradox:

    - Theres a f initeno. of stars, each with f initelifetimecontribute only finite amount of energy

    - Universe has f initeagefor stars beyond observable universe, light hasnt yet had time to reach us

    - Due to universe expansion, light received is red-shiftedcontains less energy

    2.

    Evidence of The Big Bang

    During nucleosynthesis phase, Big Bang predicts that there were for every , which can combine into & which is exactly the -abundance that is observed in stars

    It has been observed that all stars spectrum show red-shifts (is emitted frequency)for small , red-shift

    is (is only velocity alongthe line of sight, not the total velocity)

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    Hubbles Law:

    - ( is velocity of recession, & is Hubble constant) universe is expandingwhich can be visualized by using a 2D model of a balloon

    - Since & from Earth looking fromObserver

    s frame, the distant galaxy is moving away at

    velocity ( ) ()Hubbles law can be appliedto any observersall observers have the illusion that they are

    at the center of the universe, which is not correct!!!

    - Let be ageof universe & assume that rate of expansion is constant(i.e. for all time)upper bound tothe age is

    years as expansion rate must have been faster at the beginning

    Cosmic Background Radiation (CBR), detected by radio-wave antenna of Penzias & Wilson , is Big Bangsremnant;

    its a black-bodyspectrum filling all space that has been red-shifted & cooled down to due to universe expansion3. Big Bang Model & Evolution of Universe

    Time () Temp () Events Plank time

    Quantum gravitational effects which havent had any universal theory yet

    Strong & weak nuclear forces joined into 1 unified force all particles show symmetry Strong & weak nuclear forces separated; inflationbegins

    All 4 forces of nature became separated

    & are formed from quarks Nucleosynthesisnuclear physics & thermodynamics can be used to predict -abundance Matter & radiation separate end of radiation domination, start of matter domination

    previously, were scattered off from Galaxies & other large structures form in universe

    Development of Universe since the Big Bang

    According to standard& outdated Big Bang model, expansions of universe can be modeled as wherethe scale factordepends on the shape of Universe (open, flat, or closed), while & represent the distance

    between any 2 galaxies); this model of the future of Universe is shown in the table below

    Geometry Density Type Volume Expansion VisualizationHyperbolic Starts at , increases without limit Open Infinite Forever, at slowing rate Horse saddleEuclidean

    Starts at , increases to max witha decreasing rate

    Flat Infinite Forever, but rateapproaches zeroFlat sheet of paper

    extending

    SphericalStarts from , increases to a max,then decreases back to Closed Finite

    Stops, followed by

    collapse

    Surface of a sphere

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    Rough estimate of criti cal density(which is the exact density to make Universe flat):- Using Newtonian mechanics, consider a spherical cloud of dust of radius & a mass at the surface of this cloud

    which moves away from the centre with a velocity that satisfies Hubbles Law- Total energy of the mass is Using the law of escape velocity, the object

    just escapes the gravitational pull when the total energy is

    Discussion of standardBig Bang model & consideration of dark matter & dark energy:

    - Current scientific evidence suggests that , which suggests that Universe is open- However, based on the method of gravitational lensing, there seems to be much more mass which accounts for

    dark matters (which are dark as they interact with matter very weakly & dont radiate energy):

    o

    WIMPSneutrinos & exotic particles predicted by models of particle physics (e.g. supersymmetric particles)

    o

    MACHOS

    ordinary matter (e.g. brown, black, & red dwarfs or black holes)- If dark matter really exists, then there must exist dark energyto account for the accelerating expansion of

    Universe, as dark energy acts as a force that counters gravity of dark matters

    Discussion of Homogeneity of Universe

    - Milky Way is one of 20 galaxies belonging to Local Groupclusters of galaxies which make up super-clusters

    - Distribution of clusters in space is notuniformthere are areas of linear size that are empty4. Type of Galaxies

    A galaxy typically contains stars & is across

    distance between 2 stars is , while distance betweengalaxies is galaxies are classified by Hubbles criteria which is based mainly on galaxies appearance 3basic types are spiral/barred spiral, elliptical, & regular

    Spirals/barred spirals Ellipticals Irregulars

    Shape Flattened discin spirals, theres a

    central sphericalhalo (spiral arms start

    here)in barred spirals, central halo

    looks like a barwhere spiral arms start

    from the 2 ends

    Ellipsoidal shape

    stars are distributed

    uniformly

    No obvious structure

    Content Halo contains mainly of old stars;

    while spiral arms contain both young

    (formed from gas & dusts) & old stars

    Contain mainly old

    stars; no gas or dust

    Both young (formed from gas

    & dust) & old stars

    Star Formation Takes place in spiral arms No new stars! Stars forming from gas & dust

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    ___________________________________________________________________________________

    RELATIVITY

    Background Definitions in SR

    - A f rame of reference refers to a coordinate system which involves the measurement of positions (using a ruler), &

    of time (using a clock)if this observer is notaccelerated, its called an inertial frame of reference

    -

    2 observers in different inertialframes are equallyjustified in considering themselves to be at rest

    - An eventis a specific instant at a specific location in space, i.e. a coordinate of space-time

    -

    Galilean transformations of frames were used in Newtonian physics, in which & howeverthis transformation doesnt work mathematicallyif is the same for all inertial framesit must be fixed!!!

    The Basic of Special Relativity

    Since speed of light is a constantin Maxwell theory

    he believed that his theory was only accurate forobservers at rest with respect to ether(a medium)Michelson-Morley experiment:

    - Used interferometer (which relies on interference of light) to measure accurately relative speed of Earth with

    respect to ether as Earth rotates around Sunexperiment was done at different times of year

    - As Earth moves in different direction with respect to ether, detailed calculations show that rotating interferometer

    should change interference pattern by shifting interference fringes

    However, no changes in interference patterns were foundcould notprove ethers existence

    Principlesof SR:

    -

    Laws of physics are the samein all inertial frames

    - Speed of light in vacuum() is the samefor all inertial observers constancy of speed of light has now beenconfirmed by accurate experiment at CERN (neutral pions moving close to decay into 2 in differentdirections, but both are observed to move at )

    - Relativity states that no material object can exceed the speed of light; also no signal can be sent faster than Constancy of means that space & time are now linked in which time is the 4 thdimensionspace-time

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    Lorentzs tr nsform tion

    Derivation of Lorentzs Transformation Let be the unprimed frame, & be the primed frame (in which moves with velocity relative to , & vice versa)by Galilean transformation, , & :-

    Since this transformation must be fixed according to postulates of SR, we would hypothesize that & are relatedby a function of called gamma factorthus:

    } ()

    - By SRs postulate, is the same for al linertial frames if & both shoot a light-ray, then lights travelingpaths must satisfy , & substitute these into Equation () gives:o

    (negative

    root is excluded here because 2 inertial frames (where ) can give the sameresult for , onlywhen )- Substituting back, we have

    &

    -

    For relation between & , we have

    Summarization of Lorentzs Transformation:

    - Gamma factor

    -

    Position

    similarly, for velocity , we have

    notice

    that if motion is onlyalong -direction, then things happen normally in other dimensions (i.e. & )- Time

    similarly, for velocity , we have

    Simultaneity Addition of Velocity

    Relativity of Simultaneity absolute simultaneity is based on absolute time; but since time is relative, so is

    simultaneitythus 2 observers moving relative with each other:

    -

    Will notagree on simultaneityof events that take place at differentpoints in space

    - Will agree on simultaneityof events that take place at samepoints in space (as )

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    Derivation of Addition-of-Velocity Formulas:

    - Objects , , & will have subscripts , , & respectively by Lorentzs transformation, , &

    Substitute these into

    gives

    The figure shows 2 objects (& ) as viewedrom Object s frame of reference; this is a

    situation that is set up to help us deriving the

    addition-of-velocity formulas

    -

    As were interested in velocity of as observed ins frame, thus here indicates position of in its ownreference frame (i.e. ) must be the trajectory of asobserved by; thus rearranging this, we get 2 additional formulas,

    &

    Using Addition-of-Velocity Formulas:

    The most difficult thing here is to identify velocities of which objects correspond to , , & for simplicity, whendealing with addition-of-velocity problems, we need to follow threerules down below (which can be basically derived

    by re-analyzing the situation that was set up above) for 3 different types of problems

    - Use Velocity Subtractionformula to find velocity when were given:o

    velocities (& ) that describe motions of twodifferent objects & are both observed by the same(stationary)observerssince & describe sameobjects, its better to write the formula as

    -

    Use Velocity Additionformula to find velocity when were given:o

    velocities (

    &

    ) that describe motions of twodifferent objects & are observed by

    differentobservers

    -

    Use Special-Case formula to find velocity when were given:o velocities (& ) that describe motions of the same object, but are observed by 2 differentobservers

    remember that velocity is the one that is observed by a stationary observer!!! When given velociti es are in dif ferent di rection, be extremely carefu l wi th the signs of these veloci ties! ! !

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    Time Dilation Length Contraction

    Space-timegeometry has an invariance & similarly which can be proven by Lorentzs transformation- & explain why more time passes along shorter edge (e.g. at event , time

    along side is morethan proper time on side due to time dilation )- For time-like, - For space-like, Proper time () is elapsed time between two events that take place at samepoint in space in an observers frame;mathematically, is only defined for time-like trajectory let , then for alight trajectory, proper time for a light trajectory is always time stops for light!!!Proper length() is the length of an object, measured by an observer at restrelative to the objectTime Dilationlet inertial observers,(unprimed frame) & (primed frame), move relative to each other:- Dilation is a symmetriceffect (i.e. its frame dependent):

    o

    Froms frame, moves with velocity but in s frame o From s frame,moves with velocity but ins frame o

    For case , ; for case , thus time is relative& dilation is symmetricthis isnt a paradox astheres no way one can compare 2 clocks unless one of them decelerates, thus breaking the symmetry

    o Twin paradoxis not a real paradox as the problem is asymmetricdue to the acceleratedreference frame

    - Time dilation formula is thus - Small difference in time can only be measured with super-accurate atomic clocks, as it has been done in Hafele-

    Keating experiment (which compares time difference of a clock on an airplane with the one on Earth)

    Length Contractionlet inertial observers,(unprimed frame) & (primed frame), move relative to each other:- Suppose that is measuring the length of an object at restins frame at time thus:

    o

    -

    Similarly, suppose thatis measuring the length of an object at restin s frame at time thus:o

    - For case , ; for case , like time dilation, length contraction is symmetric, & length is relative(the rulers dont only look shorter, but they areshorterlooking from both frames)length contraction formula is

    thus (note that only lengths along the direction of motion get contracted!!!)

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    Muons proper time of decaying is short( ), thus cant cover the distance from where it was createdto Earths ground); however, we can still detect muons on the ground this can only be explained by SR:

    - Time dilationfrom Earthsframe, muon decays in a longertime, as (is time measured on Earth)- Length contractionfrom muonsframe, distance that Earth travels towards it is contracted, because

    (where & are distances measured by the muon & by an observer on Earth, respectively) is nowshort enough for Earthto reach the muon beforeit decays!!!

    Relativistic Mechanics

    In SR, its better to use as unit for & as unit for speed of light, & as unit for mass!!!Restenergy ( ) is amount of energy needed to produce a particle at rest relativisticenergy is is relativistic mass as approaches , approaches (only for a particle with mass); but since

    thus with constant, & with increasing, then would decrease, such that will notreach !Since - For small

    & by binomialexpansion,

    classical mechanics!

    - Notice that for a charged particle being accelerated by a potential difference , then

    Relativisticmomentum is relates to by this replaces in classical theory- For

    (i.e.

    ), then

    - For a particle at rest (i.e. ), then rest energy!- Law of conservation of momentum still applies:

    o For a free at rest to absorb a of energy , then must have afterabsorption but due toenergy conservation, which gives an impossibleanswer ( )

    o A free cant absorb a can only absorb when is bound in an atom as the atom canparticipate in sharing energy & momentum

    General Relativity

    Equivalence Principlestates that gravitational& inertialeffects (i.e. effects of acceleration) are the samethis is

    because noexperiments withina frame can be done to know exactly about the frames motion consequences:

    - An inertial frame is equivalent to a freely-falling frame in a uniform gravitational field

    - An accelerating frame (far from all masses) is equivalent to a frame at rest in a gravitational field

    - Gravitationalmass & inertial mass are the same

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    Basic of GR:

    - -o -matter (energy & mass) distorts space-timeo -geometry of space-time determines motion of mass & energy in space-time

    - Unlike Euclidean geometry, in curvedspace-time (4D), shortestpath is a geodesics:

    o This means that theres no actual gravitational force because objects & light follow geodesics which are

    completely determined by the bending of space-time

    Consequence of GR:

    (a) (b) (c)

    Red arrows represent actual paths of light rays (which are bent as shown)

    - Fi gure (a): Since light must feel the same gravitational effects as a ship in a gravitational field does (such that

    Equivalence Principle holds true) light must follow -path to hit the ships front shows the bending oflight, & that light follows paths of shortest length (geodesics)this can be used for gravitational lensing

    - Fi gure (b): Combination of SR & Equivalence Principle gives to be constanteven inside a gravitational fieldfor light ray& to be in phasetime formust pass slowerthan time for time dilation!!!

    -

    Fi gure (c): As time runs slower near Earth, a will have smaller period, higher frequency & shorter gravitational frequency-shiftblue-shift (for motion towards ), & red-shift (for motion away from )

    Test for the bending of light (i.e. gravitational lensing):

    - Involves measuring anglethat light is bent by Suns gravitydone by measuring apparent & real position of a

    starcan only be observed during totalsolar eclipse to avoid interference from Suns light

    Description of experiment

    testing bending of light

    (dotted lines represent

    imaginary lines; red lines

    represent light paths)

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    - From the figure, its clear that is the angle that the light bends due to Suns gravity experimentalvalue of is shown to be exactly the same compared to the value predicted by GR evidence for GR!!!

    - To increase accuracy, radiosignals from very distant galaxies have been used instead of using light from a star

    Pound-Rebka Experiment(test for gravitational frequency-shift)let&be emitted& observed frequency, respectively; & be the gravitational massof thus by law of conservation of energy, we have:

    - Emission from top:

    GR & Black Holes:

    - Since for , derivation of Schwarzchild Radius () can only be estimatedby Newtons law -

    (is distance of the nearobject to black holes centre)time dilationnear a black hole!!!