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Special Relativity SPECIAL RELATIVITY A Geometric Approach Course with Exercises and Answers followed by the conference Interstellar travel and antimatter Mathieu ROUAUD Physics and Chemistry Lecturer in Preparatory Classes for the Engineering Schools. Associate Professor. MSc Degree in Theoretical Physics. Phd Student in Particle Physics. You are free to: Share — copy and redistribute Adapt — remix, the material in any medium or transform, and build up on format the material Under the following terms: Attribution — You must give appropriate credit and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. Non Commercial — You may not use the material for commer - cial purposes https://creativecommons.org/licenses/by-nc/4.0/# Date of publication: September 2020 ISBN 978-2-9549309-3-0 Edition 1.2 Revision and Translation : May 2021 French book: Relativité Restreinte, Approche géométrique. Complete and High Quality Paper Books with all the Exercises Corrected on lulu.com To contact the author: [email protected] Foreword Special relativity, presented in the article published y Albert Einstein in June "905, has deeply changed our physical concepts. The &ell-established theories of the time, (e&ton)s old mechanics and Ma*&ell's rand ne& theory of electromagnetism, &ere fundamentally incompatible. In the ,rst, there is an addition la& for velocities, &hile in the second, an invariant speed is re-uired. the speed of light in vacuum. In (e&ton)s theory, in line &ith the relativity of motion introduced y Galileo, the speed of an object depends on the observational reference frame, so ho& could the speed of light in vacuum be a ,*ed fundamental constant1 2or inertial frames, special relativity reconciles mechanics and electromagnetism, at the cost of calling into -uestion the absolute nature of space and time. Space and time are no& relative and form a ne& absolute. the space'time. The theories of matter and light are thus unified in their natural spatiotemporal frame&or3. Albert Einstein)s historical approach is based on the constancy of the speed of light in a vacuum. The modern approach, &hich made it possible to build the Standard Model, is based on another logic. symmetries. This ne& approach is deeper and brea3s free from the historical bias of the early 20th century. The structure of space'time imposes a speed limit. This ma*imum speed is specific to space'time and is not lin3ed to a material object. This ne& constant is specific to the container, the space'time, and not to the content, for e*ample, light rays. This ne& vision is conceptually very different and sheds light on the true nature of physical la&s. In this boo3, &e focus on visual and graphical methods that help develop understanding &ithout the systematic use of e-uations. This geometrical approach &ill e highlighted and &ill allo& the reader to ma3e sense of the e-uations that &ill follo&. The path follo&ed is not academic, but pragmatic and utilitarian. 2rom the ,rst pages you &ill master the tools that &ill allo& you to apply special relativity independently. 6e are not studying general relativity here. 6e specify this because confusion is fre-uent bet&een the t&o theories. That said, for those &ho &ant to understand general relativity, you must ,rst have understood the special. General relativity deals &ith gravitation and is based on its o&n principles. Small notable e*ception, &e &ill sometimes ma3e analogies &ith the blac3 hole to help delimit the t&o theories. Contents # TIME DILATION AND LENGTH CONT ACTION 7 !NIT" OF TIME AND DI"TANCE # 7 F AME" $ 7 EIN"TEIN%" &O"TULATE" ' 7 THE T IANGLE OF TIME" ( 7 LENGTH CONT ACTION #) 7 "&ATIO*TEM&O AL &E "&ECTI+E EFFECT #' 7 T,IN E-&E IMENT #. 7 !"E OF E/!ATION" )# 8 Transformation of volumes and angles Exercises )2 The Crystals of the Pop Exomoon 25 / One-way ticket for Sirius 2! / Parcel delivery 2! / Twin on his way to Sirius 2! / Cr el dilemma# 2$ / % ons 2& / 'igh-speed train )o rney 2* / Satellite 2* / 'afele-+eating experiment ,-. ) "&ACETIME DIAG AM 7 ,O LDLINE" $2 7 MIN3O,"KI DIAG AM $4 7 !"E OF E/!ATION" '# 8 Equation of worldlines 8 Angle and scale factor Exercises '2 Minkowski dia(rams /5 / 0nterstellar comm nications /5 / Call for help /! / Tim1 Tam1 Tom /!. $ CHANGING EFE ENCE F AME 7 "&ACE*TIME DIAG AM '6 7 ELATI+IT7 OF "IMULTANEIT7 2) 7 CA!"ALIT7 2' 7 COM&O"ITION OF +ELOCITIE" 26 7 !"E OF E/!ATION" 4# 8 Lorentz transformation • Lorentz invariant 8 Transformation of accelerations Exercises .# Composition of "elocities $2 / Two "essels $2 / 3ow speeds limit $2. ' THE A&&EA ANCE OF THING" .2 7 DO&&LE EFFECT .4 7 &HOTOG A&H OF A MO+ING ULE .6 7 THE "TA 7 "37 "EEN F OM THE "HI& (' Exercises 62 The s icidal physicist *5 / 3aser sail *5 / Optical molasses *! / 4etection of exoplanets by 4oppler effect *& / Calc lations for the mo"in( r ler 2-- / Aberration of the light 2-2 / Composition of "elocities and accelerations in ,4 2-2 / Starry sky at the halfway point - Ma(nitude 2-2 / 7 merical sim lation of the sky 2-/ / 6 bit of math... 2-5 / Ener(y distrib tion 2-! / 7 mber of photons 2-$ / Power emitted by a star 2-&. 2 ACCELE ATED MOTION 7 "TUD7 OF AN ACCELE ATED F AME ### 7 A TIFICIAL G A+IT7 ##' 8 Horizon concept #)# 7 OUND T I& #)) 8 Photon rocket #)' Exercises #)6 'alf-time 22* / 8eality Show - 4oppler effect in an accelerated frame 22* / 'ead-to-head 2,2. 4 MET IC 7 EUCLIDEAN MET IC #$2 7 MET IC ON THE "&HE E #$( 7 MIN3O,"KI MET IC #'$ 7 MET IC OF AN ACCELE ATING F AME #'$ 7 MET IC OF A OTATING F AME #'6 7 "CH,A <"CHILD MET IC #24 Exercises #26 E clidean metric 25* / 8apidity 25* / 8indler metric 25* / 9ree fall in the rocket - 3a(ran(ian - :lack 'ole 2!- / 9all of a blue ball 2!$ / Tra)ectory of a ray of light in the Einstein;s Ele"ator 2!$ / Spherical coordinate system - Solid angles 2!* . FO! *+ECTO " #.$ 7 EUCLIDEAN +ECTO "&ACE #.( 7 MIN3O,"KI +ECTO "&ACE #(' 7 CHANGE OF COO DINATE" #66 7 FO! *+ELOCIT7 )=4 7 FO! *ACCELE ATION )#) 7 MA""*ENE G7 E/UI+ALENCE )#6 8 Four-momentum 8 '*0orce Summar> )). 7 NON*INE TIAL EFE ENCE F AME" ))6 8 Local basis and connexions 8 Covariant derivative 8 Geodesics 8 Metric effects and forces of inertia Conclusion and synthesis )$6 Exercises )'$ Chan(e of basis 2/, / 8iemann c r"ature tensor 2/, / 6 non- niformly rotating disk 2/5 / Spatial c r"atures 2/! / Pair prod ction 2/& / <a"e e= ation 2/* / Schr>dinger e= ation 25- / Electroma(netic ?eld 252 / Maxwell's e= ations 255 ( INTE ACTION" 7 FIELD C EATED A7 A CHA GE )4# 8 Force between two charges 7 ADIATION DAM&ING )44 8 Radiated ener1> 8 Damping force 7 ETA DED &OTENTIAL" )4( 8 Geometric construction of the 4-potential Exercises ).$ @nits 2$, / 8elativistic e= ation of motion 2$, / 8adiation damping /-force 2$/ / 9o r-potential ma(nitude 2$/ 6 INTE "TELLA T A+EL AND ANTIMATTE 7 INT ODUCTION ).. 7 VOYAGER & OAE" )(# 7 "LING EFFECT )($ 7 VOYAGER 3 & OBECT )(4 7 OCKET E/!ATION )(6 7 ANTIMATTE )6$ 7 B!&ITE C THE "OLA "7"TEM GA" &UM& )64 7 ANTIMATTE "TO AGE )6( 7 CONCL!"ION $=$ Exercises $=2 9ig res ,-5 / The distance of stars o"er time ,-* / Sling effect,-* / 7 merical sim lations of the slings ,2, / Calc lation of propellant masses ,2- / Planetary alignments ,22 / Motion of the stars ,22 / Pair of primor- dial black holes ,2* / Antiproton-proton collision ,,- / 'elical motion ,,- / The ma(netosphere ,,2 / Penning trap ,,,. Answers $$. Ai?lio1ra:h> 2#$ Index 2#' # I. Time Dilation and Length Contraction TIME DILATION AND LENGTH CONTRACTION In this chapter, &e introduce special relativity and &e present the ,rst geometrical tool: the triangle of times . A. Units of Time and Distance 7 !NIT" OF TIME AND DI"TANCE These t&o physical -uantities, time and distance, are of different natures. Impossible, for e*ample, to say if a second is greater or less than a meter, that ma3es no sense. 6e can use a speed to lin3 a distance to a time, but the speed depends on the observer9 this lin3 &ould therefore e perfectly arbitrary. It is always true in classical mechanics, but in special relativity &e have a novelty, &e have an invariant speed. the ma*imum speed. This fundamental constant ma3es it possible to unambiguously associate a distance &ith a time. This distance is called light-time. 2or e*ample, the light'year corresponds to the distance traveled in vacuum y light during a year. " The speed of light in vacuum is about a billion 3m:h, it is named c and is precisely ,*ed at. c ; 299 792 =58 m:s It is the speed of any electromagnetic &ave in vacuum, &hether it be radio, infrared, visible, ultraviolet, ?'rays or gamma rays. 6e specify &ell, in vacuum, ecause in a transparent material, such as air, &ater or glass, the speed is lo&er and depends on the &avelength. A light'year, denoted l.y., is &orth about "0,$$$ billion 3m. The star closest to our Sun, Pro*ima Centauri, is located about = ly. @ur Sun is > light'minutes from Earth, the Moon is one light'second, and an adult human measures bet&een % and A light' nanoseconds.
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