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A Glimpse Into the Special Theory of Relativity Siim Ainsaar January 2013 1 Why Relativity?

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A Glimpse Into the Special Theory of Relativity Siim Ainsaar January 2013 1 Why Relativity? A Glimpse into the Special Theory of Relativity Siim Ainsaar January 2013 1 Why relativity?..............................1 4.& Light­cones, sim(ltaneit" and 2 Postulates of special relativity.......2 ca(salit"......................................& 3 Basic thought experiments............2 4.) Lorent* trans ormations 3.1 Time dilation................................2 alge$raically.................................) 3.2 Length contraction.......................3 $ %ynamics.......................................& 3.3 Proper time..................................3 %.1 +o(r­velocit" and ! "orent# transformations................3 o(r­acceleration..........................) 4.1 Spacetime interval.......................3 %.2 Mass, moment(m and energ"......, 4.2 Minkowski spacetime, %.3 +orce............................................- Poincaré trans ormations.............4 ' (ptical effects................................) 4.3 !apidit".......................................4 & *lectromagnetism..........................) 4.4 #"per$olic trigonometr"..............% + Basic problems...............................) 4.% Length contraction, time dilation ) (lympia- problems........................) and velocit" addition....................% 1. /urther rea-ing............................) 1 Why relativity? frame and departing in another% 'lectromagnetism needs relativity Relativity is often seen as an intric­ for an explanation. ate theory that is necessary only when dealing with really high (hotons )thus, much of optics* are speeds or ultra-precise measure­ always relativistic and other ments. However, there are some particles often are. +nything where quite often­encountered topics that the speed of light matters , for ex­ are paradoxical if treated non­ ample, the -(. measuring the time relativistically. These are also some for a radio signal to travel from of the main sources of Olympiad satellites , uses relativity. problems on relativity. (article physics needs relativity in Thin for a moment about two several aspects. (articles cannot be charged initially stationary controlled in a modern accelerator particles. They “feel" only the elec­ without taking into account their trostatic force from each another. relativistic dynamics. The only suc­ #ut in another, moving reference cessful quantum theory predicting frame there is also the magnetic the outcomes of particle collisions, force, in general, in a different dir­ quantum field theory, is relativistic. ection$ How could force depend on /uons in cosmic rays would decay the choice of inertial reference long before reaching the ground, frame? What principles forbid the but we still detect them than s to particles from colliding in one relativistic time dilation. 1 Relativistic theory of gravity , gen­ 2 Postulates0of0special0 eral relativity , allows to formulate relativity the physics independently of 1. The laws of all physics are whether the reference frame is iner­ the same in every inertial ref­ tial or not, thus unifying time and erence frame. space even more tightly. 0t is neces­ sary for astrophysics )precession of + reference frame is inertial if and planets1 orbits, gravitational lens­ only if ob5ects onto which no force ing, blac holes*, and cosmology acts move in a straight line with )history and future of large-scale constant velocity. structures*. 2. The speed of light in vacu­ * * * um (c) is the same in every inertial reference frame. 0n the following, we shall derive the most important results of the spe­ 0n .0, after defining the second,6 the cial theory of relativity, starting metre is defined through fixing )ex­ from the fundamental postulates. actly$*c7 299,792,4;< m=s. /ost steps of the derivation are giv­ en as problems, which are also 3 Basic0thought0 good examples of what one can as experiments in relativity and exercise the reader1s ability to use the theory. 3.1 Time -ilation The most important general tech­ Problem 1. Consider a “light nique for problem­solving is rota­ cloc " that wor s as follows. + tion of /in ows i spacetime in photon is emitted towards a complex coordinates, this is de­ mirror at a nown distancel scribed in section 4. .ection 3 shows and reflected bac . 0t is detec­ the way from postulates to the use­ ted )almost* at the emitter ful techniques, its problems may be again. The time from the emis­ s ipped if concentrating purely on sion to the detection )a “tic "* Olympiad preparation. .ection 2 is is measured to bet. ?ow we mostly on inematics, the following ones develop dynamics, optics and * One second is the duration of )briefly* electromagnetism from it. 9,192,631,770 periods of the 4inally, some problems for prac­ radiation corresponding to the tising are given. transition between the two hyperfine levels of the ground state of the caesium-133 atom at rest at a temperature of 0 K. 2 loo at the cloc from a refer­ ror in our reference frame, if ence frame where the whole the distance in the stationary apparatus is moving with velo­ frame isl% city v perpendicularly to the light The answer is in the following fact. beam. +ssume that the lengths perpendicular to the motion do act 2. If the length of a station­ not change. How long is the ary rod isl" then its length in a tic for us% )Hint@ the light reference frame moving in beam follows a Aig­Aag path.* parallel to the rod with speed The answer is given by the follow­ v is l/γ . ing fact. Cengths are contracted )compressed* in the direction of motion. act 1. If the time interval between to events happening 3.3 Proper time at a stationary point ist" then + spaceship flies in a reference frame where the Problem 3. speed of the point isv the freely from (t 1 ,x1 , y1 , z1) time interval is γt " where )event B* to (t 2 , x2 , y2 , z2) the Lorentz factor )event 8*. What is the proper 1 time τ , time measured by a γ= . passenger on the spaceship , v2 1− between these events% D+nswer@ 2 c2 2 c 2 t t 2 x x 2 c τ = ( 2− 1) −( 2− 1) E √ 2 2 +nother useful quantity in relativ­ −(y 2−y 1) −(z 2−z 1) istic calculations is β=v/c . +s γ>1 , we see everything in a mov­ ! Lorentz0transformations ing vehicle ta e longer than in a sta­ tionary one , time is dilated !.1 Spacetime interval )stretched* in a moving reference 0n ordinary, Galilean relativity, frame. lengths and time intervals are abso­ lute. +s we have now seen, the pos­ 3.2 "ength contraction tulates of Einsteinian relativity im­ Problem 2. ?ow consider the ply that neither is so, once speeds same “light cloc " as in (rob­ become comparable toc. However, lem B., but moving in parallel to proper time , time in a comoving the light beam, with velocityv. frame , must clearly be independ­ What is the distance to the mir­ ent of our reference frame. There­ fore we can define a new invariant 3 quantity with the dimension of numbers. ?amely, the invariant length. quantity is ic t 2 x 2 y 2 z 2 act 3. The spacetime interval = √( Δ ) +(Δ ) +(Δ ) +(Δ ) 2 2 2 2 2 is now expressed 5ust li e (y­ s= c (Δt) −(Δx) −(Δy) −(Δz) √ thagorean theorem. .o, the 'uc­ is independent of the choice lidean distance between two events of reference frame. in the spacetime of )ict,x,y,z* is in­ 0fs is a real number, the interval is dependent of reference frame. called time-likeF ifs is imaginary, the interval is space-like. 0fs is Aero, the What transformations of 'uclidean interval is light­like. space leave lengths invariant% Rota­ tions and translations and combina­ act 4. The interval between two tions thereof$ events on the same light­ray (in vacuum) is zero & thus" act 5. )hanges of inertial refer­ light­li'e. ence frames correspond to ro­ tations and shifts in the space­ !.2 1in2ows2i spacetime4 time coordinates ict" x" y andz. Poincar5 transformations &e can say that spacetime points 0n general, such transformations are * are represented by position four­vec­ called the Poincaré transformations * μ and, if we only rotate and do not tors x =(ct,x,y,z) and the in­ shift the coordinates, the Lorentz terval calculates the length of the μ transformations. displacement four-vector Δx . However, this law of calculating the !.3 Rapi-ity length has important minus signs in #y what angle should we rotate the it, so these four­vectors form a axes% >learly, as one axis has ima­ Minkowski spacetime, not the usual ginary numbers on it, the angle Euclidean space, where lengths must also be complex. would be calculated using the usual (ythagoras1 law. Cuc ily this poses no problems in drawing the angle, as long as we &e can reuse our familiar laws of consider only one-dimensional mo­ geometry if we introduce complex tion@ it turns out to be a purely ima­ ginary angle, so its cosine * Four-vectors are customarily labelled by Greek indices written * The Poincaré transformations are as superscripts; subscripts have a also known as the inhomogeneous meaning in more advanced theory. Lorentz transformations. 4 i i e α +e − α Problem 5. Calculate cosα and cosα= )a pro5ection of the 2 sinα . unit direction vector* is real )can be drawn on the realx-axis* and its act .. cosα=γ " sinα=βγ/i . eiα−e−iα The quantity ϕ=α/i is a real di- sine sinα= is purely ima­ 2i mensionless number and is called ginary )can be drawn on the ima­ the rapidity. ginary ict-axis*. !.! 6yperbolic trigonometry Problem 4. Ta e two coordinate .ome imaginary unitsi and some systems, and !, with the minuses can be eliminated by using spatial axes parallel and the hyperbolic trigonometry. 'mploy­ α −α 6 e −e )spatial* origin of ! moving in ing the formulae sinhα= , thex-direction with velocityv. 2 eα+e−α sinhα Calculate the angle α between coshα= , tanhα= 2 coshα thex- and x'­axis. )Hint@ ma e and cosh 2α−sinh 2 α=1 you can a diagram with ict on one axis prove the following. andx on another.
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