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    Lesson 0: An introduction to special relativity

    Albert Einsteindeveloped the special theory of relativity (or special relativity) in

    1905. Before special relativity, there were a number of problems with the classical

    explanations of electromanetism which in !instein"s view contradicted the principle

    of relativity.

    #pecial relativity is based on two simple postulates$

    1. %he special principle of relativity$ %he laws of physics are the same for all

    observers, reardless of their velocity.

    &. %he speed of liht in a vacuum (c) is constant$ %hat is, everyone will always

    measure the speed of liht as bein the same (i.e. c' &99,9,*5 m+s), reardless of

    their own velocity.

    %he important point here is that the speed of light is the same for all observers.

    #uppose you measure the speed of a beam of liht travellin towards you and record

    it"s speed as c. ccordin to -ewtonian (or classical) physics, if someone else

    travellin at 1 m+s (relative to you) were then to measure the speed of a beam of liht

    travellin towards them, then they would measure the speed of liht to be c 1 m+s.

    /owever, this does not prove to be the case in practice everyone records the same

    speed of liht reardless of their velocity relative to each other. !instein explained

    this by proposin that the way you view space and time to be different from the way

    the other person views space and time. %he mathematical description of this became

    the special theory of relativity.

    #pecial relativity remained controversial for many years after it"s first publication.

    /owever, as experiments became more accurate, special relativity was accepted by

    the scientific community. espite this, !instein did not recieve a -obel pri2e for this

    wor3 he was ranted that honour for his wor3 on the photoelectric effect.

    %he special relativity simulator, Warp, is an attempt to explain some of the strane

    conse4uences of special realativity. hat does it mean that my view of space and

    time is different to your view6 ell, if you follow the next few lessons and have a

    play with Warpthen hopefully you will understand...

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    Lesson 1: Changing the View

    arp consists of a threedimensional environment that may be naviated usin the

    mouse or the 3eyboard. hen arp starts, the you should be presented with the

    followin screen$

    7n this case the ob8ect is a teapot viewed from above. %his isn"t obvious at first, so we

    need to chane our point of view. %he viewpoint can be chaned by usin the arrow

    buttons on the onsole Bar. 7t is also possible to use the 3eypad to achieve the same

    effect.

    :eys "*" and ";" move the camera hori2ontally left and riht respectively. :eys "" and "&" move the camera vertically up and down respectively.

    :eys "5" and "0" move the camera into and out of the screen respectively.

    %hat said, you miht prefer to use fixed movement axes. %his means that the arrows

    on the onsole will alwaysmove the camera alon the axes of the arp universe. 7f

    this sounds li3e a ood idea then press "r" to chane into this fixed axis system. hile

    this may not seem li3e a ood idea at first, it becomes more useful later, so 3eep it in

    mind.

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    7t is also possible to rotate the camera by clic3in and drain with the left mouse

    button within the =ain indow. %his can be a little confusin but the blue rid

    beneath the ob8ect should allow you to 3eep your bearins. 7 recommend thatwhenever you want to move around the ob8ect, you should alin the camera in the

    direction of the 2axis. %his is shown by a red line on the >rid. t the end of the line

    there is a red "" so you will 3now which direction the camera is facin. amera

    rotation can also be done usin the 3eyboard.

    :eys "?" and "@" rotate the camera up and down respectively.

    :eys "," and "." rotate the camera left and riht respectively.

    %he ob8ect orientation can also be chaned. lic3in and hold the riht mouse button.

    %he blue rid will appear aain. %hen dra the mouse. %he ob8ect will rotate about itscentre point until you are happy with its orientation, as shown below.

    lternatively, you can use the arrow 3eys to rotate the ob8ect to its desired position.

    hanin the view miht be slow and 8er3y on slower computers. Aou miht be able

    marinally improve performance by usin the followin options$

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    Lesson : E!periments at "igh #peed$

    /avin mastered the control of the camera, you are now ready to see arp do what it

    was desined to do. %hat is showin you what thins would loo3 li3e at (very) hih

    speeds. %o chane the speed you need to use the #peed Bar on the left hand side of

    the onsole Bar, as shown below.

    %he reen line indicates your current speed in the direction of the 2axis of the arp

    universe. %he 2axis is directly into the screen when you start arp, and the camera"s

    velocity is always alon this axis. By clic3in with the left mouse button on the #peed

    Bar and drain the mouse, you can chane your speed. %he line will chane si2e to

    indicate the speed. Durthermore, its colour will radually chane to yellow as you et

    close to the speed of liht. %he scale of the #peed Bar is loarithmic. %his means that

    the #peed Bar is clearer at hiher speeds. Dor example, leftmost tic3 mar3 (the one

    above the "0") indicates a speed of 0 m+s. %he rihtmost tic3 mar3 (the one above the

    "c") indicates the speed of liht (exactly &99,9&,*5 m+s ). %he mar3s inbetween

    from left to riht indicate &0E, *0E, ;0E and 0E of the speed of liht respectively.

    %his loarithmic scale allows you to vary your speed more sensitively as youapproach the speed of liht. 7nstead of usin the mouse, it is possible to chane your

    speed usin the 3eyboard, which can be more sensitive than the mouse.

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    ooahh hat"s oin on here6 Hur teapot has been hideously deformed. nd it

    chaned colour. nd, is it 8ust me, or did it et brihter too6 ell, 7"m sure the

    physicists amon you will tell us that this is all down to !instein"s %heory of #pecialFelativity. 7n fact there are three different effects oin on here.

    Corent2 %ransformations

    oppler !ffect

    /eadliht !ffect

    e will explain these one at a time in later lessons. But first, try holdin down the

    "space bar" aain. Aou should see the teapot shoot of past you. hat you are doin is

    settin time in motion. hereas before time was fro2en and you were seein a

    snapshot of the teapot, by pressin the "space bar" you increase time by a smallamount. %he teapot there for starts movin because of it"s velocity relative to the

    camera. Aou can reverse time by pressin "Feturn". nd if thins et completely out

    of hand, you can set the time bac3 to 2ero by pressin "t". #o 8ust for completeness . . .

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    Lesson %: Lorent& 'ransforms E!plained

    !rh hat a nasty soundin title. But it"s not that bad. -o really Dirst, thin3 of an

    experiment . . .

    %wo spaceships, and B, are drivin towards each other. spaceman, who is sittin

    still in space, sees both spaceships oin at 1,000,000 m+s (one million metres per

    second). %he pilot in spaceship wants to 3now how fast spaceship B is approachin

    him. 7n pre!instein physics (i.e. before special relativity) he would assume that

    spaceship B approachin him at &,000,000 m+s.

    #paceship B then slows to a halt. %he pilot decides to do an experiment and measure

    the speed of liht. /e measures the speed of a ray of liht headin towards him. /e

    ets a result of G00,000,000 m+s (that"s threehundred million metres per second).

    =eanwhile, spaceship is still oin at 1,000,000 m+s. %he pilot also decides to

    measure the speed of a ray of liht comin towards him. hat result does he et6

    1. G01,000,000 m+s (threehundred and one million metres per second)6

    &. &99,000,000 m+s (twohundred and ninety nine million metres per second)6

    G. G00,000,000 m+s (threehundred million metres per second)6

    *. !h6

    7f you said "&" or "*" then o bac3 and reread this lesson from the beinnin. 7f you said

    "1" then nearly every physicist in the world would aree with you . . . up until 1905

    and !instein that is. %he correct answer is "G" because the speed of light is the samefor all observers. %his is what special relativity is all about. 7t doesn"t matter if you

    are travelin at &00,000,000 m+s or standin still, you will always et the same result

    for the speed of liht. utchman named /endri3 ntoon Corent2 reali2ed that this

    could be explained if ob8ects in motion undero transformations called the Lorent&

    'ransformations(

    7n these e4uations x, y, 2 and t set the coordinates of space and time respectively from

    an observers point of view. lso, x", y", 2" and t" set the coordinates accordin to the

    sub8ect. %he velocity, v, is in the direction of the xaxis and c is the speed of liht. #o

    what does this mean6 ell, it means that funny thin happen to the way thins loo3.Dor example, the first imae shows a stationary lattice. %he second imae shows the

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    same lattice travelin towards us at close to the speed of liht -otice how you can

    see the bac3 of some of the lattice members despite the fact that they are in front of

    you Iery strane.

    v ' 0 m+s

    v J c

    arp simulates the Corent2 %ransformations, and it is worth pointin out that if you

    try movin around a warped ob8ect you miht find it a bit strane. Dirst of all, as youincrease the speed, the ob8ect will enerally shoot off into the distance. hy6 ell,

    loo3 at the first e4uation iven in this lesson. Aou see that if we increase the velocity,

    v, but hold the time, t, constant then x" will divere from x. Hf course in real life it"s

    impossible to increase your speed instantaneously but this is exactly what happens in

    arp. arp holds time still until you tell it otherwise. Femember in the Cesson &

    when the teapot went shootin off as you increased the speed6 ell you are now in a

    position to understand what was oin on. %he speed was increased, the time wasn"t,

    so x" became lare and hence the teapot shot off into the distance. #imple.

    %he result of all this waffle is that when you chane you speed in arp, you will also

    have to chane the time. s outlined earlier, you do this usin "space" and "return". 7f

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    you foret to do this, your ob8ect will disappear off the ede of the arp universe,

    and you"ll be terribly upset.

    nyway, the ob8ect may behave in unpredictable ways because of the way that the

    twistin at hih speeds. 7f you don"t want to et thorouhly confused when movin

    around ob8ects 7 recommend reducin the speed before movin. 7 also find that it iseasier to wor3 with the camera fixed on the axes of the arp universe as outlined in

    Cesson 1. 7f you recall, this involved pressin "r" to tole the "camera relative" mode.

    %hat said, if you do want to be completely perplexed, please try movin + rotatin the

    teapot around at hih speed and see what happens

    =ovin on, the observant amon you will have noticed that the colour of ob8ects in

    arp also chanes. %his is due to oppler #hiftin, which we tal3 about in Cesson *

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    Lesson ): All the Colours of the *ainbow + 'he

    ,oppler Effect

    H3ay, those of you with ood memories will recall that the teapot inCesson 1chaned colour as we increased the speed. %his is all down to the ood ol" oppler

    !ffect. Dor those of you who don"t 3now what that is, let me explain . . .

    Femember the last time you heard an ambulance o by6 Femember how the pitch of

    the siren chaned as the ambulance went past6 %hat"s the oppler effect. %he sound

    waves emitted by the siren et s4uashed up as the ambulance comes towards you, and

    hence the siren seems hih pitched. Ci3ewise, when the ambulance has passed, the

    sound waves are stretched out and hence you hear a lowpitched siren.

    Ciht also behaves li3e a wave (except in Kuantum =echanics, but !instein didn"t

    li3e Kuantum =echanics so we"ll pretend it doesn"t exist ()). %herefore, it is sub8ectto exactly the same effect. Ciht emitted from an ob8ect rushin towards us will have

    its waves s4uashed up. /ence, the colours will be shifted towards the blue end of the

    spectrum. onversely, liht waves from an ob8ect rushin away from us will appear

    redder. %hese chanes are called blue shift and red shift respectively. t very hih

    speeds visible liht emitted by the ob8ect can be shifted out of the visible spectrum,

    and hence the ob8ect would not be visible to the human eye

    %he oriinal wavelenth is shifted accordin to the followin formula$

    /ere, lambda is the wavelenth, v is the speed, and theta is the anle that the vector to

    the point on the ob8ect ma3es with the direction of motion (for those who are

    interested).

    arp simulates the oppler effect on fast movin ob8ects. 7n order to see this, try the

    followin$

    1. #et the speed to 2ero and arrane the ob8ect to your preference.

    &.

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    s you can see the ob8ect suddenly turns white (li3e the left hand side of the teapot

    above). %his means that the colour has been shifted out of the optical spectrum. 7n

    reality it would no loner be visible, but arp shows this as white to aid visibility.

    %he oppler effect is the reason that arp draws everythin in reen to start with.

    >reen is in the middle of the optical spectrum, so you are able to et a ood rane for

    both red and blue shiftin. /owever, if you are only interested in, say, blue shift

    (ob8ects comin towards you), then you can chane the oriinal colour of the ob8ects.

    %o do this you need to use the "" and "" 3eys, which will slowly chane the hue of an

    ob8ect until you are happy. #o if you are interested in blue shiftin you miht want to

    start of with a red teapot to ive yourself the larest possible colour rane to play

    with.

    Lse "" and "" to chane the hue of the ob8ect.

    /owever, sometimes you won"t want to see the oppler effect. fter all, those colours

    can be a bit hard on the eyes. #o . . .

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    another interestin phenomena of special relativity and is the sub8ect of the next

    lesson.

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    Lesson -: .right Lights + 'he "eadlight Effect

    e now turn our attention to one of the straner effects of special relativity, the

    headliht effect. %he best way to explain this is to imaine a torch that, in its own

    frame, emits a beam of liht in the shape of a cone. %his cone"s sides ma3e an anle of

    phi with respect to the xaxis (i.e. the torch is pointin alon the xaxis). %his is

    shown in the followin diaram.

    -ow imaine that the torch moves alon the xaxis with some velocity, v. ontrary to

    what you miht expect, the anle phi will become narrower %his contraction is

    3nown as the /eadliht effect. %he amount that phi narrows is iven by the followin

    e4uation$

    #o what does this mean in relation to arp6 ell, the important thin to note is that

    the torch isn"t ivin off any less liht. %he same amount of liht is bein focusedinto

    a smaller area. %herefore, an approachin torch will seem brihter than a stationary

    torch. Ci3ewise, a recedin torch will appear dar3er. %his doesn"t 8ust apply to torches.

    7t also applies to any ob8ect travelin towards an observer. %he chane in liht

    intensity, 7, is iven by the followin relationship$

    .

    Aou can see the headliht effect in arp by tryin the followin$

    1. rrane the ob8ect as you li3e it.

    &. uess what6 Lse "space" and "return" to alter the time and hence view of the

    ob8ect. Boy, 7 et tired of sayin that

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    7f you arraned the ob8ect so that it is speedin towards you then you should see

    somethin li3e this$

    -otice how briht the teapot is. /owever, if the ob8ect is travelin away from you

    then you should see somethin more li3e this$

    /ere the headliht effect has dar3ened everythin to such an extent that you can"t see

    much of the oriinal colour Hf course this can be a bit of a pain so you can turn the

    headliht effect off$

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    Lesson /: dds and Ends

    ell you"ve more or less seen it all. But a couple of thins still remain . . .

    1$ Coadin Hther Hb8ects

    By now you"ll be totally bored with the teapot. Cuc3ily, arp supports other ob8ects.

    Dirst you"ll need to visit the ownloadpae to download some models. Hnce you"ve

    un2ipped these onto your computer, simply clic3 on the bi yellow button in the

    middle of the console. %his will brin up a box that will allow you to load the new

    ob8ects. =ay 7 recommend the lattices, as they ma3e visualisin special relativity a lot

    easier. =ost of the ob8ects are reat fun, for example 7 bet you"ve never seen the !iffel

    tower li3e this$

    &$ :eyboard short cuts.

    Aou miht be overawed with all the 3eyboard shortcuts used by arp, but don"t

    worry #imply press D1 to brin up a little help box to remind yourself of all the 3eys

    /urrah Hr you can loo3 at the next lesson for a 4uic3 summary.

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    Lesson : #ummary

    :eyboard commands$

    2ress 31 to bring up the help screen(

    :eys "*" and ";" move the camera hori2ontally left and riht respectively.

    :eys "" and "&" move the camera vertically up and down respectively.

    :eys "5" and "0" move the camera into and out of the screen respectively.