Special relativity. x' = x vt y' = y z' = z t' = t Galilean transformation. = dx' dt. = dx. u' = dx' dt'

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1 PHYS-3 Relatiity. Notes for Physics and Higher Physics b. Joe Wolfe See also our web pages: Theories of relatiity status Galilean or Newtonian relatiity ector addition of elocities (familiar. Common sense?) usually an excellent approximation but wrong in extreme cases Special theory of relatiity (Einstein) theory of dynamics including uniform relatie motion in excellent agreement with a wide range of experiments General theory of relatiity (Einstein) theory of graitation, includes dynamics in accelerated frames works, but has competition tests hae not been so precise or seere 'Cosmological relatiity' theory of eolution of the unierse, includes expansion of space topical Special relatiity General Relatiity graitation Define: In an inertial frame, Newton's laws hold why? Mach's principle? Galilean Relatiity (Gallileo Newton 904) y x ship x' x t y' y z' z t' t Galilean transformation y' x' u' dx' dt' dx' dt dx dt u or u + u' additiity of elocities a' d x' dt' du' du dt dt a if one frame is inertial, another with constant, uniform relatie elocity is also inertial. Principle of Galilean relatiity: Mechanics is the same in two such frames.

2 Why only mechanics? What about electromagnetism? Are electrical components different if moing? Does the frequency of a tuned circuit change? C L ω LC ε ο A Capacitor: Does C? d µ ο N A Inductor: Does L? l If so, is the same? ε ο µ ο Michelson & Morley l l t t c Suppose that light traels (at c) with respect to a 'stationary' medium (called the æther). Set it up with l l l and 0, light beams return in phase. Now moe at with respect to æther, but let light trael at c in the æther ( / c in lab). t t l c + + l + l c l l c( / ) l c / t... l c 3 l m Earth around sun: 30 kms - φ π t.. c....3 radians (0.4 fringes) λ Result: 0.00 ± 0.0 fringes What about electromagnetism? Does C ε οa? Does L µ οn A d l? If so, is the same? ε ο µ ο Michelson and Morely's experiment: c is the same in inertial frames elocities are not (exactly) additie huh? Principle of special relatiity: Mechanics and Electromagnetism are the same in inertial frames. Chose Maxwell's eqns ahead of additiity of elocity. µ ο, ε o same so c the same.

3 3 Principle of special relatiity: Mechanics and Electromagnetism are the same in inertial frames. this is in agreement with Michelson and Morely's experiment: c is the same in inertial frames Common objection: But it can't be true, because then elocities wouldn't be additie. It's just not common sense. Consider the light from the headlights of a moing ehicle. EITHER i) it traels at c with respect to the ground, and so the drier must measures it to to go slower But this is like M&M's experiment OR ii) the light from the headlights of the moing ehicle must trael at c+ But we can measure light from double stars Experimentally, elocites aren't additie, at least not when one of the elocities is of order c. When << c, elocites are almost exactly additie so we don't notice, so the slight difference neer gets incorporated into common sense.

4 Most clocks are electromagnetic. Consider a ery simple one: light beam going between two mirrors. 4 Holden '6 light ray w mirror Jane timing apparatus Joe Jane is in the car Joe is on the erandah Most clocks are electromagnetic. Consider a ery simple one: light beam going between two mirrors. w Jane is in the car Joe is on the erandah t' w/c w t w + (t/) c t 4 t' 4 + t 4 t' t. ct/ t/ t' t/γ (but from the ehicle...) w Proper time t o in the rest frame. (Here it is t') t in all other frames t is γ t faster than proper time o Tests: pions, clocks in aeroplanes, accelerators

5 Pion lifetimes A slow pion in the lab has lifetime. µs proper lifetime of pion is. µs A fast pion (w.r.t Earth) has lifetime 6 µs (Earth measurement) 5 π π π Earth Earth Proper time t' t/γ where t is the lifetime measured in another frame at w.r.t. the pion. (t always proper time). t' t 0.99 c Relatiity of simultaneity and time-order Eents e and e. not simultaneous e e simultaneous Two eents that are simultaneous in one frame are not simultaneous in another e e weird, but the difference is tiny The order of occurrence can be different in two frames but only for eents at (x,t) and (x+ x,t+ t)) where x t > c Length measurements Measure a length. If object moes, must measure simultaneously, or else compensate for motion. Proper length L o is length in the rest frame

6 6 L o e e e e L o t L' L o t o t γ L' t' t o Need new transformation equations. Try linear. (Why?) x' Ax + Bt y' y (why?) z' z t' Ct + Dx At x' 0, x t. A B/ At t 0, x' γx Ax. x' γ(x t) At x 0, t' γt C γ At x' 0, t γt' & x t t/γ γt Dt γ Dγ D γ γ Lorentz transformations x' γ(x t) y' y z' z Check that L/L o γ t' γ t x Check that t/t o /γ For both, check that << c Galileo

7 What about elocities? 7 u x ' dx' dt' or dx dt dx dt γ(dx dt) γ dt dx u x u x' + + u x' u x u x Non-additiity of elocities Example: how fast is the light from Jane's headlights? (Jane's car traels at.)? u u' + + u' c + + c c Example: the Vogon star ship traels towards earth at c/. A Vogon fires a zap at Earth with speed of c/ (w.r.t the ship). At what relatie speed does the zap trael towards Earth? c/? u u' + + u' c/ + c/ + (c/)(c/) c + /4 0.8 c Example: An electron traels* at c in an accelerator. A postitron traels* at c in the opposite direction. What is their relatie speed? e e + u' u u 0.999c c + (0.999c)(0.999c).998 c c *Warning: diagram and question misleading

8 8 The twin paradox ( the clock paradox) Ernest and Algernon Prism are twin babies. Ms Prism accidentally leaes Ernest in the baggage room of a space port where he is loaded onto the Ursa Major express ( 0.99 c). The mistake discoered, he is transferred to a returning ship (0.99 c) when he is 35 light years away (as measured by Algernon on Earth). Acclerating at say g, Gen Rel effects are unimportant When Ernest returns, Algernon has aged 70 years, but at γ /\r( 0.99 ) 7, Ernest has aged only 0 years. * * * * * "Hey gramps", says Ernest, "where's my bro Algernon? He has been traelling at 0.99 c relatie to me. He should now be 0/7 7 months old?" * * * * * Who is correct? Space time diagrams (To make the geometry easier, let's use 0.66 c, so γ.33, and a closer turning point) t Joe 8 Jane 6 The twin 'paradox' These diagrams use 0.66 c, so γ.33. Joe 7 Joe 6 Joe 5 Jane 5 Jane 4 Legend anniersary greeting sent by EMR world line of other twin syncrohonise clocks t" Joe 8 Jane 6 Jane 5 Joe 7 Joe 4 Jane 3 Jane 4 Joe 3 Jane t' x" Joe Jane Joe Jane, i.e. Jane's clock says it is one year since parting Joe Jane Joe's frame on Earth x Jane's frames on ship x' hypothetical world line moing at uniform speed to Jane's second frame

9 Relatiistic Mechanics Problem. If p class m, momentum is only consered in one frame. (Check using u' aboe.) Define p γm. Check that this is consered in both. Note that, for << c, p class p 9 Work Energy Theorem in Relatiity F dp dt For force in x direction, dw Fdx dp dx dp dt.d(γm) m(dγ + γd) γ dγ γ ( / ) / / ( / ) 3/ dγ γ3 d ( /γ ) d c γ 3 dγ dw m γ + c γ dγ m dγ K dw m 0 Note: as 0, - -/ + K m dγ m (γ ) (*) γ +... binomial or Taylor expansion Write the preceding result (*) thus: γm m + K We might call this E E o + K where E o would be like a proper energy E o Example What is the minimum energy released in an annihilation collision between an electron and a positron? Minimum E o m e ( kg) (3 0 8 ms - ) J.0 Me (proper energy of electron 0.5 MeV) In principle, can make electron-positron pairs with 'modest' accelerators

10 0 Example The rest energy of a proton is 938 MeV, of a neutron 940 Me. What is the binding energy per nucleon in 4 He? c GeV m p + m n + m e GeV Difference is ~ 0 MeV ~ 5 MeV per nucleon Incidentally: data from Atomic masses: "protium" (p,e) deuterium (p,n,e) tritium (p,n,e) helium (p,n,e) H H H He m De m He au Example In the reaction p + p p + p + p + p, one of the reacting protons is at rest in the laboratory. What minimum accelerating oltage is required for the other? minimum energy collision no energy 'wasted' on motion relatie to centre of mass i.e. they al trael ~ together after the collision Before u i + After u f + u f + u f u f Let the centre of mass frame moe at, Before Lab frame C of M frame u i u' -u' + + RH p + is at rest in lab, of CM frame is u' After Lab frame C of M frame u f + u f + u f u + f + all u' f small

11 In the CM frame E i E f γm 4m u u' + + u' γ u'... 3/4 c qv KE V 5.6 GV u' + u' + u'u' m m u / c ery wasteful experiment, so: Example (As before, but): What if you collide p + and p traelling in opposite directions? Here the lab is the C of M frame, so u -u Before: + After + + all u' f small Energy before Energy after γm 4m γ u... 3/4 c for each proton Acceleration energy energy to make a new proton qv m m p 938 MeV V 938 MV

12 Example 'Solar constant' is.4 kwm -. r earth-sun km What is the rate of mass loss of the sun due to this radiation? (i.e. neglect neutrinos, solar wind...) I sun P sun 4πr P I.4πr dm dt but P de o dt dm dt P I.4πr kg.s tonnes.yr tonnes so far %yr % so far

13 A useful transformation and mnemonic 3 p γm p + (m ) γ m + (m ) m / + (mc ) m + m c 4 m / (γm ) E E (pc) + (m ) E K mc pc Mnemonic: mc Example What is the momentum of an electron that has been accelerated through 0.0 MV? i) What is? ii) What is p/m e? ii) What is p/m e? i) E i + electrical work E f m + qv γm γ + qv m + 0 MeV 0.5 MeV c ii) E f 0.0 MeV MeV E (pc) + (m ) p E (m ) c MeV c 0.5 MeV/c kgms - p/m e ms - units of speed 40 c iii) p/m e p/m e c 40

14 Particles and antiparticles electron plus positron e + e + photons proton plus anitproton p + + p photons E photons γm e E photons γm p 4 photons p + + p? fast slow p + + p p + + p + p + + p possible but cannot control photons well enough Example. A Vogon ship is approaching Earth at 0.8 c. A Klingon ship is approaching Earth at 0.8 c from the opposite direction. To an Earth obserer, it appears to hae a length of L 60 m. i) How long does the Klingon ship appear to obserers on the Vogon ship? The proper length L o γl L 00 m How much is it contracted according to Vogons? Need the relatie elocity: Klingons u x Earth Vogons undashed frame dashed frame u x ' u x u x ' is Klingon speed wrt Vogons u u x is Klingon speed wrt Earth x is Vogon speed wrt Earth Note that and u x hae opposite signs here..8c (.8c) u x ' (.8c).8c (carry sig figs) 00 m L' L o /γ' 4.8 m / u' x ii) Earth obserer see both ships to be one light hour away from Earth. It takes the Klingons 45 minutes to abandon ship. It takes the Vogons 30 minutes. They notice the impending collision now. Who will surie? (Think carefully.)

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