Chapter 39 Relativity

Transcription

1 Chapter 39 Relatiity from relatie motion to relatiity 39. The Priniple of Galilean Relatiity The laws of mehanis mst be the same in all inertial frames of referene. Galilean spae-time transformation eqations ' PO' t PO OO' y' y z' z t' t Learn how to derie the speed in one oordinate system and the transformation to another system. t' t dt' dt ; ' t d' d dt d' d dt d', ', y ' y, z ' z a' a dt' dt dt' The Speed of Light Light waes moed throgh a medim alled the ether. The speed of light was only in a speial, absolte frame at rest with respet to the ether. Downwind: ' Upwind: ' Aross wind: (orthogonal to the wind flow) '

2 39. The Mihelson-Morley periment Phase Differene Light Wae Interferene L L L L t, t L L Time Differene = t t ~ 3 Light takes longer time than Light. After rotation for 90 o, Light takes shorter time than Light. The time differene between the two eents is jst the smmation of their time differenes. Path Differene = t L Phase Differene = / 3 Shift of fringe pattern: L 3 / 39.3 instein s Priniple of Relatiity. The Priniple of Relatiity: The law of physis mst be the same in all inertial referene frames. All the laws of physis those dealing with mehanis, eletriity and magnetism, optis, thermodynamis, and so on are the same in all referene frames moing with

3 onstant eloity relatie to one another. From an eperimental point of iew, it means that any kind of eperiment performed in a laboratory at rest mst gie the same reslt when performed in a laboratory moing at a onstant eloity with respet to the first one.. The Constany of The Speed of Light: The speed of light in am has the same ale in all inertial frames, regardless of the eloity of the obserer or the eloity of the sore emitting the light. The motion of the arth does not inflene the fringe pattern obsered in the Mihelson-Morley eperiment, and a nll reslt shold be epeted Conseqenes of Speial Theory of Relatiity Here we restrit or disssion to the onepts of simltaneity, time interals, and lengths.. Simltaneity Definition of Simltaneity: two lights start at the same time and reah the same position at the same later time. Two eents that are simltaneos in one referene frame are in general not simltaneos in a seond frame moing relatie to the first. The signal from B has already swept past O. For O, the light from B emits earlier.. Time Dilation 3

4 d t', t t d d t d / t' t t', / / The time interal of the eent measred at rest is the shortest. The time interal t' is alled the proper time interal. The proper time interal is the time interal between two eents measred by an obserer who sees that the eents or at the same point in spae withot any relatie motion. If a lok is moing with respet to yo, the time interal between tiks of the moing lok is obsered to be longer than the time interal between tiks of an idential lok in yor referene frame. ample: The period of a pendlm is measred to be 3.00 s in the referene frame of the pendlm. What is the period when measred by an obserer moing at a speed of relatie to the pendlm? Instead of obserer moing at 0.960, we an take the eqialent point of iew that the obserer is at rest and the pendlm is moing at past the stationary obserer. 4

5 t t' 3 t 0.7 (s) 0.96 The flying loks lost 59±0 ns dring the eastward trip and gained 73±7 ns dring the westward trip. The earth rotates to the westward. If the flying loks moe to the eastward trip, the loks are more lose to the rest earth withot rotation and the measred time is shorter. On the ontrary, the loks moe faster than the earth s rotation to the westward trip ths the measred time is longer. after t 3. The Twin Parado Speedo: spaeraft reahes a speed of 0.95, traels to Planet X loated 0 lightyears away. 5 Lightyear: (m) Goslo: inertial frame Only Goslo who is always in a single inertial frame an make orret preditions based on speial relatiity. Upon Speedo s retrn, Goslo has aged 4 years. Goslo finds that Speedo ages only 5

6 / Length Contration The proper length L Lp of an objet is the length measred by someone at rest relatie to the objet. O at rest, O moe with a speed In O, the obserer measres the time period for O to omplete the measrement as t', while he measres the proper length L at rest. Obserer O : L t' On the ontrary, the obserer in O measres the proper time t whereas he measres the length L ' in motion. Obserer O: L' t L' t L' L L p L' L t ' 5. Spae-Time Graphs =0 All possible ftre eents lie aboe the -ais and between the red lines. ample: A Voyage to Siris An astronat takes a trip to Siris, whih is loated a distane of 8 lightyears from the arth. The astronat measres the time of the one-way jorney to be 6 years. If the spaeship moes at a onstant speed of 0.8, how an the 8-ly distane be reonile with the 6-year trip time measred by the astronat? Length ontration: 8 / lightyears Trael time: 4.8 / years 6. The Relatiisti Doppler ffet 6

7 If a light sore and a obsere approah eah other with a relatie speed f / obs f sore / 39.5 The Lorentz Transformation qation Now we look for a transformation like this form: ' t ' t' Gien a light emitting from ' 0 at t t' 0, the eent will gie the relation at later time: t' t t t t' t' ' t t t / ' t, ' t' From ' t' t' ' t t' t' / t t / t t t ' t y' z' y z t' t 7

8 Displaement Transformation: ' ' ',, t' t ' t', t t t ', t' t t ' and ' t Use t t, to derie ' t'. ample: Simltaneity and Time Dilation Reisited (a) Imagine two eents that are simltaneos and separated in spae sh that t ' 0 and ' 0 aording to an obserer O who is moing with speed respet to O. t t' ' Simltaneos in O not simltaneos in O Simltaneos in O not simltaneos in O ' (b) A moing lok is measred to rn more slowly than a lok that is at rest with respet to an obserer. ' 0 O is moing with respet to O t t' ' t' 39.6 The Lorentz Veloity Transformation qation ' t, y' y, z' z, t' t d' d dt ' dt' dt d dy' dy y y ' dt' dt d ' t', y' y, z' z, ' t t' Please derie the transform from ' to. ' ' 8

9 ample: Relatie Veloity of Two Spaeraft Two spaeraft A and B are moing in opposite diretion. An obserer on the arth measres the speed of spaeraft A to be and the speed of spaeraft B to be Find the eloity of spaeraft B as obsered by the rew on spaeraft A. B to O ' O stik on A ths O moe to the right w.r.t. O oo ' o ' o ' ' with respet to O and also to spaeraft A Relatiisti Linear Momentm Classial Definition: p m Here from the obserer at rest. t t The moing partile has its proper time Ths the momentm of the partile shall be m p m m m m t / t / Fore: dp F dt 39.8 Relatiisti nergy The ating fore gies a work that transforms to kineti energy of an objet. m p / 9

10 K Fd t 0 dp dt d dt dt t 0 dp dt dt t 0 dp d dt d dt 0 dp d d p m / dp d m m / / / 3/ m 3/ K m ' 0 m ' / / m 3/ d m ' 0 d / K m K m m m when 0, m p m p p p p m 4 m 4 m m p 4 m p 0 p m m p m p p 4 6 m / p 4 m / 4 m p 4 m p Photon has zero mass p (the partile reahing the light speed shall hae zero mass) ample: (a) Find the rest energy of a proton in the nit of ev. (b) If the total energy of a proton is three times of its rest energy. What is the speed of the proton? () Determine the kineti energy of the proton in the nit of ev. (d) What is the proton s momentm? (a) m p.6730 kg, m p ( J ) 9.40 ( ev ) (b) m p 3m p m/s () K m p m p 9 K.880 (ev) (d) p m (kg m/s) 0

11 39.9 Mass and nergy nlear and elementary-partile interations we annot se the priniple of onseration of energy onentional nlear reator: the ranim nles ndergoes fission reslting in seeral lighter fragments of atoms the total mass of the fragment atoms is less than the ranim atom by an amont m, ths giing an energy of m mass-energy transformation fsion reation: two deterim atoms ombine to form one helim atom, reslting in 9 the redtion of total mass by an amont m 4.50 kg, ths giing an energy of 3. 9 MeV ample: The 6 Po nles is nstable and ehibits radioatiity. It deays to Pb by emitting an alpha partile ( 4 He). The releant mass are m( 6 Po)= , m( 4 He)= , and m( Pb)= (a) Find the mass hange. (b) Find the energy from the fission reation. (a) kg 9 8 (b) J 39.0 The General Theory of Relatiity A beam of light shold also be bent downward by a graitational field. All the laws of natre hae the same form for obserers in any frame of referene, whether aelerated or not.. In the iinity of any point, a graitational field is eqialent to an aelerated frame of referene in graity-free spae (the priniple of eqialene).

12 Time is altered by graity. A lok in the presene of graity rns slower than one loated where graity is negligible. The freqeny of radiation emitted by atoms in the presene of a strong graitational field is redshifted to a lower freqeny. The time near the strong graitational field is shorter than that in the presene of a weak field, ths the freqeny near the strong graitational field is higher. When the freqeny is obsered near the weak field spae, the freqeny is redshifted. The presene of a mass ases a ratre of spae-time in the iinity of the mass and ratre ditates the spae-time path that all freely moing objets mst follow.