Chapter 28 Special Relativity

Transcription

1 Galilean Relatiity Chapter 8 Speial Relatiity A passenger in an airplane throws a ball straight up. It appears to oe in a ertial path. The law of graity and equations of otion under unifor aeleration are obeyed. A stationary obserer on the ground iews the path of the ball thrown to be a parabola. The ball has a eloity to the right equal to the eloity of the plane. The two obserers disagree on the shape of the ball s path Both agree that the otion obeys the law of graity and Newton s laws of otion Both agree on how long the ball was in the air Conlusion: There is no preferred frae of referene for desribing the laws of ehanis Galilean Relatiity Liitations Galilean Relatiity does not apply to experients in eletriity, agnetis, optis, and other areas Results do not agree with experients The obserer should easure the speed of the pulse as + Atually easures the speed as

2 Ether Assuption & Mihelson-Morley Expt Mehanial waes need a ediu to support the disturbane. Assue luiniferous ether to be the ediu for light waes to propagate. Present eerywhere, een in spae. Massless, but rigid ediu. Sine the earth oes through the ether, there should be an ether wind blowing. If is the speed of the ether relatie to the earth, the speed of light should hae iniu or axiu alues depending on its orientation to the wind. Mihelson-Morley Experient Designed to detet sall hanges in the speed of light due to ether wind. Used the Mihelson Interferoeter Ar is aligned along the diretion of the earth s otion through spae The interferene pattern was obsered while the interferoeter was rotated through 90 Measureents failed to show any hange in the fringe pattern The idea of an ether was disarded Light is now understood to be an eletroagneti wae, whih requires no ediu for its propagation The laws of eletriity and agnetis are the sae in all inertial fraes atie fig. atie fig.

3 The Constany of the Speed of Light Been onfired experientally in any ways, e.g. easuring the speed of photons eitted by partiles traeling near the speed of light. Explains the null result of the Mihelson-Morley experient Relatie otion is uniportant when easuring the speed of light. We ust alter our oon-sense notions of spae and tie Einstein s Priniple of Relatiity Resoles the ontradition between Galilean relatiity and the fat that the speed of light is the sae for all obserers Einstein s Postulates The Priniple of Relatiity: All the laws of physis are the sae in all inertial fraes The onstany of the speed of light: the speed of light in a auu has the sae alue in all inertial referene fraes, regardless of the eloity of the obserer or the eloity of the soure eitting the light Generalizes the priniple of Galilean relatiity, whih refers only to the laws of ehanis The results of any kind of experient perfored in a laboratory at rest ust be the sae as when perfored in a laboratory oing at a onstant speed relatie to the first one. 3

4 Speial Relatiity Speial Relatiity restrit the disussion to onepts of length, tie, and siultaneity No preferred inertial referene frae exists. Ipossible to detet absolute otion. The relatie speed between two inertial fraes ust be the sae as easured in either frae. The referene frae with the speed of light is NOT a alid frae. It does not exist. Thought Experient There is no suh thing as absolute tie, absolute length, absolute siultaneity, or absolute referene frae. A boxar oes with unifor eloity Two lightning bolts strike the ends The lightning bolts leae arks (A and B on the ar and (A and B on the ground Two obserers are present: O in the boxar and O on the ground Obserer O is idway between the points of lightning strikes on the ground, A and B Obserer O is idway between the points of lightning strikes on the boxar, A and B 4

5 Siultaneity Thought Experient The light reahes obserer O at the sae tie He onludes the light has traeled at the sae speed oer equal distanes Obserer O onludes the lightning bolts ourred siultaneously By the tie the light has reahed obserer O, obserer O has oed. The light fro B has already oed by the obserer, but the light fro A has not yet reahed hi. The two obserers ust find that light traels at the sae speed. Obserer O onludes the lightning struk the front of the boxar before it struk the bak (they were not siultaneous eents. Thought Experient Suary Two eents that are siultaneous in one referene frae are in general not siultaneous in a seond referene frae oing relatie to the first That is, siultaneity is not an absolute onept, but rather one that depends on the state of otion of the obserer In the thought experient, both obserers are orret, beause there is no preferred inertial referene frae 5

6 Tie Dilation A irror is fixed to the eiling of a ehile The ehile is oing to the right with speed An obserer, O, at rest in this syste holds a laser a distane d below the irror Obserer uses a wath to easure the tie between the eents (Δt p The two eents our at the sae plae Δt p = distanespeed = (d To an obserer on the ground, the tie between the two eents is longer! atie fig. Tie Dilation The tie interal Δt between two eents easured by an obserer who sees the two eents taking plae at different loations is longer than the tie interal Δt p between the sae two eents easured by an obserer who sees the at the sae loation. Δt p is the proper tie. A lok oing past an obserer at speed runs ore slowly than an idential lok at rest with respet to the obserer by a fator of -. And ie ersa! 6

7 Tie Dilation Generalization All proesses, physial as well as heial and biologial, slow down relatie to a lok when those proesses our in a frae oing with respet to the lok. Tie dilation is a ery real phenoena that has been erified by arious experients, e.g. uon deay Muons are unstable partiles with a half-life of Δt p =.µs when easured in a referene frae at rest with respet to the (a Relatie to an obserer on earth, uons should hae a lifetie of Δt p (b A CERN experient easured lifeties in agreeent with the preditions of relatiity Length Contration The easured distane between two points depends on the frae of referene of the obserer. The proper length, L p, of an objet is the length of the objet easured by soeone at rest relatie to the objet. The length of an objet easured in a referene frae that is oing with respet to the objet appears to be ontrated in the diretion of the relatie otion. L L P L P How is the length of a oing objet easured? atie fig. 7

8 Length Contration Obserer O easures the length of the ar to be L=AB. How does this opare with the proper length of the ar, Lp=A B as easured by O? The proper length an be easured by sending a light pulse fro A to B and bouning off a irror to return to A. The tie it takes ties is twie the proper length A B. L P L L L L Exaples 6. A neutron lies 900 s when at rest relatie to an obserer. How fast is the neutron oing relatie to an obserer who easures its life span to be 065 s? 4. How fast would a 6.0 -long sports ar hae to be going past you in order for it to appear only 5.5 long? 8

9 Relatiisti Addition of eloities Galilean relatie eloities annot be applied to objets oing near the speed of light. Relatiisti orretions are obiously needed beause no aterial objets an trael faster than the speed of light Einstein s odifiation is D eloities ab ad db ad db ad is speed of objet a with respet to a oing frae d. db is the speed of frae d with respet to frae b. The denoinator is a orretion based on length ontration and tie dilation eloity Addition 9

10 eloity Addition (ont d atie fig. Classial Moentu Conseration Suppose we are in a frae at rest with the enter of ass of the two partiles, whih hae a head-on elasti ollision. i = f = i = f = CM Frae = Total lassial oentu is onsered in this CM frae, but is it onsered when iewed fro another referene frae oing with a onstant eloity? ( before ollision ( in general ( after ollision ( 0

11 Relatiisti Moentu To properly desribe the otion of partiles within speial relatiity, Newton s laws of otion and the definitions of oentu and energy need to be generalized These generalized definitions redue to the lassial ones when the speed is uh less than To aount for onseration of oentu in all inertial fraes, the definition of oentu is odified p is the speed of the partile, is its ass as easured by an obserer at rest with respet to the ass (rest ass Conseration of Relatiisti Moentu It an be shown that the total oentu is also onsered in the oing frae and it equals ( ( (

12 Classial Energy Conseration Suppose we are in a frae at rest with the enter of ass of the two partiles, whih hae a head-on elasti ollision. i = f = i = f = CM Frae = Total lassial kineti energy is onsered in this CM frae, but it is not onsered when iewed fro a referene frae oing with a onstant eloity. ( ( before ollision in general after ollision ( ( Relatiisti Energy The definition of kineti energy requires odifiation in relatiisti ehanis KE = The ter is alled the rest energy of the objet and is independent of its speed The ter is the total energy, E, of the objet and depends on its speed and its rest energy A partile has energy by irtue of its ass alone A stationary partile with zero kineti energy has an energy proportional to its inertial ass The ass of a partile ay be opletely onertible to energy and pure energy ay be onerted to partiles

13 3 Relatiisti Energy Conseration When iewed in a frae oing with respet to the CM, the total energy before the ollision is ] ( [ ] ( [ And the total energy after the ollision is ] ( [ ] ( [ ( ( (? Can you show that both equal Can you show that this redues to the lassial for of energy when s are sall? CM Frae Energy and Relatiisti Moentu There is a siple relation between the total energy, E, and the relatiisti oentu, p E = p + ( When the partile is at rest, p = 0 and E = Massless partiles ( = 0 hae E = p This is also used to express asses in energy units ass of an eletron = 9. x 0-3 kg = 0.5 Me 4 E p

14 Exaples 5. A spae probe speeding towards the nearest star oes at 0.50 and sends radio inforation at a broadast frequeny of.00 GHz. What frequeny is reeied on the Earth? 46. The Big Bang that began the unierse is estiated to hae released 0 68 J of energy. How any stars ould half this energy reate, assuing the aerage star s ass is 4.00x0 30 kg? 38. (a What is the oentu of a 000 kg satellite orbiting at 4.00 ks? (b Find he ratio of this oentu to the lassial oentu. Pair Prodution An eletron and a positron are produed and the photon disappears A positron is the antipartile of the eletron, sae ass but opposite harge Energy, oentu, and harge ust be onsered during the proess The iniu energy required is e =.04 Me 4

15 Pair Annihilation In pair annihilation, an eletron-positron pair produes two photons The inerse of pair prodution It is ipossible to reate a single photon Moentu ust be onsered Chapter 8 Suary. Laws of physis are the sae in all inertial fraes of referene.. The speed of light is the sae for all inertial fraes. Nothing traels faster the speed of light. 3. Moing loks slow down: tie dilation. 4. Moing objets ontrat in diretion of otion: length ontration. 5. eloity addition. 6. Relatiisti oentu. 7. Relatiisti energy. 8. Pair generationannihilation. 5