The Special Theory of Relativity

Transcription

1 The Speial Theory of Relatiity

2 Galilean Newtonian Relatiity Galileo Galilei Isaa Newton Definition of an inertial referene frame: One in whih Newton s first law is alid. onstant if F0 Earth is rotating and therefore not an inertial referene frame, but we an treat it as one for many purposes. A frame moing with a onstant eloity with respet to an inertial referene frame is itself inertial. Relatiity priniple: Laws of physis are the same in all inertial frames of referene

3 Intuitions of Galilean Newtonian Relatiity Lengths of objets are inariant as they moe. Time is absolute. Mass of an objet in inariant in for inertial system Fores ating on a mass equal for all inertial frames Veloities are (of ourse) different in inertial frames Positions of objets are different in other inertial systems (oordinate transformation)

4 Relatiity priniple: The basi laws of physis are the same in all inertial referene frames r s r 0 t + r at

5 Galilean Newtonian Relatiity This priniple works well for mehanial phenomena. Howeer, Maxwell s equations yield the eloity of light; it is 3.0 x 0 8 m/s. So, whih is the referene frame in whih light traels at that speed? Sientists searhed for ariations in the speed of light depending on the diretion of the ray, but found none.

6 Galilean Newtonian Relatiity This priniple works well for mehanial phenomena. There seems to be a problem with eletrodynami phenomena E μ ε 0 0 E t 0 with μ ε 0 0 James Clerk Maxwell Light is a wae with transerse polarization and speed Problems: In what inertial system has light the exat eloity What about the other inertial systems Waes are known to propagate in a medium; where is this ether Laws of eletrodynamis do not fit the relatiity priniple?

7 The Mihelson Morley Experiment Nobel 907 Albert Mihelson Edward Williams Morley "for his optial preision instruments and the spetrosopi and metrologial inestigations arried out with their aid" Questions: What is the absolute referene point of the Ether? In whih diretion does it moe? How fast? Albert Abraham Mihelson Ether onneted to sun (enter of the unierse)? 4 Earth ~ 3 0 m / 8 ~ 3 0 m / s s } ~ 0 4 Motion of the Earth Should produe an Obserable effet

8 The Mihelson Morley Experiment axis t l + + l l ( / )

9 The Mihelson Morley Experiment axis t l ' l l /

10 The Mihelson Morley Experiment Δ / / t t t l / ' t l l l ( ) / t + + l l l Interferometer: l l l If 0, then Δt0 no effet on interferometer If 0, then Δt 0 a phase-shift introdued But this is not obsered

11 The Mihelson Morley Experiment Rotate the interferometer ΔT Δt Δt' ( l + l ) Approximate: / << l l / ΔT Numbers: ~3x0 4 m/s /s~0-4 l ~l ~ m ( l + l ) 3 ΔT Visible light: λ~550 nm f~5 x 0 4 Hz s Then: / + Phase hange (in fringes) f ΔT / + Should be obserable! Detatability: 0.0 fringe

12 Conlusion: The Mihelson Morley Experiment This interferometer was able to measure interferene shifts as small as 0.0 fringe, while the expeted shift was 0.4 fringe. Howeer, no shift was eer obsered, no matter how the apparatus was rotated or what time of day or night the measurements were made. The possibility that the arms of the apparatus beame slightly shortened when moing against the ether was onsidered, but a full explanation had to wait until Einstein ame into the piture. Hendrik A Lorentz Nobel 90 "in reognition of the extraordinary serie rendered by their researhes into the influene of magnetism upon radiation phenomena" Lorentz ontration

13 Albert Einstein A new perspetie

14 Albert Einstein On relatiity

15 Postulates of the Speial Theory of Relatiity. The laws of physis hae the same form in all inertial referene frames. Light propagates through empty spae with speed independent of the speed of soure or obserer This soles the ether problem the speed of light is the same in all inertial referene frames

16 Simultaneity One of the impliations of relatiity theory is that time is not absolute. Distant obserers do not neessarily agree on time interals between eents, or on whether they are simultaneous or not. Why not? In relatiity, an eent is defined as ourring at a speifi plae and time. Let s see how different obserers would desribe a speifi eent.

17 Simultaneity Thought experiment: lightning strikes at two separate plaes. One obserer beliees the eents are simultaneous the light has taken the same time to reah her but another, moing with respet to the first, does not.

18 Simultaneity From the perspetie of both O and O they themseles see both light flashes at the same time From the perspetie of O the obserer O sees the light flashes from the right (B) first. Who is right?

19 Simultaneity Here, it is lear that if one obserer sees the eents as simultaneous, the other annot, gien that the speed of light is the same for eah. Conlusions: Simultaneity is not an absolute onept Time is not an absolute onept

20 Time Dilation a) Obserer in spae ship Δt 0 D proper time b) Obserer on Earth speed is the same apparent distane longer l Δt Light along diagonal D + l Δt D + Δt Δt / 4 Δt D / Δt Δt 0 / γδt 0 This shows that moing obserers must disagree on the passage of time. Cloks moing relatie to an obserer run more slowly

21 Time Dilation Calulating the differene between lok tiks, we find that the interal in the moing frame is related to the interal in the lok s rest frame: Δt 0 is ther proper time (in the o-moing frame) It is the shortest time an obserer an measure with then.

22 On Spae Trael 00 light years ~ 0 6 m If spae ship traels at then it takes ~00 years to trael. But in the rest frame of the arrier: Δt Δt / 4. 5yr 0 The higher the speed the faster you get there; But not from our frame perspetie!

23 Twin Paradox The twin paradox if any inertial frame is just as good as any other, why doesn t the astronaut age faster than the Earth traeling away from him? The solution to the paradox is that the astronaut s referene frame has not been ontinuously inertial he turns around at some point and omes bak.

24 Length Contration Distane between planets is: l 0 Time for trael: Δ l t 0 Earth obserer Time dilatation Δt0 Δt / Δt γ Spae raft obserers measure the same speed but less time l Δt 0 l l 0 γ

25 Length Contration Only obsered in the diretion of the motion. No ontration, or dilatation in perpendiular diretion

26 Fantasy supertrain A ery fast train with a proper length of 500 m is passing through a 00-m-long tunnel. Let us imagine the train s speed to be so great that the train fits ompletely within the tunnel as seen by an obserer at rest on the Earth. That is, the engine is just about to emerge from one end of the tunnel at the time the last ar disappears into the other end. What is the train s speed?

27 Galilean Transformations A lassial (Galilean) transformation between inertial referene frames: View oordinates of point P in system S

28 Lorentz Transformations In relatiity, assume a linear transformation: γ as a onstant to be determined (γ lassially). Inerse transformation with - x' x' γ t' γ ( x t) Consider light pulse at ommon origin of S and S at tt 0 measure the distane in xt and x t : ( x t) γ ( t t) γ ( )t ( x' + t' ) γ ( t' + t' ) γ ( ) t' x t γ + t' γ ( ) t fill in Transformation parameter

29 Lorentz Transformations Sole further: ( x t) γ ( γ ( x' + t ) t) x' γ ' Leading to the transformations:. Time dilatation and length ontration an be deried From these Lorentz transformations

30 Lorentz Transformations Lorentz transformation length ontration time dilation. Veloity transformations an be found by differentiating x, y, and z with respet to time. is the eloity between the referene frames Note that also u y and u z transform; this has to do with the transformation (non-absoluteness) of time dt' 0 dt Verify that is maximum speed

31 Galilean and Lorentz Transformations Calulate the speed of roket with respet to Earth. u' + u u' + This equation also yields as result that is the maximum obtainable speed (in any frame).

32 Relatiisti Momentum The formula for relatiisti momentum an be deried by requiring that the onseration of momentum during ollisions remain alid in all inertial referene frames. Note: that does NOT mean that the momentum is equal in different referene frames Result

33 Relatiisti Fore Newtons seond law remains alid (without proof) r F r dp dt d dt r γm d dt r m / ) For eery physial law it has to be established how they transform in relatiity (under Lorentz transformations) ) Quatities (like F) not the same in referene frames

34 Relatiisti Mass From the momentum: Gamma and the rest mass are ombined to form the relatiisti mass:

35 Relatiisti Energy Work done to inrease the speed of a partile from 0 (i-state) to (f state): f f f f f ( p) W Fdx dx dt dp d pd beause dp d( p) pd i i dp dt i dp dt i i f i f i d f ( p) p ( γm) i f i pd 0 m / d m / 0 m / m use So: ( γ + ) ( ) m W γm + m m γ m γ γ d d ( ) / / / Kineti energy of the partile is K ( γ ) m ) Amount of kineti energy depends on inertial frame ) Amount of kineti energy redues to lassial alue at low

36 Mass and Energy The kineti energy K ( γ ) m Can be written as the total energy: Where the differene is the rest energy: The last equation is Einstein famous equation implying thast mass is equialent to energy The energy of a partile at rest. Note that m is the same as seen from all referene frames; It is an inariant upon frame transformation

37 Mass, Energy, Momentum Energy E γm Momentum p γm Combining these relations gies Hene also the following Is an inariant under Lorentz transformations E p

38 Doppler Shift for Light The Doppler shift for light for onstant in all inertial frames. λ 0 Δt0 / f0 λ Δt Δt Δt Δt Time dilatation between wae fronts 0 / ( ) Δt0 λ ( ) Δt λ 0 / + When soure moing toward obserer

39 Doppler Shift for Light Hene, one an derie the obsered frequeny and waelength: If the soure and obserer are moing away from eah other, hanges sign. Remember: higher pith, blue shift when moing toward eah other

40 Doppler Shift for Light Speeding through a red light. A drier laims that he did not go through a red light beause the light was Doppler shifted and appeared green. Calulate the speed of a drier in order for a red light to appear green. λ 500nm ; λ nm 0.6

41 Red Shifts in the Unierse λ + z 0 λ Cosmologial redshift Edwin Hubble Galaxies moing away from eah other Expansion of the unierse Big bang as starting point