Addition of velocities. Taking differentials of the Lorentz transformation, relative velocities may be calculated:

Transcription

1 Addition of veloities Taking differentials of the Lorentz transformation, relative veloities may be allated:

2 So that defining veloities as: x dx/dt, y dy/dt, x dx /dt, et. it is easily shown that: With similar relations for y and z:

3 The Lorentz veloity transformations In addition to the previos relations, the Lorentz veloity transformations for x, y, and z an be obtained by swithing primed and nprimed and hanging v to v:

4 In the lassial limit v << β 0 γ 1 z x z z y x y y x x x x v v v v v ' ' ' γ γ

5 Light veloity addition Let s onsider the ase for light. x in yor rest frame However, in Einstein s moving frame ' x x v vx 1 v v v v 1 Experimentally onfirmed by Alväger et al. at CERN in 1964

6 Spaetime When desribing events in relativity, it is onvenient to represent events on a spaetime diagram. In this diagram one spatial oordinate x, to speify position, is sed and instead of time t, t is sed as the other oordinate so that both oordinates will have dimensions of length. Spaetime diagrams were first sed by H. Minkowski in 1908 and are often alled Minkowski diagrams. Paths in Minkowski spaetime are alled worldlines.

7 Spaetime diagram

8 Partilar worldlines Slope /v Slope 1

9 Worldlines and time

10 Moving loks

11 The light one

12 Spaetime interval Sine all observers see the same speed of light, then all observers, regardless of their veloities, mst see spherial wave fronts. s x t (x ) (t ) (s )

13 Spaetime invariants If we onsider two events, we an determine the qantity Δs between the two events, and we find that it is invariant in any inertial frame. The qantity Δs is known as the spaetime interval between two events.

14 Spaetime Worldline The spae and time are entangled. The spaetime interval is invariant. ( ) ( ) s r ( t ) ( s ) ( s ) > 0 0 ( ) s < 0 Spae-like Light-like Time-like

15 Spaetime invariants Three possibilities for the invariant qantity Δs Δs 0 (or Δx Δt ) The two events an be onneted only by a light signal. The events are said to have a lightlike separation. Δs > 0 (or Δx > Δt ) No signal an travel fast enogh to onnet the two events. The events are not asally onneted. They are said to have a spaelike separation. Δs < 0 (or Δx < Δt ) The two events an be asally onneted. The interval is said to be timelike.

16 The Doppler effet The Doppler effet of sond in introdtory physis is represented by an inreased freqeny of sond as a sore sh as a train (with whistle blowing) approahes a reeiver (or eardrm) and a dereased freqeny as the sore reedes. Also, the same hange in sond freqeny ors when the sore is fixed and the reeiver is moving. The hange in freqeny of the sond wave depends on whether the sore or reeiver is moving. f f 0 On first thoght it seems that the Doppler effet in sond violates the priniple of relativity, ntil we realize that there is in fat a speial frame for sond waves. Sond waves depend on media sh as air, water, or a steel plate in order to propagate; however, light does not! + v v r s

17 Reall the Doppler effet Approahing Reeding

18 Relativisti Doppler effet Consider a sore of light (for example, a star) and a reeiver (an astronomer) approahing one another with a relative veloity v. Consider the reeiver in system K and the light sore in system K moving toward the reeiver with veloity v. The sore emits n waves dring the time interval T. The speed of light is always and the sore is moving with veloity v.

19 Relativisti Doppler effet The total distane between the front and rear of the wave transmitted dring the time interval T is: L T vt Bease there are n waves, the wavelength is given by L T vt λ n n And the reslting freqeny is n f λ T vt S R

20 Relativisti Doppler effet In the rest frame of sore: From the time dilation effet: f n 0 T0 T T 0 γ Ths: f λ n T vt f T 0 T γ vt 1 1 v f0 γ 1 v 1 v f 0 1+ v 1 v f 0

21 Sore and reeiver approahing With β v /, the reslting freqeny is given by: (sore and reeiver approahing)

22 Sore and reeiver reeding In a similar manner, it is fond that: (sore and reeiver reeding)

23 Relativisti Doppler effet For light No propagation medim is neessary. The eq. shold depend on a single relative veloity. f f 0 1± β 1 β Cf) For sond f f 0 ± v v r s

24 Relativisti Doppler effet Transverse motion / β γ γ γ f v f f T T f T T f T S R Period of the light sore Period in the K-frame Freqeny in the K-frame There is no lassial onterpart!

25 Expanding niverse The Doppler effet in astronomy The spetral lines from distant stars are all red-shifted (shifted to a longer wavelength), meaning that the stars are reeding from the earth. Hydrogen redshift Hbble s law: v H 0 d The reession speeds are observed to be proportional to distane (therefore, the reession also ors from one another). Leads to the Big-Bang (or expanding niverse) theory

26 Hbble s law v H 0 d

27 Relativisti momentm Physiists believe that Newton s nd law and therefore the onservation of momentm in a ollision where there do not exist external fores is fndamental. dp F ext 0 dt In relativity, P shold be onserved in a ollision for all referene frames.

28 Relativisti momentm Frank (fixed or stationary system) is at rest in system K holding a ball of mass m. Mary (moving system) holds a similar ball in system K that is moving in the x diretion with veloity v with respet to system K. Two balls are then thrown at veloity of 0 in eah frame sh that they ollide eah other elastially.

29 Frank s ball If we se the definition of the lassial momentm, the momentm of the ball thrown by Frank is entirely in the y diretion: p Fy m 0 The hange of momentm dring the elasti ollision as observed by Frank is p F p Fy m 0

30 Mary s ball Mary measres the initial veloity of her own ball to be ' Mx 0 ' My 0 In order to determine the veloity of Mary s ball as measred by Frank, we se the veloity transformation eqations: Mx My v 0 v / 1 x y ' x + v ' vx 1+ ' y ' v γ 1 + x

31 Mary s ball Before the ollision, the momentm of Mary s ball as measred by Frank beomes m 1 0 v / For a perfetly elasti ollision, the momentm after the ollision is The hange in momentm of Mary s ball aording to Frank is p p m 1 v M My p p p p Mx My Mx My 0 / mv mv + m 0 v / 1

32 Relativisti momentm The onservation of linear momentm reqires the total hange in momentm of the ollision to be zero. p F + p M 0 Apparently, it is not the ase. OK with the x-diretion Bt a problem with the y-diretion (or along the diretion the ball is thrown in eah system) p p F M p p Fy My m m v / Linear momentm is not onserved, if we se the onventional definition of the lassial momentm when we se the veloity transformation eqations from the speial theory of relativity. Under the Lorentz transformation, the Newton s law does not hold. Or the definition of p shold be modified to make it valid.

33 Relativisti momentm Rather than abandon the onservation of linear momentm, let s look for a modifiation of the definition of linear momentm that preserves both it and Newton s seond law. It reqires re-defining the linear momentm. Relativisti momentm: p Γm m 1 / 0 m * : veloity of the objet, not of the moving frame

34 Relativisti momentm or mass? Some physiists like to: Refer to the mass in p Γm as the rest mass m 0 and Call the term m Γ m 0 the relativisti mass. In this manner the lassial form of momentm, mv, is retained. The mass is then imagined to inrease at high speeds. Most physiists prefer to: Keep the onept of mass as an invariant, intrinsi property of an objet. We adopt this latter approah. We will se the term mass exlsively to mean rest mass. Althogh we may se the terms mass and rest mass synonymosly, we will not se the term relativisti mass.

35 Relativisti Newton s seond law We mst modify Newton s seond law to inlde or new definition of linear momentm, and fore. ( ) ) ( ) ( / 1 1 ) ( / 1 ) ( m dt d m dt d dt d Γ γ γ γ P F

36 Motion nder onstant fore Constant fore onstant aeleration??? F m dp dt d dt 1 1 ma ( 1 / ) ( γm) m / 3/ + d dt / 1 ( 1 / ) 3 γ ma 3/ / ( γma) d dt a F m ( 1 ) 3/ Not onstant! A simple replaement of m with γm does not give a orret relativisti reslt.

37 Relativisti energy De to the new idea of relativisti momentm, we mst now redefine the onepts of work and energy. The work W 1 done by a fore F to move a partile from position 1 to position along a path s is defined to be W 1 F ds K K1 1 where K 1 and K is defined to be the kineti energy of the partile at position 1 and.

38 Relativisti energy For simpliity, let the partile start from rest nder the inflene of the fore and allate the kineti energy K after the work is done. K K W d ( γm) dt dt F ds d d ( γm) dt ( γm) ds

39 Relativisti kineti energy K d ( γm) m d( γ) m d 1 / By the integration by parts K ( xdy xy ydx) m 1 m 1 / / m m d 1 γm / m m 1 / ( γ 1) m + [ ] m 1 / 0 The reslt is alled the relativisti kineti energy.

40 Relativisti kineti energy Relativisti vs lassial kineti energies K γ rel ( 1) m Kla m When / << 1 (at low speed) 1 K rel (γ 1) m m 1 m 1+ 1 K 1 m la 1

41 Relativisti and lassial kineti energies

42 Total energy and rest energy K γ m γ ( 1) m m This eqation an be rewritten E γm ( γ 1) m + m K + m The term m is alled the rest energy (denoted by E 0 ). This leaves the sm of the kineti energy and rest energy to be interpreted as the total energy of the partile. m 0 E γm K + 1 / E 1 / E 0

43 Total, kineti, and rest energies

44 The eqivalene of mass and energy By virte of the relation for the rest mass of a partile: we see that there is an eqivalene of mass and energy in the sense that mass and energy are interhangeable Ths the terms mass-energy and energy are sometimes sed interhangeably.

45 Conservation of mass-energy M E K m E + Mass-energy before: Mass-energy after: M m > ( ) 15 ~ 10 / m K m K m m M f K m M M r

46 How mh is Em? m 1kg E 0 m (1 kg)( m/s) J This eqals A 1kW heater an rn for ~10 14 se or for 3 million years!!!

47 Em Offiial poster for Em

48 Em in artoons Now that desk looks better. Everything s sqared away, yessir, sqaaaaaaared away.

49 Momentm and energy Sqaring and mltiplying it by, and rearranging the reslt / 1 m m p γ E E m m m m m m p γ γ γ β γ γ γ 0 ) ( ) ( m p E p E + +

50 Momentm and energy A sefl eqation to relate the total energy of a partile with its momentm E p + E0 ( p) + m ( ) (E p ) and m are invariant qantities E p E0 (m If 0 (or p 0), then E E 0. the rest energy If m 0 (photon), then E p. photon dispersion )

51 The speed of massless partiles For a partile with zero rest mass, sh as photon, its veloity mst be. E γ m p γm

52 Comptations in modern physis We were taght in introdtory physis that the international system of nits is preferable when doing allations in siene and engineering. In modern physis a somewhat different, more onvenient set of nits is often sed. The smallness (e) and largeness () of qantities often sed in modern physis sggests some pratial hanges.

53 Units of work and energy The work done in aelerating a harge throgh a potential differene: W qv For a proton (or an eletron), with the harge e C being aelerated aross a potential differene of 1 V, W ( C)(1 V) J

54 Eletron volt (ev) The work done to aelerate the proton aross a potential differene of 1 V old also be written as W (1 e)(1 V) 1 ev Ths ev, prononed eletron volt, is also a nit of energy. It is related to the SI (Système International) nit jole by 1 ev J MeV 10 6 ev; GeV 10 9 ev

55 Mass nit (ev/ ) Adopted from the eqivalene of mass and energy E 0 m m E 0 / (ev/ ) Proton: m p ~ 938 MeV/ Eletron: m e ~ 0.5 MeV/ ( kg) ( kg) Linear momentm nit: ev/ ( E E + p ) 0

56 Binding energy The eqivalene of mass and energy beomes apparent when we stdy the binding energy of systems like atoms and nlei that are formed from individal partiles. The potential energy assoiated with the fore keeping the system together is alled the binding energy E B. Hydrogen atom: proton-eletron Deteron: proton-netron

57 Binding energy The binding energy is the differene between the rest energy of the individal partiles and the rest energy of the ombined bond system.

58 Eletromagnetism and relativity Einstein was onvined that magneti fields appeared as eletri fields observed in another inertial frame. That onlsion is the key to eletromagnetism and relativity. Einstein s belief that Maxwell s eqations desribe eletromagnetism in any inertial frame was the key that led Einstein to the Lorentz transformations. Maxwell s assertion that all eletromagneti waves travel at the speed of light and Einstein s postlate that the speed of light is invariant in all inertial frames seem intimately onneted.

59 A ondting wire ( E + v B) F q * Eletri harge is relativistially invariant.

60 Two ondting wires A prely magneti fore effet in one referene frame F I l B Can be viewed as a prely eletri fore effet in another referene frame F qe