Relativistic Kinematics. Chapter 1 of Modern Problems in Classical Electrodynamics by Charles Brau

Transcript

1 Relativistic Kinematics Chapter of Modern Problems in Classical Electrodynamics by Charles Brau Spring 28

2 Relativistic Formalism of Electrodynamics Special relativity Lorentz transformations Electromagnetic field tensor Covariant formalism of electrodynamics Electromagnetic field of a charge moving at constant speed Spring 28 2

3 Experimental Inconsistencies Spring 28

4 Background Around mid-to-late 8' all waves were assumed to require a material medium to propagate (e.g. sound waves, ocean waves, vibration on a solid material, acoustic guitar, etc) As a consequence, the wave speed depends on the properties of the material medium where it propagates (T, P, Y, ρρ, etc) Relative motion between the observer and the material medium carrying the wave affect the measured speed of a particular wave Following on this tradition, a material medium named ether was assumed as the medium where electromagnetic radiation propagates Spring 28 4

5 Michelson-Morley Experiment An attempt to detect the existence of the ether (luminiferous medium, the light medium) Their goal was to show that different types of motion with respect to the ether give different speeds of light propagation Applied an interferometric technique (known today as Michelson interferometer) to detect small changes in the transit time along different paths Used Earth orbital speed around the sun ( Km/s) as a lower limit of the motion of the Earth through the absolute ether Spring 28 5

6 Michelson Interferometer LL LL 2 6

7 Spring 28 7

8 B A C Spring 28 8

9 tt tt = tt AA BB + tt BB CC = LL cc 2 vv 2 + LL cc 2 vv 2 = 2 LL cc 2 vv 2 tt ll = tt AA CC + tt CC AA = LL cc + vv + LL cc vv = 2 LL cc cc 2 vv 2 tt = tt ll tt tt = 2 LL cc cc 2 vv 2 2 LL cc 2 vv 2 = 2 LL cc vv cc 2 vv cc 2 /2 Spring 28 9

10 vv cc mm/ss 8 mm/ss 4 tt = 2 LL cc vv cc 2 vv cc 2 /2 2 LL cc + vv cc vv cc 2 LL vv2 cc φφ = ωω tt 2 ππ cc λλ LL vv 2 cc = 2 ππ LL λλ vv 2 cc 2 LL m 2 ππ LL λλ 8 λλ 5 9 m Spring 28

11 The Experiments on the relative motion of the earth and ether have been completed and the result decidedly negative. The expected deviation of the interference fringes from the zero should have been.4 of a fringe the maximum displacement was.2 and the average much less than. and then not in the right place. As displacement is proportional to squares of the relative velocities it follows that if the ether does slip past the relative velocity is less than one sixth of the earth s velocity. Albert Abraham Michelson, 887 Spring 28

12 Theoretical Inconsistencies Spring 28 2

13 Preliminary Concepts System of reference to describe an event: position in space (coordinate system) time (clocks) An inertial system of reference: In this system, a particle with no force acting on it will remain at rest or move at constant speed. Systems of reference: If two systems of reference move at constant speed with respect to each other, and one of them is an inertial system of reference, then the other one is also an inertial frame of reference. Spring 28

14 Galilean Transformation for two inertial systems of reference yy KK yyy KK zz OO vv oo zzz OOO xx xxx tt = tt rr = rr vv oo tt Spring 28 4

15 Velocity and Acceleration ddxx ddttt ddyy ddttt ddzz = vv xx vv yy vv zz = vv xx vv oooo vv yy vv oooo vv zz vv oooo ddttt velocity ddvvv xx ddttt ddvvv yy ddttt ddvvv zz ddttt = aaa xx aaa yy aaa zz = aa xx aa yy aa zz acceleration Invariance of Newton s Law Spring 28 5

16 Galilean Transformation as a four-vector linear transformation tt = tt yy KK yyy KK fourcomponents vector = four-vector tt xx yyy zzz rr = rr vv oo tt = tt xx vv ooxx tt yy vv ooyy tt zz vv oozz tt = zz OO vv oooo vv oooo vv oooo vv oo zzz OOO xx tt xx yy zz xxx ddttt ddxxx ddddd ddddd = ddtt ddxx vv ooxx dddd ddyy vv oooo dddd ddzz vv oooo dddd = vv oooo vv oooo vv oooo ddtt ddxx dddd dddd Spring 28 6

17 Four-Vector Position ddxxx ddttt ddtt ddxxx ddxxx vv oooo ddxx = ddxxx 2 = ddddd vv = oooo ddyy ddxxx ddddd vv oooo ddzz vv oooo vv oooo vv oooo ddxx ddxx ddxx 2 ddxx xxx μμ ddxxx μμ = ddxxνν xxνν νν= = νν= GG νν μμ ddxx νν GG νν μμ xxxμμ xx νν μμ: row νν: collumn Spring 28 7

18 Inverse Galilean Transformation tt xx yy zz = tt xx + vv ooxx tt yy + vv ooyy tt zz + vv oozz tt tt = tt rr = rr + vv oo tt = vv oooo vv oooo vv oooo zz yy OO KK vv oo zzz ttt xxx yyy zzz yyy xx KK OOO xxx ddtt ddxx dddd dddd = ddttt ddxxx + vv ooxx ddddd ddyyy + vv oooo dddd ddzzz + vv oooo dddd = vv oooo vv oooo vv oooo ddttt ddxxx ddddd ddddd Spring 28 8

19 ddxx ddtt ddttt ddxx vv ddxx oooo ddxxx = dddd vv ddxx 2 = = oooo ddddd dddd vv ddxx oooo ddddd vv oooo vv oooo vv oooo ddxxx ddxxx ddxxx 2 ddxxx ddxx αα = xx αα μμ= ddxx μμ xxxμμ = μμ= HH μμ αα ddddd μμ = μμ= HH μμ αα νν= GG νν μμ ddxx νν = νν= μμ= xx αα xxx μμ xxx μμ ddxxνν xxνν Spring 28 = νν= δδ αααα ddxx νν = ddxx αα HH μμ αα xxαα xxx μμ αα: row μμ: collumn 9

20 HH αα μμ GG μμ νν = μμ= vv oooo vv oooo vv oooo vv oooo vv oooo vv oooo = = δδ αααα Spring 28 2

21 Four-Vector Gradient Operator ttt xxx yyy zzz = xxx xxx xxx 2 xxx = xx αα xxx xx αα αα= xx αα αα= xx αα xxx xx αα xxx 2 xx αα αα= xx αα αα= xxx xx αα xx αα xx μμ = xxx μμ αα= xx αα = αα= αα HH μμ xx αα Spring 28 2

22 Four-Vector Gradient Operator for the Galilean Transformation ttt xxx yyy zzz = xxx xxx xxx 2 xxx = vv oooo vv oooo vv oooo xx xx xx 2 xx = tt + vv oo. xx yy zz Spring 28 22

23 Second-Order Four-Vector Operator for the Galilean Transformation 2 ttt 2 2 xxx 2 2 yyy 2 2 zzz 2 = 2 tt 2 + 2vv oo. tt + vv oo. 2 xx 2 2 yy 2 2 zz 2 vv oo. Spring 28 2

24 On the Galilean noninvariance of classical electromagnetism Eur. J. Phys. (29) 8 9 Giovanni Preti, Fernando de Felice, and Luca Masiero Spring 28 24

25 Special Relativity Spring 28 25

26 Spring 28 26

27 The Two Postulates of Special Relativity. Laws of physics are the same (invariant) in any inertial system of reference, there is no absolute rest. 2. Light propagates in vacuum at a constant speed Spring 28 27

28 Consequence of the Two Postulates of Special Relativity What affects the speed of a wave is the relative motion between the medium carrying the wave and the observer. If light propagates in vacuum (therefore, if there is no medium carrying the light wave), then the speed of light (in empty space) is the same for any observer even if they have a relative motion with respective to each other. For light propagating in vacuum, the speed of light is the same for any inertial system of reference. Spring 28 28

29 Two inertial systems of reference KK and KK. The axes along (x,y,z) are parallel to the corresponding axes along (x,y,z ). 2. The relative motion between the two inertial systems of reference is along the x-axis (and x -axis).. Consider that the origins OO and OO of the two systems coincide at tt = ttt = As seen by KK As seen by KK OO OOO vv oo vv oo OO OOO Spring 28 29

30 Two events as described by each inertial system of reference Event : When the origins of the two systems coincide, a pulse of light is emitted. KK: tt xx yy zz = KK : tt xxx yyy zzz = Event 2: The pulse of light reaches a detector at a certain point in space and at a certain time. KK: tt 2 xx 2 yy 2 zz 2 KK : tt 2 xx 2 yy 2 zz 2 Spring 28

31 KK: xx 2 xx 2 + yy 2 yy 2 + zz 2 zz 2 = cc 2 tt 2 tt 2 ddxx 2 + ddyy 2 + ddzz 2 = cc 2 ddtt 2 ddxx 2 + ddyy 2 + ddzz 2 cc 2 ddtt 2 = KKK: xxx 2 xxx 2 + yyy 2 yyy 2 + zzz 2 zzz 2 = cc 2 tt 2 tt 2 ddxxx 2 + ddyyy 2 + ddzzz 2 = cc 2 ddttt 2 ddxxx 2 + ddyyy 2 + ddzzz 2 cc 2 ddttt 2 = Spring 28

32 dddd 2 + dddd 2 + dddd 2 cc 2 dddd 2 = = ddddd 2 + ddddd 2 + ddddd 2 cc 2 ddddd 2 Spring 28 2

33 ddyy = ddddd ddzz = ddddd as in Galilean transformation dddd 2 + dddd 2 + dddd 2 cc 2 dddd 2 = = ddddd 2 + ddddd 2 + ddddd 2 cc 2 ddddd 2 dddd 2 cc 2 dddd 2 = ddddd 2 cc 2 ddddd 2 Spring 28

34 ddxx = γγ ddxx vv oo dddd linear modification due to symmetry ddxx = γγ ddxxx + vv oo ddddd = γγ γγ ddxx vv oo dddd + vv oo ddddd solve for: ddddd = γγ vv oo γγ 2 ddxx + γγ dddd Spring 28 4

35 dddd 2 cc 2 dddd 2 = ddddd 2 cc 2 ddddd 2 ddxx = γγ ddxx vv oo dddd ddddd = γγ vv oo γγ 2 ddxx + γγ dddd dddd 2 cc 2 dddd 2 = γγ ddxx vv oo dddd 2 cc 2 γγ vv oo γγ 2 ddxx + γγ dddd 2 Spring 28 5

36 dddd 2 : = γ 2 cc2 γγ 2 2 vv γγ2 2 oo γγ 2 = cc2 γγ 2 2 vv oo γγ 2 = vv oo cc 2 2 dddd 2 : cc 2 = γγ 2 2 vv oo cc 2 γγ 2 γγ 2 = vv 2 oo cc 2 2 ddxx dddd: = γγ 2 vv oo cc 2 vv oo γγ 2 γγ 2 vv oo = cc 2 vv oo γγ 2 vv oo 2 cc 2 = γγ 2 γγ 2 = ββ 2 ββ vv oo cc Spring 28 6

37 ddxx = γγ ddxx γγ vv oo dddd = γγ ddxx γγ ββ dd cc tt ddddd = γγ vv oo γγ 2 ddxx + γγ dddd γγ 2 = γγ vv oo γγ vv oo vv 2 oo cc 2 γγ2 = γγ vv oo cc 2 ddtt = γγ vv oo cc 2 ddxx + γγ dddd dd cc tt = γγ vv oo cc ddxx + γγ dd cc tt = γγ ββ ddxx + γγ dd cc tt Spring 28 7

38 ddxx ddxx ddxx 2 ddxx = dd cc ttt ddxxx ddddd ddddd = γγ ββ ddxx + γγ dd cc tt γγ ddxx γγ ββ dd cc tt ddyy ddzz = γγ γγ ββ γγ ββ γγ dd cc tt ddxx dddd dddd = γγ γγ ββ γγ ββ γγ ddxx ddxx ddxx 2 ddxx Spring 28 8

39 Lorentz Transformation cc ttt xxx yyy zzz xx xx xx 2 xx OO OOO vv oo xx xx xx 2 xx cc tt xx yy zz γγ = ββ 2 ββ vv oo cc ddxxx ddxxx ddxxx 2 ddxxx = γγ γγ ββ γγ ββ γγ ddxx ddxx ddxx 2 ddxx Spring 28 9

40 ddxxx = γγddxx γγ ββ ddxx = ddxxx = γγ ββ ddxx + γγ ddxx = ddxx ddxx = ββ ddxx ddxx = ββ cccc cccc ββ = θθ xx θθ xx tttttt θθ = ββ Spring 28 4

41 Inverse Lorentz Transformation cc tt xx yy zz xx xx xx 2 xx vv oo OO OOO xx xx xx 2 xx cc ttt xxx yyy zzz ββ vv oo cc γγ = ββ 2 ddxx ddxx ddxx 2 ddxx = γγ + γγ ββ + γγ ββ γγ ddxx ddxx ddxx 2 ddxx Spring 28 4

42 A Few Remarks ββ = then γγ = ddddd = dddd ddddd = dddd Although in general ddxx ddxx and ddddd = dddd, we always have: dddd 2 + dddd 2 + dddd 2 cc 2 dddd 2 = ddddd 2 + ddddd 2 + ddddd 2 cc 2 ddddd 2 = ddss 2 dddd: space-time distance Spring 28 42

43 Lorentz Transformation of the four-vector coordinates OO OOO vv oo ddxxx ddxxx ddxxx 2 ddxxx = γγ γγ ββ γγ ββ γγ ddxx ddxx ddxx 2 ddxx xxx μμ ddxxx μμ = αα= LL αα μμ ddxx αα = αα= ddxxαα xxαα LL αα μμ = xxxμμ xx αα Spring 28 4

44 Lorentz Transformation for other four-vectors xxx μμ = xxx μμ xx, xx, xx 2, xx = LL μμ αα xx αα αα= where LL αα μμ = xxxμμ xx αα contravariant tensor of rank = four-vector ddxxx μμ = LL μμ αα ddxx αα αα= VV μμ = LL μμ αα VV αα αα= Spring 28 44

45 Another Contravariant Vector The phase of a wave should be invariant φφ = φφ = constant: kk xx ωω cc cc tt = kk xx ωω cc cc tt kk = (kk,,) = kk γγ xxx + γγ ββ cc ttt ωω cc γγ cc tt + γγ ββ xx = γγ kk γγ ββ ωω cc xx γγ ωω cc γγ ββ kk cc tt ωω cc = γγ ωω cc γγ ββ kk kk = γγ kk γγ ββ ωω cc ωω cc kkk xx kk yy kk zz = γγ γγ ββ γγ ββ γγ ωω cc kk xx kk yy kk zz Spring 28 45

46 Four-Vector K contravariant tensor of rank kk μμ = ωω cc kk xx kk yy kk zz kk μμ = αα= LL αα μμ kk αα Spring 28 46

47 Proper Time cc 2 ddtt 2 dddd 2 dddd 2 dddd 2 = ddss 2 = cc 2 ddττ 2 cc 2 vv xx 2 vv yy 2 vvzz 2 = cc 2 ddττ dddd 2 ββ 2 = ddττ dddd 2 ddττ = dddd γγ Spring 28 47

48 Four-Vector Velocity contravariant tensor of rank vv μμ ddddμμ ddττ = ddττ cc dddd dddd ddyy ddzz = γγ ddtt cc dddd dddd dddd dddd = γγ γγ cc dddd dddd ddyy γγ dddd γγ ddzz dddd vvv μμ = αα= LL αα μμ vv αα Spring 28 48

49 Four-Vector Momentum contravariant tensor of rank pp μμ mm dddd μμ ddττ = mm ddττ cc dddd dddd ddyy ddzz = mm γγ ddtt cc dddd dddd dddd dddd = γγ mm cc γγ mm dddd dddd ddyy γγ mm dddd γγ mm ddzz dddd = EE cc pp xx pp yy pp zz EE γγ mm cc 2 pp ii γγ mm ddxx ii dddd EE 2 cc pp xx 2 cc pp yy 2 cc ppzz 2 = mm 2 cc 4 ppp μμ = LL αα μμ pp αα Spring 28 αα= 49

50 pp ii γγ mm vv ii EE 2 cc pp xx 2 cc pp yy 2 cc ppzz 2 = mm 2 cc 4 EE 2 cc γγ mm vv xx 2 cc γγ mm vv yy 2 cc γγ mm vv zz 2 = mm 2 cc 4 EE 2 = mm 2 cc 4 + ββ 2 ββ 2 EE = mm cc 2 ββ 2 Spring 28 5

51 Four-Vector Force contravariant tensor of rank FF μμ ddddμμ ddττ = γγ dd ddtt EE cc γγ mm dddd dddd γγ mm dddd dddd γγ mm dddd dddd = γγ cc ddtt dd ddxx γγ mm γγ ddtt ddtt γγ mm dd ddtt γγ mm dd ddtt ddee ddyy γγ ddtt ddzz γγ ddtt = γγ cc PP FF xx FF yy FF zz Spring 28 5

52 Linear Transformations: xxx μμ = xxx μμ xx, xx, xx 2, xx = νν= AA νν μμ xx νν where AA νν μμ xxxμμ xx νν contravariant tensor of rank = four-vector xxx μμ ddxxx μμ = ddxxνν xxνν νν= xxx μμ VV μμ = νν= VVνν xxνν covariant tensor of rank = four-vector xx νν xxx μμ = xxx μμ νν= xx νν WWW μμ = xx νν νν= xxx μμ WW νν Spring 28 52

53 Inner Product of a Contravariant and a Covariant tensor of rank xxx μμ VVV μμ = VVνν WWW xxνν νν= μμ = xx αα αα= xxx μμ WW αα xxx μμ xx αα μμ= VVV μμ WWW μμ = μμ= νν= xx νν VVνν αα= xxx μμ WW αα = μμ= νν= xxx μμ xx νν αα= xx αα xxx μμ VVνν WW αα = νν= αα= δδ νν αα VV νν WW αα = νν= VV νν WW νν invariant!! Spring 28 5

54 Example : Charge Conservation =. JJ + ρρ =. JJ + cc ρρ cc tt = cc ρρ cc tt + JJ xx + JJ yy yy + JJ zz zz = cc ρρ xx + JJ xx xx + JJ yy xx 2 + JJ zz xx Spring 28 = μμ= JJμμ xxμμ covariant contravariant JJ μμ = cc ρρ JJ xx JJ yy JJ zz 54

55 Four-Vector Current Density J contravariant tensor of rank JJ μμ = cc ρρ JJ xx JJ yy JJ zz JJJ μμ = LL μμ αα JJ αα αα= Spring 28 55

56 Consider a charge qq at rest at a particular point rr in space. What are the charge density and current density for a frame of reference moving along x-axis at a constant speed vv = ββ cc? Spring 28 56

57 ρρ rr, tt = qq δδ rr rr & JJ rr, tt = (,,) JJ μμ rr, tt = cc ρρ JJ xx JJ yy JJ zz = cc qq δδ rr rr JJJ μμ = LL μμ αα JJ αα = αα= γγ γγ ββ γγ ββ γγ cc qq δδ rr rr = γγ cc qq δδ rr rr γγ ββ cc qq δδ rr rr Spring 28 57

58 δδ rr rr = δδ xx xx δδ yy yy δδ zz zz cc tt tt xx xx yy yy zz zz II, = LL μμ αα xxx αα xx αα = αα= γγ + γγ ββ + γγ ββ γγ cc tt tt xxx xx yy yy zz zz = γγ cc tt tt + γγ ββ xxx xxx γγ xxx xxx + γγ ββ cc tt ttt yy yy zz zz δδ rr rr = δδ γγ xxx xxx + γγ ββ cc tt ttt δδ yyy yyy δδ zzz zzz = γγ δδ xxx xxx + vv tt ttt δδ yyy yyy δδ zzz zzz Spring 28 58

59 JJJ μμ = γγ cc qq δδ rr rr γγ ββ cc qq δδ rr rr δδ rr rr = γγ δδ xxx xxx + vv tt ttt δδ yyy yyy δδ zzz zzz JJJ μμ = cc qq δδ xxx xxx + vv tt ttt δδ yyy yyy δδ zzz zzz ββ cc qq δδ xxx xxx + vv tt ttt δδ yyy yyy δδ zzz zzz ρρ = qq δδ xxx xxx + vv tt ttt δδ yyy yyy δδ zzz zzz JJJ xx = vv qq δδ xxx xxx + vv tt ttt δδ yyy yyy δδ zzz zzz Spring 28 59

60 ρρ rr, ttt = qq δδ xxx xxx + vv tt ttt δδ yyy yyy δδ zzz zzz JJ xx rr, ttt JJ yy rr, ttt JJ zz rr, ttt = vv qq δδ xxx xxx + vv tt ttt δδ yyy yyy δδ zzz zzz ttt = xx = rr tt = vv tt yyy zzz ρρ rr, ttt = qq δδ xxx + vv tt δδ yyy yyy δδ zzz zzz = qq δδ rr rrr tt JJJ rr, tt = vv qq δδ xxx + vv tt δδ yyy yyy δδ zzz zzz = vv qq δδ rrr rrr tt Spring 28 6

61 Example 2: Lorenz Gauge =. AA + μμ oo εε Φ =. AA + Φ/cc cc tt = Φ/cc cc tt + AA xx + AA yy yy + AA zz = Φ/cc xx + AA xx xx + AA yy xx 2 + AA zz xx = μμ= AAμμ xxμμ contravariant AA μμ = Φ/cc AA xx AA yy AA zz Spring 28 covariant 6

62 Four-Vector Potential A contravariant tensor of rank AA μμ = Φ/cc AA xx AA yy AA zz AAA μμ = αα= LL αα μμ AA αα Spring 28 62

63 Consider a charge qq moving at a constant speed vv = ββ cc along a straight line parallel to the x-axis.. What are the scalar potential and vector potential created by this moving charge? 2. What are the electric field and magnetic field created by this moving charge? Spring 28 6

64 Hard Way 2 Φ rr, tt + μμ oo εε 2 Φ rr, tt tt 2 = ρρ rr, tt εε ρρ rr, tt = qq δδ xx + vv tt δδ yy yy δδ zz zz = qq δδ rr rr tt 22 AA rr, tt + μμ oo εε 2 AA rr, tt tt 2 = μμ oo JJ rr, tt JJ rr, tt = vv qq δδ xx vv tt δδ yy yy δδ zz zz = vv qq δδ rr rr tt Spring 28 64

65 Easier Way: Start with an inertial frame of reference K moving in the same way as the charge ρρ rr = qq δδ rrr rrr Φ rr = + ρρ rrrr 4 ππ εε rr rrrr ddvvvv electrostatic problem = 4 ππ εε qq rr rr Spring 28 65

66 JJ rrr = magnetostatic problem AA rrr = μμ + oo JJ rrrr 4 ππ rr rr dddddd = Spring 28 66

67 AA μμ II, = LL μμ αα AAA αα αα= LL αα II, μμ = γγ + γγ ββ + γγ ββ γγ AAA αα = Φ /cc AAA xx AAA yy AAA zz = Φ /cc AA μμ = γγ Φ /cc γγ ββ Φ /cc Φ rr, tt = γγ Φ = AA rr, tt = γγ ββ Φ /cc γγ qq 4 ππ εε rrr rrr = γγ qq vv 4 ππ εε cc 2 rrr rrr Spring 28 67

68 rrr rrr cc tt ttt xxx xxx yyy yyy zzz zzz = LL μμ αα xx αα αα xx = αα= γγ γγ ββ γγ ββ γγ cc tt tt xx xx yy yy zz zz = γγ cc tt tt γγ ββ xx xx γγ xx xx γγ ββ cc tt tt yy yy zz zz rrr rrr = xxx xxx 2 + yyy yyy 2 + zzz zzz 2 = γγ xx xx γγ ββ cc tt tt 2 + yy yy 2 + zz zz 2 Spring 28 68

69 tt = xx = Φ rr, tt = = γγ qq 4 ππ εε rrr rrr γγ qq 4 ππ εε γγ 2 xx vv tt 2 + yy yy 2 + zz zz 2 AA rr, tt = γγ qq vv 4 ππ εε cc 2 rrr rrr = γγ qq vv 4 ππ εε cc 2 γγ 2 xx vv tt 2 + yy yy 2 + zz zz 2 Spring 28 69

70 HW: From the previous expressions for the vector and scalar potentials, prove:. AA rr, tt + μμ oo εε Φ rr, tt = HW: Calculate the electric field of a charge moving at constant speed EE rr, tt = Φ rr, tt AA rr, tt HW: Calculate the magnetic field of a charge moving at constant speed BB rr, tt = AA rr, tt Spring 28 7

71 Metric Tensor and Invariants dddd 2 = cc 2 dddd 2 dddd 2 dddd 2 dddd 2 = ddxx 2 ddxx 2 ddxx 2 2 ddxx 2 ddxx = ddxx ddxx = ddxx ddxx 2 = ddxx 2 ddxx = ddxx dddd μμ = νν= gg μμμμ ddxx νν dddd 2 = νν= dddd νν dddd νν gg μμμμ = Spring 28 Metric Tensor 7

72 kk μμ = ωω cc kk xx kk yy kk zz kk μμ = νν= gg μμμμ kk νν = ωω cc kk xx kk yy kk zz μμ= kk μμ kk μμ = ωω cc 2 kk 2 2 xx kk yy 2 kkzz = cccccccccccccccc Spring 28 72

73 JJ μμ = cc ρρ JJ xx JJ yy JJ zz JJ μμ = νν= gg μμμμ JJ νν = cc ρρ JJ xx JJ yy JJ zz JJ μμ JJ μμ = cc ρρ 2 JJ 2 2 xx JJ yy 2 JJzz = cccccccccccccccc μμ= Spring 28 7

74 AA μμ = Φ/cc AA xx AA yy AA zz AA μμ = νν= gg μμμμ AA νν = Φ/cc AA xx AA yy AA zz AA μμ AA μμ = μμ= 2 Φ cc 2 2 AAxx AA yy 2 AAzz = cccccccccccccccc Spring 28 74

75 pp μμ = EE cc pp xx cc pp yy cc pp zz pp μμ = νν= gg μμμμ pp νν = EE cc pp xx cc pp yy cc pp zz pp μμ pp μμ = EE 2 cc pp 2 xx cc pp 2 xx cc pp 2 zz = cccccccccccccccc μμ= Spring 28 75

76 dddd νν = μμ= gg νννν ddxx μμ dddd νν = gg νννν dddd μμ μμ= dddd 2 = dddd νν dddd νν = gg ννμμ ddxx μμ dddd νν νν= νν= μμ= = νν= gg νννν dddd μμ dddd νν μμ= gg νννν = gg νννν = Spring 28 76

77 Tensors of Rank contravariant ddxx μμ gg νννν, gg νννν covariant dddd μμ xx μμ μμ gg νννν, gg νννν xx μμ μμ Spring 28 77

78 Contravariant Tensor of Rank 2 TT αααα VV αα WW ββ VV αα = xxx αα μμ= xx μμ VVμμ = μμ= LL μμ αα VV μμ xxx ββ WW ββ = xx νν WWνν = νν= νν= LL νν ββ WW νν xxx αα xxx ββ TTT αααα = VV αα WW ββ = μμ= xx μμ VVμμ νν= WWνν xxνν = μμ= LL μμ αα VV μμ νν= LL νν ββ WW νν = μμ= LL μμ αα νν= LL νν ββ TT μμνν tt ααββ = LL TT LL Spring 28 78

79 Electromagnetic Field Tensor xx αα = αα AA ββ = Φ/cc AA xx AA yy AA zz FF αααα αα AA ββ ββ AA αα FF αααα = FF ββββ FF αααα = Spring 28 79

80 Electric Field as a component of the electromagnetic field tensor EE rr, tt = Φ rr, tt AA rr, tt AA μμ = Φ/cc AA xx AA yy AA zz ii, 2, EE ii = Φ AAii xxii cc AA = xx ii cc AAii xx = cc AA AAii xxii xx EE ii cc = AA AAii xxii xx = ii AA AA ii = FF iii Spring 28 8

81 Magnetic Field as a component of the electromagnetic field tensor ii, jj, kk, 2, BB rr, tt = AA rr, tt BB ii = jj= kk= εε iiiiii AA kk xx jj = jj= kk= AA μμ = εε iiiiii jj AA kk Φ/cc AA xx AA yy AA zz Spring 28 BB = 2 AA AA 2 = FF 2 BB 2 = AA AA = FF BB = AA 2 2 AA = FF 2 8

82 Components of the Electromagnetic Field Tensor EE ii cc = FFiii BB = 2 AA AA 2 = FF 2 BB 2 = AA AA = FF BB = AA 2 2 AA = FF 2 FF αααα αα AA ββ ββ AA αα = cc EE EE 2 EE EE cccc ccbb 2 EE 2 ccbb cccc EE cccc 2 cccc Spring 28 82

83 How the Electromagnetic Field Tensor changes under a Lorentz transformation: FF ααββ = LL FF LLtt ααββ FF = cc EE EE 2 EE EE cccc ccbb 2 EE 2 ccbb cccc EE cccc 2 cccc LL = γγ γγ ββ γγ ββ γγ For a frame of reference K moving along x-axis at a constant speed vv = ββ cc Spring 28 8

84 FF = cc EE EE EE 2 EE ccbb ccbb 2 EE 2 EE ccbb 2 ccbb ccbb ccbb = LL FF LL tt EEE = EE EEE 2 = γγ EE 2 ββ cc BB BB = BB BB 2 = γγ BB 2 + ββ cc EE EEE = γγ EE + ββ cc BB 2 BB = γγ BB ββ cc EE2 Spring 28 84

85 Note: Galilean transformation leads to different and incorrect relations: vv : particle velocity with respect to K FF = qq EEE + qq vvv BBB = qq EE + qq vv BB vv = vv + vv = qq EE + qq vv + vv BB = qq EE + vv BB + qq vv BB EE = EE + vv BB BB = BB Spring 28 85

86 Electric and Magnetic Fields of a Moving Charge Spring 28 86

87 Consider a charge qq moving at a constant speed vv = ββ cc along a straight line parallel to the x-axis. What are the electric and magnetic fields created by this moving charge? Spring 28 87

88 Consider that the charge is at rest in the reference frame K EEE = qq 4 ππ εε xxx xxx xxx xxx 2 + yyy yyy 2 + zzz zzz 2 /2 BB = EEE 2 = qq 4 ππ εε yyy yyy xxx xxx 2 + yyy yyy 2 + zzz zzz 2 /2 BB 2 = EEE = qq 4 ππ εε zzz zzz xxx xxx 2 + yyy yyy 2 + zzz zzz 2 /2 BB = Spring 28 88

89 EEE = EE BB = BB EEE 2 = γγ EE 2 ββ cc BB EEE = γγ EE + ββ cc BB 2 BB 2 = γγ BB = γγ BB 2 + ββ cc EE BB ββ cc EE2 EE = EE BB = BBB EE 2 = γγ EE 2 + ββ cc BB BB 2 = γγ BBB 2 ββ cc EEE EE = γγ EE ββ cc BB 2 BB = γγ BBB + ββ cc EEE2 Spring 28 89

90 EE = EE BB = BBB EE 2 = γγ EE 2 + ββ cc BB BB 2 = γγ BBB 2 ββ cc EEE EE = γγ EE ββ cc BB 2 BB = γγ BBB + ββ cc EEE2 EE = EE EE 2 = γγ EEE 2 EE = γγ EEE BB = BB 2 = BB = γγ ββ cc EEE γγ ββ cc EEE2 = ββ cc EE = ββ cc EE2 Spring 28 9

91 EE = EE = qq 4 ππ εε xxx xxx xxx xxx 2 + yyy yyy 2 + zzz zzz 2 /2 EE 2 = γγ EEE 2 = qq 4 ππ εε γγ yyy yyy xxx xxx 2 + yyy yyy 2 + zzz zzz 2 /2 EE = γγ EEE = qq 4 ππ εε γγ zzz zzz xxx xxx 2 + yyy yyy 2 + zzz zzz 2 /2 Spring 28 9

92 BB = BB 2 = γγ ββ cc EE ββ = cc qq 4 ππ εε γγ zzz zzz xxx xxx 2 + yyy yyy 2 + zzz zzz 2 /2 BB = γγ ββ cc EEE2 = ββ cc qq 4 ππ εε γγ yyy yyy xxx xxx 2 + yyy yyy 2 + zzz zzz 2 /2 Spring 28 92

93 cc tt ttt xxx xxx = yyy yyy zzz zzz γγ γγ ββ γγ ββ γγ cc tt tt xx xx yy yy zz zz tt = ttt = xx = xx = cc tt ttt xxx xxx = yyy yyy zzz zzz γγ cc tt ββ xx γγ xx vv tt yy yy zz zz Spring 28 9

94 EE = qq γγ xx vv tt 4 ππ εε γγ 2 xx vv tt 2 + yy yy 2 + zz zz 2 /2 EE 2 = qq 4 ππ εε γγ yy yy γγ 2 xx vv tt 2 + yy yy 2 + zz zz 2 /2 EE = qq 4 ππ εε γγ zz zz γγ 2 xx vv tt 2 + yy yy 2 + zz zz 2 /2 Spring 28 94

95 γγ 2 xx vv tt 2 + yy yy 2 + zz zz 2 = ββ 2 xx vv tt 2 + yy yy 2 + zz zz 2 ββ 2 ssssss θθ 2 xx, yy, zz vv tt, yy, zz θθ dd xx vv ssssss 2 θθ yy yy 2 + zz zz 2 xx vv tt 2 + yy yy 2 + zz zz 2 dd xx vv tt, yy yy, zz zz Spring 28 95

96 EE rr, tt = qq dd ββ 2 4 ππ εε dd ββ 2 ssssss 2 θθ /2 classical Coulomb term The electric field points away from the charge at present time (t). However, it is not isotropic. relativistic correction θθ 9 ββ 2 ββ =.99 ββ =.9 ββ =.7 The electric field amplitude depends on the direction away from the vv θθ charge. ββ 2 The amplitude shows higher strength for directions perpendicular to the direction of propagation. Spring 28 ββ =. &.2 96

97 Spring 28 97

98 BB = BB 2 = ββ cc EE BB = ββ EE cc BB = ββ cc EE2 BB rr, tt = μμ 4 ππ qq vv dd dd ββ 2 ββ 2 ssssss 2 θθ /2 classical Biot-Savart term relativistic correction Spring 28 98

99 Maxwell s Equations in terms of the Electromagnetic Field Tensor FF αααα αα AA ββ ββ AA αα = cc JJ μμ = EE EE 2 EE EE cccc ccbb 2 EE 2 ccbb cccc EE cccc 2 cccc cc ρρ JJ xx JJ yy JJ zz sources fields Spring 28 99

100 The Four Inhomogeneous Maxwell s Equations. EE rr, tt = ρρ rr, tt εε Gauss s Law BB rr, tt μμ oo εε EE rr, tt = μμ oo JJ rr, tt Ampere s Law Spring 28

101 EE ii = cc FF iii. EE rr, tt = ρρ rr, tt εε ii= xx ii EE ii = xx ii cc FF iii = cc ii FF iii ii= ii= + cc FF = cc αα FF ααα αα= ρρ rr, tt = JJ cc = ρρ rr, tt εε = JJ εε cc αα FF ααα = μμ JJ αα= Spring 28

102 BB rr, tt μμ oo εε EE rr, tt = μμ oo JJ rr, tt jj= kk= BB kk εε iiiiii xx jj μμ EE ii oo εε = jj= kk= εε iiiiii jj BB kk EE ii cc cc tt = εε iiiiii jj BB kk + EE ii cc jj= kk= = μμ oo JJ ii Spring 28 2

103 jj= kk= εε iiiiii jj BB kk + EE ii cc = μμ oo JJ ii BB = FF 2 BB 2 = FF BB = FF 2 EE ii cc = FF ii ii = 2 BB BB 2 + FF = 2 FF 2 + FF + FF + FF = αα FF αα = μμ oo JJ αα= αα FF αα = μμ JJ Spring 28 αα=

104 jj= kk= εε iiiiii jj BB kk + EE ii cc = μμ oo JJ ii BB = FF 2 BB 2 = FF BB = FF 2 EE ii cc = FF ii ii = 2 BB BB + FF 2 = FF 2 + FF 2 + FF FF 22 = αα FF αα2 = μμ oo JJ 2 αα= αα FF αα2 = μμ JJ 2 Spring 28 αα= 4

105 jj= kk= εε iiiiii jj BB kk + EE ii cc = μμ oo JJ ii BB = FF 2 BB 2 = FF BB = FF 2 EE ii cc = FF ii ii = BB 2 2 BB + FF = FF + 2 FF 2 + FF + FF = αα FF αα = μμ oo JJ αα= αα FF αα = μμ JJ Spring 28 αα= 5

106 The Four Inhomogeneous Maxwell s Equations (Gauss s and Ampere s Laws) can be written as: αα= αα FF ααββ = μμ JJ ββ fields sources Spring 28 6

107 αα FF αααα = μμ JJ ββ FF αααα αα AA ββ ββ AA αα Lorenz gauge αα FF αααα = αα αα AA ββ ββ AA αα = αα αα AA ββ ββ αα AA αα = αα αα AA ββ = μμ JJ ββ αα αα AA ββ = μμ JJ ββ αα αα 2 d Alembertian, which is an invariant Spring 28 7

108 The Four Homogeneous Maxwell s Equations. BB rr, tt = Gauss s Law of Magnetism EE rr, tt + BB rr, tt = Faraday s Law Spring 28 8

109 . BB rr, tt = ii= BB = FF 2 BB 2 = FF BB = FF 2 xx ii BB ii = BB xx + BB2 xx 2 + BB xx = FF2 xx FF xx 2 FF2 xx = FF FF + FF 2 = FF FF + FF 2 = Spring 28 9

110 EE rr, tt + BB rr, tt = jj= kk= EE kk BBii εε iiiiii + cc xxjj cc = jj= εε iiiiii jj EE kk /cc + BB ii = kk= jj= kk= εε iiiiii jj EE kk /cc BB ii = Spring 28

111 jj= kk= εε iiiiii jj EE kk /cc BB ii = BB = FF 2 BB 2 = FF BB = FF 2 EE ii cc = FFiii ii = 2 EE /cc EE 2 /cc BB = 2 FF + FF 2 + FF 2 = Spring 28

112 jj= εε iiiiii jj EE kk /cc BB ii = kk= BB = FF 2 BB 2 = FF BB = FF 2 EE ii cc = FFiii ii = 2 EE /cc EE /cc BB 2 = FF + FF + FF = Spring 28 2

113 jj= kk= εε iiiiii jj EE kk /cc BB ii = BB = FF 2 BB 2 = FF BB = FF 2 EE ii cc = FFiii ii = EE 2 /cc 2 EE /cc BB = FF FF + FF 2 = Spring 28

114 ii = FF FF + FF 2 = 2 ii = 2 FF + FF 2 + FF 2 = 2 ii = 2 FF + FF + FF = 2 ii = FF 2 + FF FF = 2 Spring 28 4

115 The Four Homogeneous Maxwell s Equations (Gauss s Law of Magnetism and Faraday s Law) can be written as: αα FF ββββ + ββ FF γγγγ + γγ FF αααα = or εε δδδδδδδδ εε δδδδδδδδ αα FF ββββ = 2 Spring 28 5

116 Electromagnetic Theory JJ μμ = cc ρρ JJ xx JJ yy JJ zz AAμμ = Φ/cc AA xx AA yy AA zz FF αααα αα AA ββ ββ AA αα = cc EE EE 2 EE EE cccc ccbb 2 EE 2 ccbb cccc EE cccc 2 cccc αα FF αααα = μμ JJ ββ Spring 28 αα FF ββββ + ββ FF γγγγ + γγ FF αααα = 6

117 A Few Notes on Relativistic Mechanics and Field Theory Chapter 2 of Modern Problems in Classical Electrodynamics by Charles Brau Spring 28 7

118 Lagrangian of Discrete Particles LL = LL ii qq ii, qq ii ii dd ddtt LL qq ii = qq ii tt 2LL SS = qqii, qq ii tt ddtt Spring 28 8

119 Lagrangian of Continuous Fields and Particles: φφ kk : AA μμ & JJ μμ LL = LL ii qq ii, qq ii LL = L φφ kk, ββ φφ kk dddd ii dd ddtt LL qq ii = qq ii ββ L ββ φφ kk = L φφ kk tt 2LL SS = qqii, qq ii tt ddtt tt 2 SS = dddd L φφkk, ββ φφ kk tt dddd = cc tt tt 2L φφkk, ββ φφ kk dd 4 xx Spring 28 9

120 Relativistic Lagrangian of Fields and Particles L = 4 μμ FF μμμμ FF μμμμ JJ μμ AA μμ ββ L ββ AA αα = L AA αα HW: L ββ AA αα = μμ FF ββββ ββ L ββ AA αα = μμ ββ FF ββββ L AA αα = JJ αα ββ FF ββββ = μμ JJ αα Spring 28 2

121 Hamiltonian of Particles HH = ii pp ii qq ii LL ii qq ii, qq ii pp ii qq ii Spring 28 2

122 Relativistic Hamiltonian of Fields and Particles HH = ii qq ii qq ii LL ii qq ii, qq ii H αα ββ = L αα AA γγ ββ AA γγ gg αα ββ L = μμ FF ααγγ ββ AA γγ gg αα ββ 4 μμ FF μμμμ FF μμμμ JJ μμ AA μμ Spring 28 22

123 H αα ββ = μμ FF ααγγ ββ AA γγ gg αα ββ 4 μμ FF μμμμ FF μμμμ JJ μμ AA μμ = μμ FF ααγγ FF ββγγ + γγ AA ββ + 4 μμ FF μμμμ FF μμμμ gg αα ββ + JJμμ AA μμ gg αα ββ = μμ FF ααγγ FF ββγγ + 4 μμ FF μμμμ FF μμμμ gg αα ββ μμ FF ααγγ γγ AA ββ + JJ μμ AA μμ gg αα ββ = TT αα ββ μμ FF ααγγ γγ AA ββ + JJ μμ AA μμ gg αα ββ TT αα ββ μμ FF ααγγ FF ββγγ + 4 μμ FF μμμμ FF μμμμ gg αα ββ Spring 28 2

124 TT αα ββ μμ FF ααγγ FF ββγγ + 4 μμ FF μμμμ FF μμμμ gg αα ββ TT ααββ = uu ccgg xx ccgg yy ccgg zz ccgg xx TT xxxx TT yyxx TT zzzz ccgg yy TT xxyy TT yyyy TT zzzz ccgg zz TT xxzz TT yyyy TT zzzz ββ = αα TT αα = uu+ ccgg xx + 2 ccgg yy + ccgg zz = cc uu tt +. SS ββ = ii αα TT ααii = ccgg ii + jj TT jjjj Spring 28 24