special relativity notes

1Lecture 9 Lecture 9 Covariant electrodynamics Covariant electrodynamics WS2010/11 WS2010/11 : : ‚‚Introduction to Nuclear and Particle Physics Introduction to Nuclear and Particle Physics ‘‘ 2Consider Lorentz transformations – pseudo-orthogonal transformations in 4-dimentional vector space (Minkowski space) 4-vectors: with contravariant components (upper indices) Note: covariant components (lower indices) are defined by 1. Lorentz Group 1. Lorentz Group The transformation between covariant and contravari ant components is z x y x x x ct x −− −−== == −− −−== == −− −−== == == ==3 2 1 0 , , , (1) (2) (3) (4)



















            == == 4321 4 3 2 1 ),,,( xxxx xxxxTNotation: T = transposition 31. Lorentz Group 1. Lorentz Group Pseudometric tensor : (5) For any space-time vector we get: where the vector x’ is the result of a Lorentz transformation ΛΛ ΛΛµµ µµ ν ν ν ν :: :: such that (6) (7) (8) (9) •Pseudo-orthogonality relation: which implies that the pseudometric tensor is Lorentz invariant ! 4For the transformation in x1 direction with velocity the transformation matrix is given by where , and 1 4is the 4x4 unitary matrix



1. Lorentz Group 1. Lorentz Group In matrix form: The inverse Lorentz transformation reads: (10) (11) (12) (13) (14) From (9), (6) we obtain: with determinant (15) 5Properties of Lorentz transformations: 1) Two Lorentz transformations applied successively appear as a single Lorentz transformation. In matri x form: 2) The neutral element is the 4x4 unitary matrix 1 4 for a Lorentz boost with υ υ υ υ =0 3) For each ΛΛ ΛΛexists the inverse transformation : 4) The matrix multiplication is associative



the Lorentz transformation is associative also, ( i.e. the order in which the operations are perfor med does not matter as long as the sequence of the operands is not changed.)1. Lorentz Group 1. Lorentz Group (16) (17) (18) (19) (20) (21) 62. Lorentz Group: scalars, vectors, tensors 2. Lorentz Group: scalars, vectors, tensors 1) Lorentz-Scalars transform as: Lorentz transformation ΛΛ ΛΛ (22) ΨΨ ΨΨ: e.g. electric charge 2) Lorentz-Vectors are defined by the transformation properties as: (23) (24)  Transformation of covariant components: 72. Lorentz Group: vectors, tensors 2. Lorentz Group: vectors, tensors 3) Lorentz-Tensor (25) (26) (27) Transformation of the covariant components of a Lor entz tensor is defined as:  Transformation of contravariant-covariant component s: 4) Higher tensor products (Kronecker products) (28) e.g. tensor of 3rd order: A µ µ µ µ is a vector-component Tνν ννρ ρ ρ ρ is a tensor-component 82. Lorentz Group: vectors, tensors 2. Lorentz Group: vectors, tensors 5) Covariant trace (29) (30) (31) Consider e.g. the trace of a tensor of 2nd order: After Lorentz transformation it has to transform as a scalar: 6) Fundamental Levi-Civita-Pseudo-tensor (or permut ation tensor) if (µ,ν,ρ,σ) (µ,ν,ρ,σ) (µ,ν,ρ,σ) (µ,ν,ρ,σ) is an even permutation of (0,1,2,3) if (µ,ν,ρ,σ) (µ,ν,ρ,σ) (µ,ν,ρ,σ) (µ,ν,ρ,σ) is an odd permutation of (0,1,2,3) in all other cases (32) Pseudo-tensor, since After Lorentz transformation we get: (33) Cf. Eq.(27) 92. Lorentz Group: vectors, tensors 2. Lorentz Group: vectors, tensors  Covariant differentiation: (34) (35) (36) (37) ΨΨ ΨΨ’ is a Lorentz scalar and the 4-derivative is a covariant 4-vector : The 4-divergence of a Lorentz vector is a Lorentz scalar: (38) Since Note: the total derivative is a scalar: 10 2. Lorentz Group: vectors, tensors 2. Lorentz Group: vectors, tensors If one introduces the 4-vector (39) (41) (40) Thus the D’Alembert-Operator is also invariant under Lorentz transformations , i.e. (42) (43) In general: The scalar product of two Lorentz vectors is a Lore ntz scalar: ít follows from (38): 11 The continuity equation: New notation: ρis the charge density and j1, j 2,j 3are the 3-components of the currect In covariant notation we get: •In a system with a charge distribution at rest we have: As a covariant component of a 4-vector, jµµ µµ‘must be also invariant under Lorentz transformations with velocity in x1-direction :3. Vector current 3. Vector current (44) (45) (46) (47) (48) and transformes as with (49) 12 3. Vector current 3. Vector current The Lorentz transformation of the volume from frame ΣΣ ΣΣto ΣΣ ΣΣ‘ reads: (50) (51) One obtains in any frame the charge invariance: Q= 13 4. The four 4. The four --potential potential Consider the vector potential A and scalar potential ΦΦ ΦΦ (52) (53) The Lorentz gauge reads: The wave equation: One can also write in covariant notation :(54) (55) (56) using that Then Eq. (53) reads: The Lorentz covariance of Eq. (52) can be also written in the form: (57)
0is the electric constant, 0is the magnetic constant. 14 4. The four 4. The four --potential potential Consequences: •Lorentz invariance of the continuity equation and ga uge condition: If in the frame ΣΣ ΣΣ then in the frame ΣΣ ΣΣ‘ we also have (57) (58) Since eq. (53) is covariant, the same result will b e in Σ Σ Σ Σ and ΣΣ ΣΣ‘ (59) (60) thus This is the covariance of classical electrodynamics ! 15 5. Plane 5. Plane --waves waves (61) Consider plane-waves in vacuum in the inertial system Σ: Σ: Σ: Σ: (62) (63) Due to the covariance of the homogenuous wave equation a transformation to another system ΣΣ ΣΣ’ gives: (64) Accordingly, the phase of the plane wave should be also invariant: as in case of a point-like excitation, where the wa ve fronts are spherical surfaces moving with the velocity of light c. (65) with 16 5. Plane 5. Plane --waves waves (66) Since the phase in Eq. (64) is an invariant scalar p roduct, the k
must be the covariant components of a 4-vector , which transform as: (67) Using the dispersion relation: (68) (69) and denoting the angles φφ φφand φφ φφ with the direction of k and k  with respect to the direction of v (i.e. the x-direction in case of (28)) we obtain: 17 5. Plane 5. Plane --wave wave (70) Equation (68) describes the Doppler-effect , which apart from a longitudinal effect , (71) for  << 1 and φφ φφ= 0 , , also implies a transversal effect for which is a typical relativistic phenomenon. This ef fect was shown experimentally in 1938 within the investigation of radiation from m oving H-atoms. Well known is the longitudinal effect in the radiation of distant galaxies ( red shift ) which demonstrates that these objects are moving away fro m us. Furthermore, both phenomena describe the relativistic aberration of light , i.e. the apparent change of position of light sources due to the motion of the earth around the sun relative to stars. 18 6. Transformation of the fields E and B 6. Transformation of the fields E and B Once knowing the fields A and ΦΦ ΦΦ, one can compute the fields E and B via (72) (73) (74) Let‘s rewrite Eq. (72) in covariant form with coordi nates xand the components of the 4-potential A. For example we obtain: Eq. (73) suggests to introduce the antisymmetric field-strength tensor of second rank (75) 19 6. Transformation of the fields E and B 6. Transformation of the fields E and B (76) (77) The contravariant components are (78) Now we know the transformation properties of the fi elds E and B since the contravariant components transform as (25) For the special Lorentz boost (14) we obtain: transformation in x1 direction with velocity (14): 20 6. Transformation of the fields E and B 6. Transformation of the fields E and B (79) (80) (81) For a Lorentz-Boost with velocity vvin arbitrary direction holds that the parallel components (in direction of v) are conserv ed : while the transverse components transform as: The inversion is obtained – in analogy to the coordinate transform ation - by replacing v  −v. Equations (80) and (81) demonstrate the connection b etween the electromagnetic fields E and B in different frames. 21 7. 7. Maxwell equations Maxwell equations Now we can rewrite the Maxwell equations for the ele ctromagnetic field in covariant form. We focus on the case of the vacuum and recall the Maxwell equations in conventional notation: (82) (83) The equations (82) are the homogenuous Maxwell equations. They can be fulfilled by introducing scalar and vector potentia ls. The equations (83) describe the ‚ creation‘ of the fields from electric charges and currents . As seen before these equations can be written in covariant form of 4-tensor structure. The component s of the field strength appear in the field-strength tensor (76), i.e. we h ave to express the equations in terms of this tensor. 22 7. 7. Maxwell equations Maxwell equations and where († F)
is the dual tensor to F
. The first equation we can evaluate by writing the field-strength tensor in the form (84) via the potentials and take care of the Lorentz gauge : (84) (85) (86) These equations apparently correspond to the inhomogenuous Eqs. (83). When rewriting the dual tensor in terms of the potential s we obtain: By calculating the 4-divergence of (86) we find tha t due to the antisymmetry of the Levi-Civita tensor and the exchange p roperties of the derivatives we get: (87) Indeed, we can construct two independent covariant equations with first order derivatives: 23 7. 7. Maxwell equations Maxwell equations where (x) is an arbitrary scalar field . In fact, the field-strength tensor does not depend on such gauge transformations as seen from The advantage of the Maxwell equations (85) and (87) is that they are gauge independent. Indeed, a change in the gauge for the potentials im plies This implies that the 4-potentials A
und A˜
describe the same physical fields. The invariance of (85) and (87) with respect to gauge t ransformations (88) is the starting point of the standard model of elementary particle physics .(88) (89) 24 8. Coulomb field 8. Coulomb field (90) The field of a charge qat rest in the frame ΣΣ ΣΣ is: (91) In a frame moving with velocity v = ( v, 0, 0) relative to ΣΣ ΣΣwe obtain: Here x, y , z  has been written explicitly as a function of x, y, z using the Lorentz transformation of the coordinates. The field appears as in ΣΣ ΣΣ as a central field but is no longer isotropic. The factor 2in the square root differentiates the x- direction relative the y- and z-direction. 25 8. Coulomb Field 8. Coulomb Field (93) since the charge q is moving and thus generating a current . For illustration we consider the case  >> 1: i) Close to the x-axis ( y, z  0) we get which implies a reduction of the field strength by a factor −2. ii) In the plane parallel to the y − z-plane through q we get: (94) which implies an enhancement of the transverse field strength by a factor of  .



The radial field lines are thus thinner in the dire ction of motion whereas they are enhanced in transverse direction: An observer measures also the magnetic field: (92) 26 Covariant electrodynamics Covariant electrodynamics Summary: The basic equations of electrodynamics are covarian t with respect to Lorentz transformations and have th e same form in all inertial systems thus following th e Einstein principle of relativity.