Electromagnetic. G. A. Krafft Jefferson Lab Jefferson Lab Professor of Physics Old Dominion University Physics 804 Electromagnetic Theory II

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1 Physics 704/804 Electromagnetic Theory II G. A. Krafft Jefferson Lab Jefferson Lab Professor of Physics Old Dominion University

2 4-Vectors and Proper Time Any set of four quantities that transform as the coordinate displacements is called a (contravariant) 4-vector ( ) = γ ( ) = γ = = 3 3 proper (invariant!) time r r d τ = 1 v v / c dt

3 Other 4-vectors velocity 4-vector r dt d r U = c, = c γ, γ dτ dτ U U = c ( ) Energy-momentum 4-vector p = mu E r r p p = m c = m v v c r 4 E = p c + m c

4 Invariance of wave phase r r r r ω t k = ωt k r ( ω, kc ) for photons is a 4-vector ( kc z ) ( ω / ) ω = γ ω k = γ k c z k = k = y k k y z relativistic Doppler Shift ( 1 cos ) ω = γω θ sinθ sinθ tan θ = = cosθ γ cosθ ( )

5 Tensor Analysis Lorentz Transformations leave invariant the space-time interval (, ) = ( ) ( ) ( ) ( ) ds y y y y y and more generally the Lorentz scaler products of 4-vectors v w = v w v w v w v w Goal in Special Relativity; write laws of physics in terms of quantities with specific Lorentz transformation properties, for eample invariants, 4-vectors, 4-tensors, etc. ( 0 1 3,,, ) = d = d Einstein i summation convention on

6 Contravariant (4)-vectors: Transform like coordinate differentials A = A Eamples: 4-velocity, energy-momentum 4-vector, k for photons Not only possibility. Consider the gradient operator = uses inverse matri to contravariant vectors. Quantities transforming similarly to gradiant called covariant vectors B = B

7 Second Rank 4-Tensors Contravariant rank two tensors transform like the outer product of two contravariant (4)-vectors: γ F = F γ Covariant rank two tensors transform like theouter product of two covariant (4)-vectors: γ H = H γ Mied rank two tensors transform like: γ H = H γ Not necessarily true that H = H

8 Tensor Contraction Summing upper and lower indices leaves tensor with rank two lower. For mied tensor yields an invariant γ γ H = H = H = H γ γ to the linear algebra theorem that the trace of a matri is invariant under similarity transformations Tr ( 1 SMS ) = Tr ( M ) Similar eample: inner product is outer product of covariant and contravariant components followed by a contraction γ v w = v w = v w = v w γ v w = = v w γ γ

9 Write invariant interval Metric Tensor ( ) ( ) ( ) ( ) ( ) ds = d d d d = g d d µ ν g gives the scalar product by v w = g v w. µν µν µν µ ν Because v w = w v, g is symmetric. We'll use Minkowski metric g µν g µν = inverse matri 1 = γ g g γ = γ, gγ g = γ γ = γ = 0 γ

10 Inverse Matri g µν = metric may be used to provide an association between contravariant and covariant components r = g = 0, F = g F ( ), r = 0 0, r =

11 Lorentz Invariant Operators 4di 4-divergence 0 A r r A = A = + A 0 wave operator = 0 ( ) define column vector consisting of contravariant components X = as a matri equation gx = 3 3

12 Note ggx = g X = IX = X If ( X, Y ) t X Y Lorentz Group The Lorentz scalar product is (, ) (, ) t a b = a gb = ga b = a gb Elements of the Lorentz group of matrices leavethe Lorentz scalar product invariant X = LX Y = LY t t t ( ) for a X Y = LX gly = X L gly = X Y ll X, Y if t LgLL = g as a matri ti equation

13 Now Components of the Lorentz Group ( t ) ( t) LgL = L ( g) ( L) = ( L) ( g) Det Det Det Det Det Det ( L ) Det =± 1 Lorentz Group has 4 components + Det + parity usual component with identity + Det parity time reverse + parity Det parity usual parity Det + parity time reverse Component containing identity may be constructed from infinitesimal i i generators (like all llli Lie groups)

14 t ( 1 ) ( 1 ) Infinitesimal Generators = t E g E g E g ge HOT g ( ) t t E g + ge = 0 ge = ge ge antisymmetric 6 Possibilities S1 = S = S3 = K 1 = K = K3 =