Special relativity in electromagnetism M.C. Chang Dept of Phys Special relativity in electromagnetism Supplement to Jackson Chap 11 Refs: Special relativity, by A.P. French Electromagnetic fields and waves, by Lorrain and Corson

Special relativity in classical mechanics (I) x y z x’ y’ z’ S S’ v

Special relativity in classical mechanics (II) x y z x’ y’ z’ S S’ v

Transformation of electric force (I) x y z x’ y’ z’ S S’ q q v Vector notation x y z a central force in one frame is no longer a central force in another Magnetic-like force exists for all moving central force, such as gravity!

Transformation of electric force (II) x y z x’ y’ z’ S S’ q q v Vector notation

The force between two moving charges Force exerted by qa on qb θa x y z S qa qb r vb va θb Force exerted by qb on qa just switch the subscripts a and b and let r → -r

Electric field of a moving charge θ x y z S q r u

Transformation of electric and magnetic field From the transformation of force, we can obtain the transformation of E and B field The result is: From which we can also get That is, A=(ψ, A) also forms a 4-vector!

Motion in uniform E and B field E⊥B Can remove either E or B by jumping on a different frame x’ y’ z’ B’ S’ x y z E B S E<B E×B drift (～ Hall effect) x’ y’ z’ S’ E’ x z E B E>B S y E not ⊥ B Cannot remove either E or B by jumping on a different frame Note: Show that E2–B2 and E．B are Lorentz invariants

Electric charges on a straight line Static charges x y z Charge density λ b … … + + + + + + + + + + + + + + + + + + + + Moving charges x y z Charge density λ b … … + + + + + + + + + + + + + + + + + + + +

A straight current wire x y z Charge density λ b … - - - - - - - - - - - - - - - - - - - - … + + + + + + + + + + + + + + + + + + + + electric fields from positive and negative charges are cancelled magnetic field , consistent with Ampere’s law typically, v～1 mm/sec, ∴ β～10-12 It’s tiny but crucial. Special relativity can be important at low velocity too! (see below for another example)

Shockley-James paradox (Shockley and James, PRLs 1967) A simpler version (Vaidman, Am. J. Phys. 1990) A charge and a solenoid: E B S q Poynting vector

Resolution of the paradox: needs to consider relativistic effect Penfield and Haus, Electrodynamics of Moving Media, 1967 S. Coleman and van Vleck, PR 1968 Jackson, Classical Electrodynamics, the 3rd ed. A stationary current loop in an E field Smaller mass m Gain energy Lose energy momentum flow “hidden momentum” E Larger mass Force on a magnetic dipole (Jackson, p.189) magnetic charge model current loop model

Gravitomagnetism (Heaviside 1893) Valid for weak field, low velocity Electromagnetic wave → gravity wave Frame-dragging effect (Lense andThirring, 1918) … A rotating stick can generate gravity wave And many more …

Relativistic notations 4-vectors Space-time (ct, x) energy-momentum (E/c, p) scalar potential-vector potential (φ, A) 4-dim space-time operator (below) charge density-charge current (below) … but not the usual velocity, acceleration, force…

Covariant 4-vector and contravariant 4-vector Vectors that transform like x are called contravariant vectors; Vectors that transform like are called covariant vectors. Their notations are distinguished by the position of the index. inverse Raising and lowering of the index g (called a metric tensor) converts a contravariant vector to a covariant vector, and vice versa

Inner product between two 4-vectors The inner product is invariant under Lorentz transformation because and transform oppositely. Relativistic invariants Conversely, any linear transformation that leaves x2 (or other inner product) invariant must be a Lorentz transformation (including spatial rotation). Analogy: any linear transformation that leaves |x|2 (or other inner product) invariant must be a rotation.

Covariant form of the electromagnetic field No relation with the same word in “covariant vector” Covariant form of the electromagnetic field Transformation of the field strength tensor or

local time, or “propre” time Covariant form of the Maxwell equations Covariant form of the Lorentz force equation The usual velocity v is not part of a 4-vector since t is not invariant under the Lorentz transformation 4-velocity local time, or “propre” time

Relativistic electrodynamics Exactly the same as Maxwell’s electrodynamics!