Relativistic Electrodynamics An inertial frame = coordinate system where Newton's 1st law of motion - the law of inertia - is true. An inertial frame moves with constant velocity with respect to any other inertial frame. Special Relativity 1. Postulate of relativity. The laws of physics are the same in all inertial frames. 2. Postulate of the absolute speed of light. The speed of light in vacuum is the same in all inertial frames. Overthrows a basic assumption of classical physics - that lengths and time intervals are absolute quantities, i.e., the same for all observes. The Lorentz Transformations (follow from postulate 2) This is the relativistic transformation between inertial frames. If inertial frame F' (x',y',z',t') is moving with constant velocity vî relative to another inertial frame F (x,y,z,t) then x' = γ (x vt), y' = y, z' = z, t ' = γ (t vx / c 2 ), γ = 1 1 v2 c 2 1
We then have c 2 t ' 2 x' 2 y' 2 z' 2 = c 2 t 2 x 2 y 2 z 2 = invariant This quantity is the spacetime interval. All theories must also satisfy postulate 1 under the Lorentz transformations, i.e., the equations have the same form in different inertial frames. The property is called covariance. Minkowski Space 4-Vectors, Scalars and Tensors Minkowski space is the 4-dimensional spacetime. Define the 4-vector x µ by x µ = x 0 x 1 x 2 x 3 = ct x y z The Lorentz transformation Greek indices( µ ) take values 0,1,2,3. x µ x' µ is linear, i.e., x' µ = Λ µ ν x ν, Λ µ ν = matrix of constants We are also using Einstein summation convention. Thus, ct ' = x' 0 = Λ 0 ν x ν = Λ 0 0 x 0 + Λ 0 1 x 1 + Λ 0 2 x 2 + Λ 0 3 x 3 = Λ 0 0 ct + Λ 0 1 x + Λ 0 2 y + Λ 0 3 z 2
We have Λ µ ν = Vectors and Tensors γ βγ 0 0 βγ γ 0 0 0 0 1 0 0 0 0 1, β = v c Lorentz vectors and tensors are defined with respect to Lorentz transformations in the same way that 3-dimensional vectors are defined with respect to rotations. A 4-vectors way as A tensor x µ T µν a µ = set of 4 components that transform in the same under Lorentz transformations, a' µ = Λ µ ν a ν as x µ x ν, T ' µν = Λ ρ = set of 16 components that transform in the same way µ Λ ν σ T ρσ aµ It is useful to define a quantity and a µ = g µν a ν 1 0 0 0 0 1 0 0 g µν = 0 0 1 0 0 0 0 1 where = metric tensor 3
in special relativity. This gives a µ = a 0 a 1 a 2 a 3 a 0 a 1 = a 2 a 3 In this formalism, there are two types of 4-vectors - a 4-vector with an upper index is called a contravariant vector and on with a lower index is called a covariant vector. This formalism with two types of vectors replaced the older formalism where one coordinate was an imaginary number (ict). In a similar way, T µν = g µρ g µσ T ρσ T 00 = T 00, T ij = T ij, T 0i = T 0i, T i0 = T i0 where we use Roman indices (i,j,k,...) which take the values 1,2,3. We also have a' µ = a ν ( Λ 1 ) ν µ The Lorentz Product The Lorentz product of two vectors (scalar product in Minkowski space) is defined It is easy to show that a b = a µ b µ = a µ b µ = g µν a µ b ν = a 0 b 0 a b 4
a b a' b' = a b i.e., is a Lorentz scalar or it is invariant under Lorentz transformations. This is the same as the invariance of the spacetime interval, i.e., x x = x µ x µ = x µ x µ = g µν x µ x ν = x 0 x 0 x x = c 2 t 2 x 2 y 2 z 2 is invariant. Kinematics of a Point Particle Worldline = trajectory of particle (P) in Minkowski τ cdτ = dx µ dx µ = c 2 (dt) 2 (dx) space. Proper time of P = 2, where dx µ is the infinitesimal displacement along the worldline. dτ and τ thus are Lorentz scalars. u = dx If we let / dt coordinate in F, then dτ = dt 1 u 2 / c 2 = velocity of P in frame F, where t = time Clearly, dτ is the time interval experienced by P in its rest frame Fp as P moves by in F, i.e., since. 4-velocity and 4-momentum dx µ dτ = dt p u p = 0 Define 4-velocity: η µ = dxµ dτ η0 = γ c, η = γ u 5
Since it is a 4-vector we have η' µ = Λ µ ν η ν Define 4-momentum: p µ = mη µ where m = rest mass, which is a Lorentz scalar. This is a 4-vector also. Energy and Momentum We have or p µ = p 0 p 1 p 2 p 3 = E / c p x p y p z E = mη 0 c = γ mc 2, p = γ m u E 2 = p 2 c 2 + m 2 c 4, β = u c = pc E m = 0 E = pc These definitions can actually be derived by using postulate 1 with respect to conservation of energy and momentum in all inertial frames, i.e., µ P' final = Λ µ ν ν P final = Λ µ ν ν P initial or conservation laws are covariant. µ = P' initial 6
Relativistic Dynamics The way to write the equations of physics that guarantees covariance is to write the equations as relations involving only 4-vectors, scalars and tensors. For example, if an equation has the form (in F) T 1 µν = T 2 µν then in F' it has the same form, i.e., T ' 1 µν = Λ ρ µ Λ σ ν T 1 ρσ = Λ ρ µ Λ σ ν T 2 ρσ = T ' 2 µν The Equation of Motion of a Particle in Covariant Form We write(in covariant form) dp µ where dτ = K µ = Minkowski force K = d p dτ = d p / dt dτ / dt = γ F, K 0 = 1 c Electromagnetism in Covariant Form de dτ = γ The Lorentz Force and the Field Tensor In any particular frame F, the equation ( E, B) of motion of a charge q moving under the influence of fields is F u c 7
F d p dτ = F = q E + q u B The force is called the Lorentz force. We want to write this as a tensor equation, i.e., we want to identify the proper K µ that gives this equation of motion. We have and K = γ F = γ qe + qu B F u ( ) = qη 0 E c + q η B K i = qη 0 E i K 0 = γ c = q η E c c + qε ijkη j B k The brilliant insight of Minkowski was to combine ( E, B) in an antisymmetric Lorentz tensor - the electromagnetic field tensor, denoted by and defined by F µν (x) F ij = ε ijk B k, F 0i = F i0 = Ei The field tensor is antisymmetric, i.e., matrix we have F µν = c, F 00 = 0 0 E x / c E y / c E z / c E x / c 0 B z B y E y / c B z 0 B x E z / c B y B x 0 F µν = F νµ. Written as a 4X4 8
In terms of the field tensor we have K µ = qη ν F µν η µ K µ Since the 4-velocity and the Minkowski force are Lorentz vectors, F µν must also be a tensor (quotient law). Just as the Lorentz product of two vectors similar product of a vector and a tensor a ν T µν we have the covariant equation of motion dp µ dτ = K µ = qη ν F µν a ν b ν is a scalar, the is a 4-vector. Thus, One can also define another antisymmetric tensor called the dual field tensor, as G µν = 1 2 ε µναβ F αβ ε µναβ where is the completely antisymmetric tensor in 4 dimensions, i.e., ε µναβ = 0 unless all 4 indices are different and ε µναβ is +1 or -1, respectively, if µναβ is an even or odd permutation of 0123 so that 0 B x B y B z B x 0 E z / c E y / c G µν = B y E z / c 0 E x / c B z E y / c E x / c 0 9
Maxwell's Equation in Covariant Form We have a 4-vector current cρ J J µ x = J y J z The continuity equation (conservation of charge) in covariant form is J µ x µ = µj µ = 0 Clearly, x µ = µ transforms as a covariant vector so that µ J µ is a Lorentz scalar. This corresponds the the old form of the continuity equation J = ρ t Two of the Maxwell equations relate fields and sources - Gauss's law(a scalar equation with respect to spatial rotations) and the Ampere-Maxwell law(a vector equation with respect to spatial rotations). The two equations together make one covariant 4-vector equation 10
F µν x ν = ν F µν = µ 0 J µ corresponding to E = ρ, B 1 E ε 0 c 2 t = µ 0J The other two Maxwell equations involve only the fields, not charge. With respect to spatial rotations, these equations are a scalar equation (Gauss law for B ) and a vector equation (Faraday's law). With respect to Lorentz transformations, the two equations make one covariant vector equation G µν x ν = ν G µν = 0 Thus, the Maxwell equations reduce to the very compact form ν F µν = µ 0 J µ, ν G µν = 0 indicating there is something very deep going on here! The 4-vector Potential The 4-vector potential is given by A µ = V(x,t) / c A x A y A z 11
with F µν = µ A ν ν A µ corresponding to E = V A Transformation Rules t, B = A Using the relation F ' µν = Λ ρ µ Λ ν σ F ρσ we can derive the transformation rules for ( E, B). Summarizing all transformation rules we have: x' = γ (x vt), y' = y, z' = z, t ' = γ (t vx / c 2 ) p' x = γ (p x ve / c 2 ), p' y = p y, p' z = p z, E ' = γ (E vp x ) E ' x = E x, E ' y = γ (E y vb z ), E ' z = γ (E z + vb y ) B' x = B x, B' y = γ (B y + ve z / c 2 ), B' z = γ (B z ve y / c 2 ) The field transformations mix electric and magnetic fields. In special relativity, (t,x,y,z) combine ( E, into B) a 4-vector, (E,px,py,pz) also combine into a 4-vector, while combine into the field tensor. 12
Magnetism is a relativistic effect - the Lorentz transformation implies the existence of a velocity dependent force on q, perpendicular to the velocity in any frame where the charge sources are in motion. As we have seen, relativity(covariance) demands certain interactions between particles and fields. The Energy-Momentum Tensor The covariant form of electromagnetic theory is ideal for analyzing energy and momentum because these quantities are combined by special relativity. Derivations are very complex, but we will only need the results. The equation of motion for a single point charge is dp µ dτ = qη νf µν We can generalize this equation, to describe any charge J µ (x) (cρ, J distribution in a volume. Let be the 4-vector current ). The Minkowski force on the charge in a volume d 3 x is obtained by the substitution qη ν J ν d 3 x Therefore, if µ (x) denotes the momentum density of the charged matter, then the equation of motion is 13
µ (x) τ = J ν F µν (01) i.e., J ν F µν is the rate of change of 4-momentum, per unit volume, of the charged matter, with respect to proper time. Local energy and momentum conservation implies that µ / τ can be written as a divergence in Minkowski spacetime, i.e., where field. T µν T µν J ν F µν = x µ is the energy-momentum flux tensor of the electromagnetic Equation (01) is a kind of continuity equation. The standard continuity equation ν J ν = 0 expresses conservation of charge, a scalar quantity. In the absence of charge, i.e., for J ν = 0, the equation T µν / x µ = 0 expresses energy and momentum conservation - a 4-vector quantity - of the electromagnetic field. In the presence of charged matter, there may be transfer of energy and momentum between the field and the matter. Then the local conservation of energy and momentum is described by (01). To understand this equation and learn the interpretation of T µν, let us investigate the components of (01). The 0 component is J E = 1 T 00 0i T (02) c c t x i 14
J E is the rate at which work is done, per unit volume, on the charged matter at x. In order for equation (02) to be a statement of conservation of energy, T 00 / t must be the rate of change of field energy, per unit volume, i.e., T 00 is the field energy density. Also, (ct 0i ) / x i must be the divergence of the energy flux, i.e., ct 0i is the Poynting vector. In other words, (02) is just Poynting's theorem expressing the local conservation of energy. The spatial components of (01) are T i0 ρe i + ε ijk J j B k = 1 (03) c t x j Equation (03) must express local conservation of 3-momentum. The LHS is the force per unit volume acting on the charged matter, i.e., the rate of change of its 3-momentum density. In order for (03) to be a statement of conservation of momentum, (T i0 / c) / t must be the rate of change of field momentum density, that is, T i0 / c is the field momentum density. Also, T ij / x j must be the divergence of the momentum flux. With these interpretations for the three parts of (03), the rate of change of momentum in a small volume at x equals the total flux of momentum into the volume. T ij (x) is the flux (momentum per unit area per unit time) in the j th direction, of the i th component of momentum. Now that we have identified the meaning of T µν, we must write T µν in terms of the electromagnetic field F µν. T ij 15
The defining equation is (01). It turns out that T µν is given by T µν = 1 F µρ F ν ρ 1 µ 0 4 gµν F ρσ F ρσ (04) Since g µν is a tensor, this is a covariant equation. A little bit of algebra shows that (04) does satisfy the continuity equation (01). We get T µν = 1 F F ν µρ ρ x ν µ 0 x 1 F ρσ µρ F + + Fσµ ν 2 x µ x σ x ρ where we have used g µν x = ν gµν ν = µ = ν F ρ, x µ x = µ J ν 0 ρ The square bracket is zero using Maxwell's equations. Now we can go back to the interpretation of T µν and ( E, B) write the energy and momentum density and flux in terms of. The field energy density is u = T 00 = 1 E 2 µ 0 2c + B2 2 2 which is correct. The energy flux is S = ê i ct 0i = 1 E B µ 0 which agrees with the standard Poynting vector. 16
The electromagnetic field carries momentum as well as energy. The field momentum density must be T i0 / c as discussed above. By (04) the energy-momentum flux is symmetric - T µν = T νµ. Therefore, T 0i = T i0 = S i / c. Thus, Π( x,t) S( x,t) the momentum density in the electromagnetic field is / c 2. 17