Finite size corrections to the classical radiation reaction. Tams - PowerPoint PPT Presentation

Finite size corrections to the classical radiation reaction. Finite size corrections to the classical radiation reaction. Tamás Herpay(KFKI-RMKI) Collaborators: Péter Forgács, Péter Kovács Outline • Overview of the radiation reaction of a point-like charged particle, regularization of the infinite electromagnetic self-energy • The rigid model for an extended charge distribution. • Relativistic multipole expansion of the four-current. • Regularization and renormalization of a charged point-like dipole. • The equation of motion of a spherically symmetric rigid charge distribution. • Summary

Finite size corrections to the classical radiation reaction. Radiation reaction or self-interaction, Abraham–Dirac–Lorentz equation External Force e− Acceleration γ Lorentz Force Maxwell Eqs. γ z( ) τ Radiated Field Equation of motion including back-reaction force E + e B + 2 a = e � v × � 3 e 2 ˙ Non-relativistic limit: m � c � � a e 2 Covariant equation: ma µ = e cF µν v ν + 2 � g µν + v µ v ν � ˙ a ν 3 c Abraham–Dirac–Lorentz equation for a point-like charge

Finite size corrections to the classical radiation reaction. Radiation reaction or self–interaction, Abraham–Dirac–Lorentz equation External Force g µν = diag ( − 1 , 1 , 1 , 1) Minkowski metric: Acceleration z ( τ ) World-line: v µ ( τ ) = ˙ z µ ( τ ) Four-velocity: Lorentz Force Maxwell Eqs. a µ ( τ ) = ˙ v µ ( τ ) Four-acceleration: Radiated Field Equations External force + radiation reaction ma µ ( τ ) = f µ ext + eF µν self ( x = z ( τ )) v ν ( τ ) , where F µν is determined by the Maxwell equations � ∂ ν F µν d τ v µ ( τ ) δ 4 [ x − z ( τ )] self ( x ) = 4 π j µ ( x ) = e 4 π ǫ µνρ ∂ µ F νρ ( x ) = 0 By virtue of the Dirac delta function (point-particle limit): F µν self ( x → z ( τ )) = ∞ m 0 + e 2 e 2 � � a µ ( τ ) = e µν + 2 cF ext � � a ν , lim g µν + v µ v ν ˙ 2 ǫ c 2 3 ǫ → 0 where ǫ > 0 regularize the self-force.

Finite size corrections to the classical radiation reaction. Regularization Infinite electromagnetic self-energy, regularization For a point-like particle with charge e , ǫ → 0 ( m 0 + e 2 νµ v µ + 2 e 2 m phys a ν = eF ext � � a µ m phys = lim 2 ǫ ) , g νµ + v ν v µ ˙ 3 = ⇒ finite equation of motion, ADL equation. However, the the finite part of the mass depends on the regularization method. = ⇒ Scheme dpendence. Generalization • Point–particle with higher multipole (dipole, quadrupole, etc.) moments. • Scalar self–force. • Gravitational self–force. We need a general, scheme independent method to determine, which part of self–force can be absorbed into the bare mass.

Finite size corrections to the classical radiation reaction. Regularization Regularization methods • Separating the singularity of A µ self on the world-line self = 1 + 1 • A µ � A µ ret + A µ � � A µ ret − A µ � (Dirac) adv adv 2 2 self = 1 + 1 • A µ A µ ret + f µ A µ ret − f µ � � � � sing sing 2 2 • Regularization of Green function Λ G µν ( x , x ′ ) ∼ δ � ( x − x ′ ) 2 � � − Λ ( x − x ′ ) 2 / 2 � −→ √ exp (Coleman) 2 π • Regularization of the stress-energy tensor • Excluding a small neighborhood of world-line from the integral of the stress-energy tensor • Point-like particle −→ Extended charge distribution • Wald (2008) : Regularization, renormalization is not needed if the charge and mass of the charged body follows a special scaling rule in the point-particle limit.

Finite size corrections to the classical radiation reaction. Relativistic models of an extended charge distribution There is no consistent description for an extended relativistic charged particle! Models • Dirac’s bubble: an elastic, conducting shell. The self-field should satisfies the boundary conditions on the surface of the bubble. Unstable for certain deformation (Gnadig et al.). • Spherically symmetric, rigid charge distribution (Nodvik) . • Soliton solutions. Rigid model (Nodvik) • Rigidity defined only in the momentary rest frame of the body. • Spherically symmetric charge distribution: f ( x ) ≡ f ( r ) . • Finite size R : f ( r ) = 0 for r > R . No conflict with the special relativity if R is sufficiently small: | a | R ≪ 1 , | ω | R ≪ 1( = c )

Finite size corrections to the classical radiation reaction. Rigid model Four-current density in the rigid model � ∞ d τ � v µ − � v µ a ν − ω µν ( τ ) � [ x ν − z ν ( τ )] � f j µ ( x ) = q � [ x − z ( τ )] 2 � δ ( v ( τ ) · [ x − z ( τ )]) −∞ � r 2 f ( r ) dr =1 q = ⇒ total charge, f ( r ) normalized = ⇒ 4 π v α a β = ⇒ Thomas precession, ω µν = ⇒ rotation of the body � ∞ � ∞ ∞ � J µ ( x ) = q v µ ( τ ) δ (4) ( x − z ( τ ))d τ + ( − 1) n m ν 1 ...ν n µ ( τ ) ∂ ν 1 ...∂ ν n δ (4) ( x − z ( τ ))d τ −∞ −∞ n = 1 m νµ ( τ ) : antisymmetric dipole moment tensor, m ν 1 ...ν n µ ( τ ) : higher multipoles p 1 p 2 p 3  0      − p 1 µ 3 − µ 2 m νµ = q   0 3 � r 2 � ( v µ a ν − a µ v ν + ω µν ) , Rest frame: m µν =       ,    − p 2 − µ 3 µ 1   0         − p 3 µ 2 − µ 1   0  where � r 2 � = 4 π � r 4 f ( r ) dr ∼ O ( R 2 ) form factor. p i ∼ a i and µ i ∼ ω i : the usual electric and magnetic dipole three-vectors.

Finite size corrections to the classical radiation reaction. Equation of motion Energy-momentum balance = ⇒ Equation of motion = ⇒ d ∂ ν [ T µν M + T µν p µ M + p µ = f µ � � ext , EM ] = 0 d τ self where the four momentums � p µ d σ ν ( τ ) T µν in the point-like case: p µ M ( τ ) = m 0 v µ ( τ ) self ( τ ) = EM , σ ( τ ) x µ The self field outside the body From the Maxwell equations (in Lorentz gauge): � A µ self ( x ) = 4 π j µ ( x ) subs. multipole series J µ ( x ) ←− k µ In the monopole-dipole approximation self ( x ) = qv µ ( τ r ) � m νµ ( τ r ) � ∼ 1 ρ, 1 A µ + ∂ ν ρ ρ ρ 2 The stress-energy tensor is biliniar in F µν τ r ρ � � self = 1 ρ − g µν 1 ∼ 1 ρ 2 , 1 ρ 3 , 1 ρ 4 , 1 ρ 5 , 1 z( ) T µν F µρ F ν 4 F ρσ F ρσ τ 4 π ρ 6

Finite size corrections to the classical radiation reaction. Bound and radiated momentum Bound and radiated momentum (Teitelboim) Separation of T µν self into radiative and bound part: T µν T µν 6 ( − 2) ( k ,τ r ) ( − n ) ( k ,τ r ) T µν � = T µν rad + T µν k µ self = + ρ 2 ρ n bound n = 3 Properties: k µ ∂ ν T µν rad ( x ) = ∂ ν T µν for x � z ( τ ) bound ( x ) = 0 and T µν T µν T µν rad ∼ O ( ρ − 2 ) , bound ∼ O ( ρ − 3 ) rad k ν = 0 , Radiative part: no energy-momentum flux across the light-cone and the four-momentum flows out to infinity. z( ) τ Bound part: the energy-momentum flux is bounded to the particle, no contributions to the momentum flux at infinity.

Finite size corrections to the classical radiation reaction. Bound and radiated momentum Bound and radiative momentum The four-momentum has also two parts: p µ self ( τ ) = p µ bound ( τ ) + p µ rad ( τ ) • the regular p µ rad non-local (depends on the past history of the charge) • all singularity in the localized p µ bound = ⇒ The bound part is combined with the particle (original) four-momentum d bound ] ≡ d ext − d d τ [ p µ M + p µ d τ p µ T = f µ d τ p µ rad = f µ ext + f µ self Radiated four-momentum σ � p µ d 4 xT µν rad ( τ ) = ( rad ) θ ( ρ − ǫ ) v ν δ ( v ( τ ) · [ x − z ( τ )]) τ r ε � � � d ρρ 2 T µν = d τ r d Ω ( rad ) v ν δ ( ρ − ǫ ) θ ( τ − τ r ) � τ � 2 � 3 q 2 a 2 v µ + q 2 v µ ¨ � m νρ a ν v ρ + ... � + terms with two m µν d τ r = −∞ ˜

Finite size corrections to the classical radiation reaction. Mass renormalization Bound momentum for a point-like charged dipole � p µ d 4 xT µν bound ( τ ) = ( bound ) θ ( ρ − ǫ ) v ν δ ( v ( τ ) · [ x − z ( τ )]) � � divergent terms with two m µν and qm µν ∼ 1 ǫ 3 , 1 + q 2 v µ 1 = ǫ 2 2 ǫ + 2 3 q 2 a µ + terms with m µν ∼ ǫ 0 � � The total four-momentum of the particle is a complicated expression M + q 2 p µ T = p µ M + p µ p µ 2 ǫ v µ + [ divergent terms with m µν ] = bound 2 3 q 2 a µ + [ finite terms with m µν ] � Mv µ + Renormalization condition The physical mass can be defined by p µ T , in the rest frame as follows p µ T = δ µ 0 m phys m phys = m 0 + [ finite and infinite terms from p 0 = ⇒ bound ] This condition makes the equation motion finite and independent from the regularization sceme!