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

z( ) τ e−

γ γ

Acceleration Maxwell Eqs. Lorentz Force Radiated Field External Force

Equation of motion including back-reaction force Non-relativistic limit: m a = e E + e c v× B+2 3e2˙

• a

Covariant equation: maµ = e cFµνvν+2 3 e2 c gµν +vµvν ˙ aν 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

Minkowski metric: gµν = diag(−1, 1, 1, 1) World-line: z(τ) Four-velocity: vµ(τ) = ˙ zµ(τ) Four-acceleration: aµ(τ) = ˙ vµ(τ)

Acceleration Maxwell Eqs. Lorentz Force Radiated Field External Force

Equations External force + radiation reaction maµ(τ) = f µ

ext +eFµν self (x = z(τ))vν(τ),

where Fµν is determined by the Maxwell equations ∂νFµν

self (x) = 4πjµ(x) = e4π

• dτvµ(τ)δ4 [x−z(τ)]

ǫµνρ∂µFνρ(x) = 0 By virtue of the Dirac delta function (point-particle limit): Fµν

self (x → z(τ)) = ∞

lim

ǫ→0

• m0 + e2

2ǫ

• aµ(τ) = e

cFext

µν + 2

3 e2 c2

• gµν +vµvν
• ˙

aν, 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, mphys = lim

ǫ→0(m0 + e2

2ǫ ), mphysaν = eFext

νµ vµ + 2e2

3

• gνµ +vνvµ
• ˙

aµ =⇒ 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 Aµ

self on the world-line

• Aµ

self = 1

2

• Aµ

ret +Aµ adv

• + 1

2

• Aµ

ret −Aµ adv

• (Dirac)
• Aµ

self = 1

2

• Aµ

ret +f µ sing

• + 1

2

• Aµ

ret −f µ sing

• Regularization of Green function

Gµν(x,x′) ∼ δ

• (x−x′)2

−→ Λ √ 2π exp

• −Λ(x−x′)2/2
• (Coleman)
• Regularization of the stress-energy tensor
• Excluding a small neighborhood of world-line from the integral
• f 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 jµ(x) = q ∞

−∞

dτvµ −vµaν −ωµν(τ)[xν−zν(τ)]f

• [x−z(τ)]2

δ(v(τ)·[x−z(τ)]) q =⇒ total charge, f(r) normalized =⇒ 4π

• r2f(r)dr=1

vαaβ =⇒ Thomas precession, ωµν =⇒ rotation of the body Jµ(x) = q ∞

−∞

vµ(τ)δ(4)(x−z(τ))dτ+

∞

• n=1

(−1)n ∞

−∞

mν1...νnµ(τ)∂ν1 ...∂νnδ(4)(x−z(τ))dτ mνµ(τ): antisymmetric dipole moment tensor, mν1...νnµ(τ): higher multipoles mνµ = q 3r2(vµaν −aµvν +ωµν), Rest frame: mµν =                p1 p2 p3 −p1 µ3 −µ2 −p2 −µ3 µ1 −p3 µ2 −µ1                , where r2 = 4π

• r4f(r)dr ∼ O(R2) form factor.

pi ∼ ai and µi ∼ ωi: 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 ∂ν[Tµν

M +Tµν EM] = 0

=⇒ d dτ

• pµ

M +pµ self

• = f µ

ext,

where the four momentums pµ

self (τ) =

• σ(τ)

dσν(τ)Tµν

EM,

in the point-like case: pµ

M(τ) = m0vµ(τ)

The self field outside the body From the Maxwell equations (in Lorentz gauge): Aµ

self (x) = 4πjµ(x)

←−

• subs. multipole series Jµ(x)

In the monopole-dipole approximation Aµ

self (x) = qvµ(τr)

ρ +∂ν mνµ(τr) ρ

• ∼ 1

ρ, 1 ρ2 The stress-energy tensor is biliniar in Fµν Tµν

self = 1

4π

• FµρFν

ρ −gµν 1

4FρσFρσ

• ∼ 1

ρ2 , 1 ρ3 , 1 ρ4 , 1 ρ5 , 1 ρ6

τr kµ

z( )

τ ρ

xµ

Finite size corrections to the classical radiation reaction. Bound and radiated momentum

Bound and radiated momentum (Teitelboim) Separation of Tµν

self into radiative and bound part:

Tµν

self =

Tµν

(−2)(k,τr)

ρ2 +

6

• n=3

Tµν

(−n)(k,τr)

ρn = Tµν

rad +Tµν bound

Properties: ∂νTµν

rad(x) = ∂νTµν bound(x) = 0

for x z(τ) and Tµν

radkν = 0,

Tµν

rad ∼ O(ρ−2),

Tµν

bound ∼ O(ρ−3)

Radiative part: no energy-momentum flux across the light-cone and the four-momentum flows out to infinity. Bound part: the energy-momentum flux is bounded to the particle, no contributions to the momentum flux at infinity.

kµ kµ

z( )

τ

Finite size corrections to the classical radiation reaction. Bound and radiated momentum

Bound and radiative momentum The four-momentum has also two parts: pµ

self (τ) = pµ bound(τ)+pµ rad(τ)

• the regular pµ

rad non-local (depends on the past history of the charge)

• all singularity in the localized pµ

bound

=⇒ The bound part is combined with the particle (original) four-momentum d dτ[pµ

M +pµ bound] ≡ d

dτpµ

T = f µ ext − d

dτpµ

rad = f µ ext +f µ self

Radiated four-momentum pµ

rad(τ) =

• d4xTµν

(rad)θ(ρ−ǫ)vνδ(v(τ)·[x−z(τ)])

=

• dτr
• dΩ
• dρρ2Tµν

(rad)vνδ(ρ−ǫ)θ(τ−τr)

= τ

−∞

dτr 2 3q2a2vµ +q

• 2vµ ¨

mνρaνvρ +...

• +terms with two mµν
• τr

σ ε

˜

Finite size corrections to the classical radiation reaction. Mass renormalization

Bound momentum for a point-like charged dipole pµ

bound(τ) =

• d4xTµν

(bound)θ(ρ−ǫ)vνδ(v(τ)·[x−z(τ)])

=

• divergent terms with two mµν and qmµν ∼ 1

ǫ3 , 1 ǫ2

• +q2vµ 1

2ǫ +2 3q2aµ +

• terms with mµν ∼ ǫ0

The total four-momentum of the particle is a complicated expression pµ

T = pµ M +pµ bound

= pµ

M + q2

2ǫ vµ +[divergent terms with mµν] + 2 3q2aµ +[finite terms with mµν] Mvµ Renormalization condition The physical mass can be defined by pµ

T, in the rest frame as follows

pµ

T = δµ0mphys

=⇒ mphys = m0 +[finite and infinite terms from p0

bound]

This condition makes the equation motion finite and independent from the regularization sceme!

Finite size corrections to the classical radiation reaction. Finite size

Finite size For an extended charge distribution, any spherically symmetric charge distribution can be build up by spherical shells. Outside a shell with radius r1: Fµν

shell(r1,x) = Fµν self

• q −→ qshell(r1), mνµ −→ mνµ

shell(r1)

• θ

(x−z(τ))2 −r1

• Fµν

body(x) =

R dr1Fµν

shell(r1,x)

where the charge and the dipole moment of the shell are qshell = qf(r1), mνµ

shell(r1) = qf(r1)r2 1(vµaν −aµvν +ωµν)/3 = qf(r1)r2 1 ˜

mνµ/3 The stress-energy tensor is bilinear in Fµν

body(x) and so

Tµν

body =

R dr1 R dr2Tµν

self [q2 → q2f(r1)f(r2), qmνµ →

2r2

1

3 q2f(r1)f(r2) ˜ mνµ, mνµmρσ → r2

1r2 2

9 q2f(r1)f(r2) ˜ mνµ ˜ mρσ]θ (x−z(τ))2 −r1

• +(r1 ↔ r2)

The θ function is a natural regularization in Tµν

body!

Finite size corrections to the classical radiation reaction. Finite size equation of motion

Equation of motion Performing the same calculatitions with Tµν

body as in the case of charged dipole

particle, finally we have the equation of motion dpµ

T

dτ = f µ

ext +f µ self

with pµ

T = Mphysvµ + 2

3q2aµ − 2 3q2 1 2R˙ ηµν + r1 r2 R ˙ ωµν

• vν +O(R2)

Mphys = m0 + q2 2 1 R+ 1 6 1 2Rηµνηµν + r1 r2 R ωµνωµν

• +O(R3)

f µ

self

= 2 3q2a2vµ + 2 3R2

• (v· ¨

a)vµ −2a2˙ aµ −2... ωµνvν −a2 ˙ ωµνvν +...

• where

ηµν = vµaν −aµvν, 1 R = 32π2 R

r1=0

dr1r2

1f(r2 1)

R

r2=r1

dr2r2f(r2

2)

• r1

r2 R = 32π2 3 R

r1=0

dr1r4

1f(r2 1)

R

r2=r1

dr2r2f(r2

2)

Finite size corrections to the classical radiation reaction. Finite size equation of motion

Mass renormalization Mphys is determined by our renormalization condition, pµ

T = δµ0mphys =⇒ Mphys = m0 + q2

2 1 R+ 1 6 1 2Rηµνηµν + r1 r2 R ωµνωµν

• Renormalization is needed to correctly identify the physical mass

even for a finite equation! However, Wald said that no regularization and renormalization is needed! Stricly speakig, he rescales the mass, charge and the size of the particle with a (small) parameter λ, as follows q → λq, m0 → λm0, R → R/λ. Therefore, the first two order in λ of the ADL equation is free of any singularity λm0aµ = λqFµν

extvν +λ2q2 gµν +vµvν ˙

aµ +O(λ3), where the divergent contribution is hidden in the O(λ3) term. We think that the above perturbative scaling method is nothing just an another regularization scheme.

Finite size corrections to the classical radiation reaction. Conclusion

Conclusion The separation of the stress-energy tensor into bound and radiative parts is a very useful tool to identify the total four momentum of the particle. Requiring an appropiete renormalization condition for the bound momentum, the equation of motion is finite and unique. This systematic method can be applyed both for point-like and extended particles. Renormalization or the definition of the physical mass is needed even for an extended particle.