Exploring Born-Infeld electrodynamics usingplasmas

David A Burton & Haibao Wen(with Raoul MGM Trines1, Adam Noble2, Timothy J Walton)

Department of Physics, Lancaster Universityand the Cockcroft Institute of Accelerator Science and Technology, UK

XIX International Fall Workshop on Geometry and PhysicsPorto

6-9 September 2010

1Central Laser Facility, Rutherford Appleton Laboratory, UK2SUPA, University of Strathclyde, UK

Abraham-Lorentz-Dirac equation

What is the right equation of motion for a classical point charge?[Dirac, Proc. Roy. Soc. (1938)]

mAa = qF extab xb +

1

6πq2(ηab + xaxb)

DAb

dτ(1)

N.B. ε0 = µ0 = c = 1

Problems :

I Infinite negative bare mass to compensate for infiniteself-energy of the electron

I Abraham-Lorentz-Dirac equation is third order in timederivatives of the electron’s worldline

I Acceleration in regions where there is no external fieldI Runaway solutions

[Recent derivation of ALD equation : DAB, J Gratus, RW Tucker, Ann. Phys. 322 3, 599 (2007)]

Constitutive equations

D = D[E,B], H = H[E,B]

In particular,

I in the vacuum in classical Maxwell electrodynamics :

D = E, H = B, (2)

or G = F (3)

I non-linear when QED vacuum effects are included(Euler-Heisenberg electrodynamics)

I Could electrodynamics be fundamentally non-linear?

Constitutive equations

D = D[E,B], H = H[E,B]

In particular,

I in the vacuum in classical Maxwell electrodynamics :

D = E, H = B, (2)

or G = F (3)

I non-linear when QED vacuum effects are included(Euler-Heisenberg electrodynamics)

I Could electrodynamics be fundamentally non-linear?

Vacuum Born-Infeld equations

dF = 0, d ? G = 0, (4)

G =1√

1− κ2X − κ4Y 2/4

(F − κ2Y

2? F

)(5)

X = ?(F ∧ ?F ), Y = ?(F ∧ F ) (6)

κ is a new constant of nature[Born et al Proc. Roy. Soc. (1934)]

I The field of a point charge at rest is finite, and the charge’sself energy is finite

I Appears in low-energy string field theory [Fradkin et al, PLB (1985)]

I Considerable interest from a string theory perspective[e.g. Gibbons et al]

I Investigations in waveguides and in background magneticfields [e.g. Ferraro et al, PRL (2007); Tucker et al, EPL (2010)]

High intensity lasers

Two examples :

I VULCAN (UK) : 1022 W/cm2 (2016 upgrade to 10PW,intensity 1023 W/cm2)

I ELI (2012, located in Hungary, Czech Republic, Romania, +1EU) : intensity ∼ 1025 W/cm2 or field strength 1016 V/m

Motivation :

I Numerous applications : fast ignition, compact electronacceleration, material science,. . .

I Opportunity to test QED vacuum phenomena in a controllableenvironment (Schwinger limit 1018 V/m)

Plasma waves

A sufficiently intense and short laser pulse with sufficiently highpeak frequency propagating through a plasma creates a wave(wake behind the pulse)

I The wake contains a large longitudinal electric field(∼ 1011 V/m) =⇒ Laser Wakefield Accelerator.

[Tajima et al, PRL (1979);

Mangles et al, Nature (2004)]

Image courtesy of W. Mori

Electric waves in a cold Born-Infeld plasma

I Maxwell’s equations

dF = 0, d ? G = −q(?N − ?Nion)

where

G =1√

1− κ2X − κ4Y 2/4

(F − κ2Y

2? F

)I Employing an action principle =⇒ Lorentz force equation

∇V V =q

mιV F , g(V ,V ) = −1

Electric waves in a cold Born-Infeld plasma

I Metric tensor g in ion rest frame (i.e. laboratory frame)

g = −dx0 ⊗ dx0 + dx1 ⊗ dx1 + dx2 ⊗ dx2 + dx3 ⊗ dx3

I Seek solutions in ζ = x3 − v x0 with F = E (ζ) dx0 ∧ dx3

(numerous simplifying assumptions...)

I Adapted basis in the “wave frame”

e1 = v dx3 − dx0, e2 = dx3 − v dx0 = dζ

and for electrons moving slower than wave

V = µe1 − (µ2 − γ2)1/2e2

where µ = µ(ζ) > 0

Electric waves in a cold Maxwell plasma

For example, when κ = 0

1

γ2

d2µ

dζ2=

q2

mnionγ

2

(v µ√µ2 − γ2

− 1

)

and E = m/(qγ2)dµ/dζ, γ = 1/√

1− v2, e.g. for v = 0.8

–0.6

–0.4

–0.2

0

0.2

0.4

0.6

1 1.5 2 2.5 3 3.5

dµ

dζ

µ

Cold Maxwell plasma wave-breaking limit

E

ζ

Maximum amplitude oscillation for κ = 0 :

E 0max =

mωp

|q|√

2(γ − 1)

Oscillation frequency for γ À 1

ω0 ≈ π

2√

2γωp

[Akheizer et al, Sov. Phys. JETP 3 (1956)]

Cold Born-Infeld plasma wave-breaking limit

EBImax =

1

κ

√1− [κ2(E 0

max)2/2 + 1]−2

Note limv→1 EBImax = 1/κ

ωBI ≈ ω0

[1−

(κmωpc

2q

)2

γ

][DAB, RMGM Trines, TJ Walton, H Wen, arXiv:1006.2246 [physics.plasm-ph] (submitted)]

Cold plasma on TM

Relativistic Vlasov equation

I Not in thermal equilibrium

I Vlasov equation for 1-particle distribution f (x , x) ∈ Λ0(TM)

Lf = 0

L = xa

(∂

∂xa− ΓbV

ac xc ∂

∂xb− q

mF bV

a∂

∂xb

)I Electron number 4-current

Na[f ](x) =

∫Ex

xaf (x , x) νx

where Ex = {xa ∈ R4 | gVab(x , x)xaxb = −1, x0 > 0} and νx is

a measure on Ex

Vlasov equation for a jump

E = {(x , x) ∈ TM| gVab(x , x)xaxb = −1, x0 > 0}

Lf = 0 =⇒ d(f ω) ' 0 (pulled back to unit hyperboloid E)

where ω ∈ Λ6(TM), dω ' 0 and ιLω = 0

d(f ω) ' 0 =⇒∫

∂Bf ω = 0

=⇒ dλ ∧ [f ω] = 0

where λ = 0 is the interface between 1 and 2

Waterbag distributions

x3

x1 x2

Waterbag distributions

1-particle distribution f described by

Σ : M× S2 → TM,

(x , ξ) 7→ (x , x = Vξ(x))

where dλ ∧ ω = 0 is satisfied by

∇VξVξ =

q

mιVξ

F , g(Vξ,Vξ) = −1

x3

x1 x2

Axisymmetric waterbags

I For electrons moving slower than the wave

Vξ =[µ(ζ) + A(ξ1)] e1 + ψ(ξ1, ζ) e2

+ R sin(ξ1) cos(ξ2)dx1 + R sin(ξ1) sin(ξ2)dx2

with

ψ = −√

[µ+ A(ξ1)]2 − γ2[1 + R2 sin2(ξ1)]

for 0 < ξ1 < π, 0 ≤ ξ2 < 2π and constant R

Electric waves in a warm Maxwell plasma

1

γ2

d2µ

dζ2=− q2

mnionγ

2 − q2

m2πR2α

π∫0

([µ+ A(ξ1)]2

− γ2[1 + R2 sin2(ξ1)]

)1/2

sin(ξ1) cos(ξ1) dξ1

with

2πR2

π∫0

A(ξ1) sin(ξ1) cos(ξ1) dξ1 = −nionγ2 v

α

and E = m/(qγ2)dµ/dζ

Maximum electric field

x3

x1 x2

For width ¿ 1 ¿ height, and v → 1

E 2max ≈

m2ω2p

q2

√9mc2

20kBT‖eq

where T‖eq is an effective longitudinal “temperature”.

I Does the wave actually break? Only tip reaches v

Maximum electric field

x3

x1 x2

For width ¿ 1 ¿ height, and v → 1

E 2max ≈

m2ω2p

q2

√9mc2

20kBT‖eq

where T‖eq is an effective longitudinal “temperature”.

I Does the wave actually break? Only tip reaches v

Summary

I Understanding radiation reaction is important forcontemporary and future acceleration concepts.

I Abraham-Lorentz-Dirac equation is pathological

I Could Born-Infeld electrodynamics be the answer?I Test using forthcoming laser facilities and astrophysical

systems?I Explore large-amplitude waves in Born-Infeld plasmas

I Cold plasmas[DAB, RMGM Trines, TJ Walton, H Wen, arXiv:1006.2246[physics.plasm-ph] (submitted)]

I Problems remain to be ironed out in warm plasmas [DAB, ANoble, arXiv:0908.4498 [physics.plasm-ph]J. Phys. A Math. Theor. 43 075502 (2010)]

We thank Robin Tucker for useful discussions.