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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
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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)]
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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?
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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?
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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)]
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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)
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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
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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
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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
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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
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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)]
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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)]
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Cold plasma on TM
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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
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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
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Waterbag distributions
x3
x1 x2
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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
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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
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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
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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
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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
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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.