Exploring Born-Infeld electrodynamics using plasmas

Transcription

1 Exploring Born-Infeld electrodynamics using plasmas David A Burton & Haibao Wen (with Raoul MGM Trines 1, Adam Noble 2, Timothy J Walton) Department of Physics, Lancaster University and the Cockcroft Institute of Accelerator Science and Technology, UK XIX International Fall Workshop on Geometry and Physics Porto 6-9 September Central Laser Facility, Rutherford Appleton Laboratory, UK 2 SUPA, University of Strathclyde, UK

2 Abraham-Lorentz-Dirac equation What is the right equation of motion for a classical point charge? [Dirac, Proc. Roy. Soc. (1938)] ma a = qfab ext ẋ b + 1 6π q2 (η ab + ẋ a ẋ b ) DAb dτ (1) N.B. ε 0 = µ 0 = c = 1 Problems : Infinite negative bare mass to compensate for infinite self-energy of the electron Abraham-Lorentz-Dirac equation is third order in time derivatives of the electron s worldline Acceleration in regions where there is no external field Runaway solutions [Recent derivation of ALD equation : DAB, J Gratus, RW Tucker, Ann. Phys , 599 (2007)]

3 Constitutive equations In particular, D = ˆD[E, B], H = Ĥ[E, B] in the vacuum in classical Maxwell electrodynamics : D = E, H = B, (2) or G = F (3) non-linear when QED vacuum effects are included (Euler-Heisenberg electrodynamics)

4 Constitutive equations In particular, D = ˆD[E, B], H = Ĥ[E, B] in the vacuum in classical Maxwell electrodynamics : D = E, H = B, (2) or G = F (3) non-linear when QED vacuum effects are included (Euler-Heisenberg electrodynamics) Could electrodynamics be fundamentally non-linear?

5 Vacuum Born-Infeld equations df = 0, d G = 0, (4) ( ) 1 G = F κ2 Y F (5) 1 κ 2 X κ 4 Y 2 /4 2 X = (F F ), Y = (F F ) (6) κ is a new constant of nature [Born et al Proc. Roy. Soc. (1934)] The field of a point charge at rest is finite, and the charge s self energy is finite Appears in low-energy string field theory [Fradkin et al, PLB (1985)] Considerable interest from a string theory perspective [e.g. Gibbons et al] Investigations in waveguides and in background magnetic fields [e.g. Ferraro et al, PRL (2007); Tucker et al, EPL (2010)]

6 High intensity lasers Two examples : VULCAN (UK) : W/cm 2 (2016 upgrade to 10PW, intensity W/cm 2 ) ELI (2012, located in Hungary, Czech Republic, Romania, +1 EU) : intensity W/cm 2 or field strength V/m Motivation : Numerous applications : fast ignition, compact electron acceleration, material science,... Opportunity to test QED vacuum phenomena in a controllable environment (Schwinger limit V/m)

7 Plasma waves A sufficiently intense and short laser pulse with sufficiently high peak frequency propagating through a plasma creates a wave (wake behind the pulse) The wake contains a large longitudinal electric field ( V/m) = Laser Wakefield Accelerator. [Tajima et al, PRL (1979); Mangles et al, Nature (2004)] Image courtesy of W. Mori

8 Electric waves in a cold Born-Infeld plasma Maxwell s equations df = 0, d G = q( Ñ Ñion) where G = ( 1 F κ2 Y 1 κ 2 X κ 4 Y 2 /4 2 ) F Employing an action principle = Lorentz force equation V Ṽ = q m ι V F, g(v, V ) = 1

9 Electric waves in a cold Born-Infeld plasma Metric tensor g in ion rest frame (i.e. laboratory frame) g = dx 0 dx 0 + dx 1 dx 1 + dx 2 dx 2 + dx 3 dx 3 Seek solutions in ζ = x 3 v x 0 with F = E(ζ) dx 0 dx 3 (numerous simplifying assumptions...) Adapted basis in the wave frame e 1 = v dx 3 dx 0, e 2 = dx 3 v dx 0 = dζ and for electrons moving slower than wave where µ = µ(ζ) > 0 Ṽ = µe 1 (µ 2 γ 2 ) 1/2 e 2

10 Electric waves in a cold Maxwell plasma For example, when κ = 0 1 d 2 µ γ 2 dζ 2 = q2 m n ionγ 2 ( ) v µ µ 2 γ 1 2 and E = m/(qγ 2 )dµ/dζ, γ = 1/ 1 v 2, e.g. for v = 0.8 d µ d ζ µ

11 Cold Maxwell plasma wave-breaking limit E ζ Maximum amplitude oscillation for κ = 0 : Oscillation frequency for γ 1 Emax 0 = mω p 2(γ 1) q ω 0 π 2 2γ ω p [Akheizer et al, Sov. Phys. JETP 3 (1956)]

12 Cold Born-Infeld plasma wave-breaking limit Emax BI = 1 1 [κ κ 2 (Emax) 0 2 /2 + 1] 2 Note lim v 1 E BI max = 1/κ ( ) ω BI ω [1 0 κmωp c 2 γ] 2q [DAB, RMGM Trines, TJ Walton, H Wen, arxiv: [physics.plasm-ph] (submitted)]

13 Cold plasma on TM

14 Relativistic Vlasov equation Not in thermal equilibrium Vlasov equation for 1-particle distribution f (x, ẋ) Λ 0 (T M) Lf = 0 ( L = ẋ a x a ΓbV acẋ c ẋ b q ) m F bv a ẋ b Electron number 4-current N a [f ](x) = E x ẋ a f (x, ẋ) ν x where E x = {ẋ a R 4 g V ab (x, ẋ)ẋ a ẋ b = 1, ẋ 0 > 0} and ν x is a measure on E x

15 Vlasov equation for a jump E = {(x, ẋ) T M g V ab (x, ẋ)ẋ a ẋ b = 1, ẋ 0 > 0} Lf = 0 = d(f ω) 0 (pulled back to unit hyperboloid E) where ω Λ 6 (T M), dω 0 and ι L ω = 0 d(f ω) 0 = B f ω = 0 = dλ [f ω] = 0 where λ = 0 is the interface between 1 and 2

16 Waterbag distributions ẋ 3 ẋ 1 ẋ 2

17 Waterbag distributions 1-particle distribution f described by Σ : M S 2 T M, (x, ξ) (x, ẋ = V ξ (x)) where dλ ω = 0 is satisfied by Vξ Ṽ ξ = q m ι V ξ F, g(v ξ, V ξ ) = 1 ẋ 3 ẋ 1 ẋ 2

18 Axisymmetric waterbags For electrons moving slower than the wave Ṽ ξ =[µ(ζ) + A(ξ 1 )] e 1 + ψ(ξ 1, ζ) e 2 + R sin(ξ 1 ) cos(ξ 2 )dx 1 + R sin(ξ 1 ) sin(ξ 2 )dx 2 with ψ = [µ + A(ξ 1 )] 2 γ 2 [1 + R 2 sin 2 (ξ 1 )] for 0 < ξ 1 < π, 0 ξ 2 < 2π and constant R

19 Electric waves in a warm Maxwell plasma 1 d 2 µ π γ 2 dζ 2 = q2 m n ionγ 2 q2 m 2πR2 α 0 ( [µ + A(ξ 1 )] 2 ) 1/2 γ 2 [1 + R 2 sin 2 (ξ 1 )] sin(ξ 1 ) cos(ξ 1 ) dξ 1 with π 2πR 2 and E = m/(qγ 2 )dµ/dζ 0 A(ξ 1 ) sin(ξ 1 ) cos(ξ 1 ) dξ 1 = n ionγ 2 v α

20 Maximum electric field ẋ 3 ẋ 1 ẋ 2 For width 1 height, and v 1 E 2 max m2 ω 2 p q 2 9mc 2 20k B T eq where T eq is an effective longitudinal temperature.

21 Maximum electric field ẋ 3 ẋ 1 ẋ 2 For width 1 height, and v 1 E 2 max m2 ω 2 p q 2 9mc 2 20k B T eq where T eq is an effective longitudinal temperature. Does the wave actually break? Only tip reaches v

22 Summary Understanding radiation reaction is important for contemporary and future acceleration concepts. Abraham-Lorentz-Dirac equation is pathological Could Born-Infeld electrodynamics be the answer? Test using forthcoming laser facilities and astrophysical systems? Explore large-amplitude waves in Born-Infeld plasmas Cold plasmas [DAB, RMGM Trines, TJ Walton, H Wen, arxiv: [physics.plasm-ph] (submitted)] Problems remain to be ironed out in warm plasmas [DAB, A Noble, arxiv: [physics.plasm-ph] J. Phys. A Math. Theor (2010)] We thank Robin Tucker for useful discussions.