Charged relativistic fluids and non-linear electrodynamics
aa r X i v : . [ m a t h - ph ] J a n Charged Relativistic Fluids and Non-linearElectrodynamics
T. Dereli ∗ Department of Physics, Ko¸c University, 34450 Istanbul, Turkey
R. W. Tucker † The Cockcroft Institute, Daresbury, UKandDepartment of Physics, Lancaster University, Lancaster, UK
Abstract
The electromagnetic fields in Maxwell’s theory satisfy linear equationsin the classical vacuum. This is modified in classical non-linear electro-dynamic theories. To date there has been little experimental evidencethat any of these modified theories are tenable. However with the ad-vent of high-intensity lasers and powerful laboratory magnetic fieldsthis situation may be changing. We argue that an approach involvingthe self-consistent relativistic motion of a smooth fluid-like distribu-tion of matter (composed of a large number of charged or neutralparticles) in an electromagnetic field offers a viable theoretical frame-work in which to explore the experimental consequences of non-linearelectrodynamics. We construct such a model based on the theory ofBorn and Infeld and suggest that a simple laboratory experiment in-volving the propagation of light in a static magnetic field could beused to place bounds on the fundamental coupling in that theory.Such a framework has many applications including a new descriptionof the motion of particles in modern accelerators and plasmas as well ∗ [email protected] † [email protected] s phenomena in astrophysical contexts such as in the environment ofmagnetars, quasars and gamma-ray bursts. Classification Numbers: 02.40.Hw , 03.50.De , 41.20.-q2
Introduction
Maxwell’s theory of classical electromagnetic phenomena employs linear par-tial differential equations to describe the behaviour of fields in source-freeregions of the vacuum. The extension of the theory to fields in materialmedia may involve non-linear modifications arising from the complex inter-actions between distributions of charge at a fundamental level. However,with the advent of high-power lasers and high-field gradients in plasmas onemay be approaching regimes where the linear nature of Maxwell vacuumelectrodynamics breaks down with attendant implications for electrodynam-ics in media. Certainly one expects vacuum polarization induced by quantumprocesses to intrude when electric field strengths approach 1 . × V/cm.This is still some orders of magnitude greater than current field intensitiesin pulsed lasers so it is of interest to enquire whether classical effects of non-linear vacuum electrodynamics [1], [2] may yield experimental signatures be-fore the need to accommodate quantum phenomena. The role of non-linearvacuum electrodynamics at a fundamental level may offer new insights intothe problem of classical radiation reaction on particles and high-intensityfield-particle interactions in plasmas.One difficulty in assessing the significance of non-linear vacuum electro-dynamics is in constructing a tractable generalisation of Maxwell’s theorythat is amenable on some scale to experimental verification. In this note wesuggest that an approach involving the self-consistent relativistic motion ofa smooth fluid-like distribution of matter (composed of a large number ofcharged or neutral particles) in an electromagnetic field offers a viable the-oretical framework in which to explore experimental consequences. Such aframework has many applications including the motion of particles in mod-ern accelerators and plasmas as well as phenomena in astrophysical contextssuch as in the environment of magnetars, quasars and gamma-ray bursts.In the following it is assumed that the electromagnetic field is a 2-form F onspace-time with a metric tensor field g = − e ⊗ e + X k =1 e k ⊗ e k (1)where { e a } is a local co-frame with dual basis { X a } for a = 0 , , ,
3. Fur-thermore it is assumed that locally F can be expressed in terms of the 1-form A by F = d A and that electrically charged matter interacts with the field1ia a U (1) covariant interaction giving rise to a regular 4-current density3-form J U . Singular sources, such as point charges contribute a singulardistributional current J De . The generalized Maxwell system for the field F is taken to be d F = 0 , (2) d ⋆G = J De + J U (3)where ⋆ denotes the Hodge map associated with g . The 2-form G is relatedto F by a constitutive relation which for non-linear vacuum electrodynamicsis non-linear. In this note it is assumed that such a relation is local and takesthe form ⋆G = Z ( F, g ) (4)for some tensor Z . Furthermore it is assumed that the 3-form J U maydepend locally on F, g and a unit time-like 4-vector field V describing asmooth flow of matter on space-time: J U = Z ( F, V, g ) . (5)In the following the 1 − form e V is related directly to the vector field V by themetric. It is defined by the relation e V ( X ) = i X e V = g ( V, X ) for all vectorfields V and i X a is abbreviated i a .In the absence of matter the constitutive tensor Z can be derived froman action of the form S [ A, g ] = Z M ˆ L ( F, ∇ F, · · · , g ) ⋆ L ( F, ∇ F, · · · , g ). If one further restricts toLagrangians of the form ˆ L ( F, ∇ F, · · · , g ) = L( X, Y, g ) where X = ⋆ ( F ∧ ⋆F )and Y = ⋆ ( F ∧ F ) then ⋆G = Z ( F, g ) = 2 ⋆ F L X + 2 F L Y (7)where L X = ∂ L ∂X and L Y = ∂ L ∂Y . The vacuum stress-energy-momentum tensor T ( NLED ) follows from metric variations of S [ A, g ] as T ( NLED ) = ( ⋆τ a ) ⊗ e a where τ a = M ⋆ e a + N τ ( LED ) a (8)2ith M = ( L − X L X − Y L Y ), N = 2 L X and τ ( LED ) a = 12 ( i a F ∧ ⋆F − i a ⋆F ∧ F ) . (9)The dependence of the forms τ ( LED ) a on F is the same as that in Maxwell’slinear electrodynamics in the vacuum. Consider matter with proper mass-energy density ρ , proper charge density ρ e and convective electric 4-current J U = ρ e ⋆ e V . Its equation of motion isgiven by ∇ · T ( total ) = 0 (10)for some total stress-energy-momentum tensor T ( total ) = T ( NLED ) + T ( fluid ) . (11)For a fluid without dissipation but thermodynamic pressure p it will be as-sumed that T ( fluid ) = ( ρ + p ) ˜ V ⊗ ˜ V + p g . (12)It follows immediately that ^ ∇ · T ( fluid ) = V ∇ · (( ρ + p ) V ) + ( ρ + p ) ∇ V V + f d p . (13)The divergence of T ( NLED ) is more complicated and it is convenient to intro-duce some abbreviations. For any vector field Q with ortho-normal compo-nents Q a write τ ( LED ) Q = τ ( LED ) a Q a so ∇ · T ( NLED ) = d M + ⋆τ ( LED ) f dN − i ˆ J F (14)where ˆ J = J U − ⋆ ( dN ∧ ⋆F ) − ⋆ ( dL ∧ F ) (15)with L = 2 L Y . Equation (10) yields d M + ⋆τ ( LED ) f dN − i ˆ J F V ∇ · (( ρ + p ) V ) + ( ρ + p ) ∇ V V + f d p = 0 (16)3hich upon contracting with V gives the tangential component continuityequation ( p + ρ ) ∇ · V = i V d M − i V ⋆τ ( LED ) f dN − V ( ρ ) − i V i ˆ J F . (17)Substituting this into (16) yields, in terms of the projection operator Π V =(1 + ˜ V i V ), the relativistic fluid equation of motion( ρ + p ) ^ ∇ V V = Π V P (18)where the total pressure 1-form P = i ˆ J F − d M − d p − ⋆τ ( LED ) f dN . The proper mass-energy density ρ ( ρ m , p ) can be expressed in terms of theproper mass density ρ m and the pressure p given a specific internal energyfunction E ( ρ m , p ): ρ ( ρ m , p ) = ρ m (1 + E ( ρ m , p )) (19)The thermodynamic temperature T and entropy S of the fluid are definedvia the relation T d S = d E + p d (cid:18) ρ m (cid:19) (20)which may be expressed in terms of d ρ m and d p . Born-Infeld non-linear vacuum electrodynamics has much to recommend it[3], [4]. Aside from its historic significance it is thought to encapsulate aspectsof effective string theory [5], [6] including electromagnetic duality covariance.Here it will be adopted in a gravity free environment and its salient featuresexplored in the context of the relativistic fluid. The Lagrangian takes theform L(
X, Y, g ) = ǫ κ (1 − p ∆( X, Y )) (21)with ∆(
X, Y ) = 1 − κ X − κ Y X = ǫ κ √ ∆ (22)4 L Y = ǫ Y √ ∆ (23)and is governed by a new constant of nature κ . It follows that the vacuumconstitutive relation is ⋆G = ǫ κ √ ∆ (cid:18) ⋆F + κ Y F (cid:19) (24)and the fluid system can be rewritten as( ρ + p ) ^ ∇ V V = Π V ( i ˆ J F + i η F − d p − ξ ) (25)where ξ ≡ d M + ⋆τ ( LED ) f dN , ˜ η ≡ ⋆ ( d N ∧ ⋆F ) + ⋆ ( d L ∧ F ) . With J U = ρ e V one has Π V i ˆ J F = ρ e i V F and (3) yields:( ρ + p ) ^ ∇ V V = ρ e i V F + Π V ( i η F − d p − ξ ) . (26)In Maxwell electrodynamics L = X and hence η = ξ = 0 and the system reduces to: V · V = − , d F = 0 , d ⋆ G = ρ e ⋆ ˜ V , G = ǫ F , ( ρ + p ) ∇ V ˜ V = ρ e F ( V ) − d p , ( ρ + p ) ∇ · V = − V ( ρ )exhibiting flow under the Lorentz force ρ e F ( V ) and thermodynamic pressuregradients.By contrast, in the Born-Infeld electrodynamics, even in the absence ofelectrically charged matter couplings contributing to ˆ J via the U (1) electriccurrent J U , there is a non-zero Born-Infeld electro-dynamic pressure Π V ( ξ − i η F ) contributing to the total pressure on the fluid:( ρ + p ) ^ ∇ V V = − Π V ( ξ − i η F ) − Π V d p . (27) The fundamental constant κ has SI dimensions [ Q T M L ] and ǫ is the permittivity of freespace. ρ + p ) ∇ · V = − V ( ρ ) + i V ( ξ + i η F ) (28)and Born-Infeld field equations in the absence of singularities d F = 0 , d ⋆G = 0 (29)constitute the equations for a coupled U (1)-neutral, relativistic Born-Infeldthermodynamic fluid [7], [8]. If one can neglect collisions and internal energy,one has a U (1)-neutral cold, thermodynamically inert fluid (dust) satisfying(29), (27) and (28) with p = 0 , ρ = ρ m . The remaining electrodynamicpressures may arise whenever η and ξ are non-zero for fields F such that d X = 0 and d Y = 0.Such pressures can arise from solutions in Born-Infeld electrodynamicsin Minkowski space-time with plane propagating waves superposed with auniform static magnetic field in vacuo [9]. A particular case is a magnetic fieldtransverse to the direction of propagation of a plane wave with an arbitrarysmooth longitudinal profile E ( z ): F = E ( z − vt ) d ( z − vt ) ∧ dx − B dy ∧ dz describing the electric and magnetic fields in an inertial frame: e = − vc E ( z − vt ) ˆ x , b = E ( z − vt ) c ˆ y − Bc ˆ x where v = c √ c κ B . Thus the static magnetic field with magnitude B slows down the propagatingelectromagnetic field with amplitude proportional to E to a phase speed v < c in vacuo . Since this retardation is cumulative it may be amenable toexperimental analysis with laboratory magnetic fields. If one sets κ ≃ ǫ r e in terms of the classical radius of the electron then the Born-Infeld electronmodel bounds κ < − . The wave transit time difference between whenthe static field is switched on and off in a magnet region of length L is τ = L κ | B | . r = e πǫ m e c ≃ . E −
6o for κ < − , L in metres and | B | in Tesla τ < L | B | − pswhere 1ps= 10 − sec. This suggests that a terrestrial experiment could beused to place bounds on the coupling κ . A general model of a charged fluid interacting with an electromagnetic fieldwhose vacuum properties are governed by a Lagrangian generalizing Maxwell’stheory has been devised. It exhibits new pressure gradients of a purelyelectrodynamic origin in addition to those expected from Maxwell’s theory.These forces may exist even when the fluid is electrically neutral in the vac-uum. The particular case of a Born-Infeld fluid has been chosen to illustratethe existence of these forces when the fluid moves in a background staticmagnetic field on which a plane wave of arbitrary longitudinal profile prop-agates. The properties of this wave offer a means to bound the fundamentalBorn-Infeld coupling. Once one has bounds on κ it is proposed that theframework above offers a new and intriguing avenue to explore the effects ofnon-linear vacuum electrodynamics in high field regimes that may becomeaccessible to observation before the breakdown of classical electrodynamics.7 cknowledgment RWT is grateful to colleagues at the Cockcroft Institute for valuable dis-cussions, to the EPSRC for a Springboard Fellowship for financial supportfor this research which is part of the Alpha-X collaboration. We thank Ko¸cUniversity for its hospitality where part of this research is carried out andthe Turkish Academy of Sciences (TUBA) for a travel grant.
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