The electromagnetic fields in Maxwell’s theory satisfy linear equations in the classical vacuum. This is modified in classical non-linear electrodynamic theories. To date there has been little experimental evidence that any of these modified theories are tenable. However with the advent of high-intensity lasers and powerful laboratory magnetic fields this situation may be changing. We argue that an approach involving the self-consistent relativistic motion of a smooth fluid-like distribution of matter (composed of a large number of charged or neutral particles) in an electromagnetic field offers a viable theoretical framework in which to explore the experimental consequences of non-linear electrodynamics. We construct such a model based on the theory of Born and Infeld and suggest that a simple laboratory experiment involving the propagation of light in a static magnetic field could be used to place bounds on the fundamental coupling in that theory. Such a framework has many applications including a new description of the motion of particles in modern accelerators and plasmas as well as phenomena in astrophysical contexts such as in the environment of magnetars, quasars and gamma-ray bursts.

Classification Numbers: 02.40.Hw , 03.50.De , 41.20.-q

## 1 Introduction

Maxwell’s theory of classical electromagnetic phenomena employs linear partial differential equations to describe the behaviour of fields in source-free regions of the vacuum. The extension of the theory to fields in material media may involve non-linear modifications arising from the complex interactions between distributions of charge at a fundamental level. However, with the advent of high-power lasers and high-field gradients in plasmas one may be approaching regimes where the linear nature of Maxwell vacuum electrodynamics breaks down with attendant implications for electrodynamics in media. Certainly one expects vacuum polarization induced by quantum processes to intrude when electric field strengths approach V/cm. This is still some orders of magnitude greater than current field intensities in pulsed lasers so it is of interest to enquire whether classical effects of non-linear vacuum electrodynamics [1], [2] may yield experimental signatures before the need to accommodate quantum phenomena. The role of non-linear vacuum electrodynamics at a fundamental level may offer new insights into the problem of classical radiation reaction on particles and high-intensity field-particle interactions in plasmas.

One difficulty in assessing the significance of non-linear vacuum electrodynamics is in constructing a tractable generalisation of Maxwell’s theory that is amenable on some scale to experimental verification. In this note we suggest that an approach involving the self-consistent relativistic motion of a smooth fluid-like distribution of matter (composed of a large number of charged or neutral particles) in an electromagnetic field offers a viable theoretical framework in which to explore experimental consequences. Such a framework has many applications including the motion of particles in modern accelerators and plasmas as well as phenomena in astrophysical contexts such 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 on space-time with a metric tensor field

(1) |

where is a local co-frame with dual basis for . Furthermore it is assumed that locally can be expressed in terms of the 1-form by and that electrically charged matter interacts with the field via a covariant interaction giving rise to a regular 4-current density 3-form . Singular sources, such as point charges contribute a singular distributional current . The generalized Maxwell system for the field is taken to be

(2) |

(3) |

where denotes the Hodge map associated with . The 2-form is related to by a constitutive relation which for non-linear vacuum electrodynamics is non-linear. In this note it is assumed that such a relation is local and takes the form

(4) |

for some tensor . Furthermore it is assumed that the 3-form may depend locally on and a unit time-like 4-vector field describing a smooth flow of matter on space-time:

(5) |

In the following the form is related directly to the vector field by the metric. It is defined by the relation for all vector fields and is abbreviated .

In the absence of matter the constitutive tensor can be derived from an action of the form

(6) |

involving some Lagrangian 0-form . If one further restricts to Lagrangians of the form where and then

(7) |

where and . The vacuum stress-energy-momentum tensor follows from metric variations of as where

(8) |

with , and

(9) |

The dependence of the forms on is the same as that in Maxwell’s linear electrodynamics in the vacuum.

## 2 Charged Thermodynamic Fluids

Consider matter with proper mass-energy density , proper charge density and convective electric 4-current . Its equation of motion is given by

(10) |

for some total stress-energy-momentum tensor

(11) |

For a fluid without dissipation but thermodynamic pressure it will be assumed that

(12) |

It follows immediately that

(13) |

The divergence of is more complicated and it is convenient to introduce some abbreviations. For any vector field with ortho-normal components write so

(14) |

where

(15) |

with . Equation (10) yields

(16) |

which upon contracting with gives the tangential component continuity equation

(17) |

Substituting this into (16) yields, in terms of the projection operator , the relativistic fluid equation of motion

(18) |

where the total pressure 1-form

The proper mass-energy density can be expressed in terms of the proper mass density and the pressure given a specific internal energy function :

(19) |

The thermodynamic temperature and entropy of the fluid are defined via the relation

(20) |

which may be expressed in terms of and .

## 3 Born-Infeld Fluids

Born-Infeld non-linear vacuum electrodynamics has much to recommend it [3], [4]. Aside from its historic significance it is thought to encapsulate aspects of effective string theory [5], [6] including electromagnetic duality covariance. Here it will be adopted in a gravity free environment and its salient features explored in the context of the relativistic fluid. The Lagrangian takes the form

(21) |

with

(22) |

(23) |

and is governed by a new
constant of nature^{3}^{3}3The fundamental constant has SI
dimensions is the
permittivity of free space. and . It follows that the vacuum
constitutive relation is

(24) |

and the fluid system can be rewritten as

(25) |

where

With one has and (3) yields:

(26) |

In Maxwell electrodynamics

and hence and the system reduces to:

exhibiting flow under the Lorentz force and thermodynamic pressure gradients.

By contrast, in the Born-Infeld electrodynamics, even in the absence of electrically charged matter couplings contributing to via the electric current , there is a non-zero Born-Infeld electro-dynamic pressure contributing to the total pressure on the fluid:

(27) |

This equation together with the above continuity equation:

(28) |

and Born-Infeld field equations in the absence of singularities

(29) |

constitute the equations for a coupled -neutral, relativistic Born-Infeld thermodynamic fluid [7], [8]. If one can neglect collisions and internal energy, one has a -neutral cold, thermodynamically inert fluid (dust) satisfying (29), (27) and (28) with . The remaining electrodynamic pressures may arise whenever and are non-zero for fields such that and .

Such pressures can arise from solutions in Born-Infeld electrodynamics in Minkowski space-time with plane propagating waves superposed with a uniform static magnetic field in vacuo [9]. A particular case is a magnetic field transverse to the direction of propagation of a plane wave with an arbitrary smooth longitudinal profile :

describing the electric and magnetic fields in an inertial frame:

where

Thus the static magnetic field with magnitude slows down the propagating electromagnetic field with amplitude proportional to
to a phase speed in vacuo. Since this retardation is cumulative it may be amenable to experimental analysis
with laboratory magnetic fields.
If one sets in terms of
the classical radius of the electron^{4}^{4}4m then the
Born-Infeld electron model bounds . The wave
transit time difference between when the static field is switched
on and off in a magnet region of length is

So for , in metres and in Tesla

where ps sec. This suggests that a terrestrial experiment could be used to place bounds on the coupling .

## 4 Conclusions

A general model of a charged fluid interacting with an electromagnetic field whose vacuum properties are governed by a Lagrangian generalizing Maxwell’s theory has been devised. It exhibits new pressure gradients of a purely electrodynamic origin in addition to those expected from Maxwell’s theory. These forces may exist even when the fluid is electrically neutral in the vacuum. The particular case of a Born-Infeld fluid has been chosen to illustrate the existence of these forces when the fluid moves in a background static magnetic field on which a plane wave of arbitrary longitudinal profile propagates. The properties of this wave offer a means to bound the fundamental Born-Infeld coupling. Once one has bounds on it is proposed that the framework above offers a new and intriguing avenue to explore the effects of non-linear vacuum electrodynamics in high field regimes that may become accessible to observation before the breakdown of classical electrodynamics.

## Acknowledgment

RWT is grateful to colleagues at the Cockcroft Institute for valuable discussions, to the EPSRC for a Springboard Fellowship for financial support for this research which is part of the Alpha-X collaboration. We thank Koç University for its hospitality where part of this research is carried out and the Turkish Academy of Sciences (TUBA) for a travel grant.

## References

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