Born–Infeld model

In theoretical physics, the Born–Infeld model is a particular example of what is usually known as a nonlinear electrodynamics. It was historically introduced in the 30s to remove the divergence of the electron's self-energy in classical electrodynamics by introducing an upper bound of the electric field at the origin. The Born-Infeld electrodynamics possesses a whole series of physically interesting properties:

First of all the total energy of the electromagnetic field is finite and the electric field is regular everywhere.

Second it displays good physical properties concerning wave propagation, such as the absence of shock waves and birefringence. A field theory showing this property is usually called completely exceptional and Born-Infeld theory is the only completely exceptional regular nonlinear electrodynamics.

Finally (and more technically) Born-Infeld theory can be seen as a covariant generalization of Mie's theory, and very close to Einstein's idea of introducing a nonsymmetric metric tensor with the symmetric part corresponding to the usual metric tensor and the antisymmetric to the electromagnetic field tensor.

During the 1990s there was a revival of interest on Born-Infeld theory and its nonabelian extensions as they were found in some limits of string theory.

The model is named after physicists Max Born and Leopold Infeld who first proposed it.

Equations

We will use the relativistic notation here as this theory is fully relativistic.

The Lagrangian density is

\mathcal{L}=-b^2\sqrt{-\det\left(\eta+{F\over b}\right)}+b^2

where η is the Minkowski metric, F is the Faraday tensor and both are treated as square matrices so that we can take the determinant of their sum; b is a scale parameter. The maximal possible value of the electric field in this theory is b, and the self-energy of point charges is finite. For electric and magnetic fields much smaller than b, the theory reduces to Maxwell electrodynamics.

In 4-dimensional spacetime the Lagrangian can be written as

\mathcal{L}=-b^2\sqrt{1-\frac{E^2-B^2}{b^2}-\frac{(\vec{E}\cdot\vec{B})^2}{b^4}}+b^2

where E is the electric field, and B is the magnetic field.

In string theory, gauge fields on a D-brane (that arise from attached open strings) are described by the same type of Lagrangian:

\mathcal{L}=-T\sqrt{-\det\left(\eta+2\pi\alpha'F\right)}

where T is the tension of the D-brane.

Partial differential equation

$\displaystyle (1-u_t^2)u_{xx} +2u_xu_tu_{xt}-(1+u_x^2)u_{tt}=0$

There are many travelling wave exact solutions of Born Infeld pde^[1]

$u[1] := 2\arctan\left[\sqrt{\frac{a_0+b_1}{a_0-b_1}}\tan\left(\frac{1}{2}\sqrt{a_0^2-b_1^2}\eta\right)\right] + \frac{\pi}{2}$

$u[2] := 2\arctan\left[\sqrt{\frac{a_0+b_1}{a_0-b_1}}\cot\left(\frac{1}{2}\sqrt{a_0^2-b_1^2}\eta\right)\right] + \frac{\pi}{2}$

$u[3] := 2\arctan\left[\frac{\sqrt{a_1^2+b_1^2}}{a_1}\tan\left(\frac{1}{2}\sqrt{a_1^2+b_1^2}\eta\right) + \frac{b_1}{a_1} \right]$

$u[4] := 2\arctan\left[\frac{\sqrt{a_1^2+b_1^2}}{a_1}\cot\left(\frac{1}{2}\sqrt{a_1^2+b_1^2}\eta\right) + \frac{b_1}{a_1} \right]$

$u[5] := 2\arctan\left[\frac{\sqrt{a_1^2+b_1^2}}{a_1}\left(\coth(\sqrt{a_1^2+b_1^2}\eta)+\csc(\sqrt{a_1^2+b_1^2}\eta)\right) + \frac{b_1}{a_1}\right]$

$u[6] := 2\arctan\left[\sqrt{a_1^2+b_1^2}\frac{\cosh(\sqrt{a_1^2+b_1^2}\eta)+1}{a_1\sinh(\sqrt{a_1^2+b_1^2}\eta)} + \frac{b_1}{a_1}\right]$

$u[7] := 2\arctan\left[\frac{\sqrt{a_1^2+b_1^2}}{a_1}\frac{\sinh(\sqrt{a_1^2+b_1^2}\eta}{\cosh(\sqrt{a_1^2+b_1^2}\eta)+1} + \frac{b_1}{a_1}\right]$

$u[8] := 2\arctan\left[\sqrt{\frac{a_0+b_1}{b_1-a_0}}\left(\cot(\sqrt{b_1^2-a_0^2}\eta+\sec(h\sqrt{b_1^2-a_0^2}\eta)\right)\right] + \frac{\pi}{2}$

$u[9] := 2\arctan\left[\sqrt{\frac{a_0+b_1}{b_1-a_0}}\frac{\sinh(\sqrt{b_1^2-a_0^2}\eta)}{\cosh(\sqrt{b_1^2-a_0^2}\eta)+1}\right]$

Born Infeld equation animation3

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Born Infeld equation animation10

References

↑ Yuanxi Xie and Jiashi Tang,New Explicit Exact Solutions of the Born Infeld Equation,2005

Born-Infeld theory on arxiv.org