Dilaton photoproduction in a magnetic dipole field of pulsars and magnetars

According to Einstein–Maxwell-dilaton theory, the dilaton field ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\psi }$$\end{document} can be produced by electromagnetic fields with non-zero Maxwell invariant. So electromagnetic wave propagating in an external electromagnetic field is a typical source of dilaton radiation. For study dilaton photoproduction in astrophysical conditions it’s interesting to consider plane elliptically polarized electromagnetic wave propagating in the electromagnetic field of magnetic dipole m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{m}$$\end{document} of pulsars and magnetars. The dilation field equation is solved in case |ψ|≪1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\varvec{\psi }| {\mathbf {\ll 1}}$$\end{document}. The angular distribution dilaton radiation is studied in every point of space. It’s shown that spectral composition of dilatons is similar to spectral composition of plane electromagnetic wave. Amount of dilaton energy radiated in time and all directions is greatest in condition (B12-B22)(mx2-my2)≥0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({{\varvec{B}}}_{\textbf{1}}^{\textbf{2}}-{{\varvec{B}}}_{\textbf{2}}^{\textbf{2}})({{\varvec{m}}}_{{\varvec{x}}}^{\textbf{2}}-{{\varvec{m}}}_{{\varvec{y}}}^{\textbf{2}})\ge 0,$$\end{document} where B1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\varvec{B}}}_{\textbf{1}}$$\end{document} and B2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\varvec{B}}}_{\textbf{2}}$$\end{document} are electromagnetic wave amplitudes along the axes of polarization ellipse. This condition is valid for many neutron star systems.

According to [15], the action of Einstein-Maxwell-Dilaton theory can be written as where Ψ is a dilaton field, a 0 , a 1 and K are gauge constants and F nm is Maxwell tensor.
In string theory K = 1, the five-dimensional Kaluza-Klein theory results in the value In this work the constant K is arbitrary.
The field equations in Minkowski spacetime obtained from action (1) have the form: where E is the electric field strength and B is magnetic induction.
As dilaton field hasn't been discovered, one can assume that dilaton field is weak in Solar system and |ψ| ≪ 1.In this case equation ( 3) can be expressed as According to equations ( 3) -( 4), the invariant B 2 − E 2 is the source of the dilaton field.
Beside this invariant is equal to zero for wave zone of every electromagnetic wave, dilaton photoproduction is possible from near zone or in area where there is a superposition of electromagnetic fields with non-zero invariant B 2 − E 2 .

II. BASIC EQUATION AND ITS SOLUTION
Consider a plane elliptically polarized electromagnetic wave with frequency ω propagating among axis z.Fields E and B of electromagnetic wave have the following form: where k = ω/c, B 1 and B 2 are the electromagnetic wave amplitudes among principal axes of the polarization ellipse.
For example, if B 1 = B 2 then the wave (5) has circle polarization; if B 1 = 0 or B 2 = 0 then the wave ( 5) is linear polarized.In other cases wave has elliptical polarization.
Assume that there is a neutron star with radius R S rotating around magnetic dipole momentum m in the origin of the axis.The magnetic induction B of the neuron star for r > R S has the form: For now a few hundred neutron stars have been discovered [16,17] whose rotating axis doesn't coincide with the axis of magnetic dipole momentum.Such stars radiate electromagnetic waves in magnetic dipole approximations (pulsars and magnetars).But there must be neutron stars whose dipole momentum axis coincides with the rotation axis.In this case there is no electromagnetic radiation and their magnetic field (6) must be static.
Substituting superposition of electromagnetic fields ( 5) and ( 6) to expression (4) and discarding the time-independent terms, one can obtain Exact solution of the Eq. ( 7) is found to be It follows from this expression that spectral composition of dilatons coincides with the spectral composition of plane electromagnetic wave.
Rewrite expression (8) in spherical coordinates: It should be noted that in limit θ → 0 dilaton field has no singularity Thus, dilaton field has finite value everywhere out of neutron star.Using the expression (9) one can study angular distribution of the dilaton radiation.

III. ANGULAR DISTRIBUTION OF THE DILATON RADIATION
By definition [7,18], the amount of energy dI emitted by the source per unit time through the solid angle dΩ is given by the formula: where W is the energy flux density vector associated with the components of the stressenergy tensor T ik by the relations: For free dilatonic field the stress-energy tensor T ik has the form: It follows that for a 0 > 0 dilaton energy density is positive for every distribution of electromagnetic fields.
Angular distribution of dilaton radiation produced by elliptically polarized electromagnetic wave (5) propagating in magnetic field of neutron star ( 6) can be calculated by formula: In the expression above it is taken into account that the dilaton field ψ has a special point in θ = 0.
Substituting (8) to ( 14) one can obtain angular distribution averaged over the period of electromagnetic wave T = 2π/ω: As it was shown, a plane elliptically polarized electromagnetic wave propagating in an electromagnetic field of magnetic dipole produces a dilaton wave, whose amplitude has a special point in θ = 0.But dilaton ψ has a finite value at this point.That's why the dilaton field has finite value in an area where r > R S .
Simple analysis shows that the amount of dilaton energy I per unit of time averaged over the period of electromagnetic wave is the greatest in condition (B 2 1 − B 2 2 )(m 2 x − m 2 y ) > 0. This condition is valid for many superpositions of electromagnetic waves and magnetic dipole momentums of neuron stars.