Charged particles in curved spacetime

Authors: K Mehdizadeh O Jalili

Publish Date: 2015/11/28

Volume: 10, Issue: 1, Pages: 47-52

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Abstract

Considering the dynamics of geometry and the matter fields dynamical equations of geometry and the matter fields are rederived The solutions of these equations are studied We focus on a charged particle and explain the axiomatic approach to drive the electromagnetic selfforce on its motion then the energy conservation is considered A new mathematical concept which is introduced in axiomatic approach in general is discussedIn considering the plasma around a neutron star we encounter a charged gas in the curved space In statistical formalism of such problem usually we just need the first velocity moment of Vlasov or Boltzmann equation In the equation of first velocity moment we study the average motion of particles in all elements It is clear that every particle’s trajectory is not in full adjustment to the element’s trajectory Some physical phenomena come back to this mismatching For example the first velocity moment cannot explain the Landau damping Apart from this the very charged particle motion is an important problem that contains some mathematical complexity Radiation field of a charged particle motion in curved space in general can be divided into two parts electromagnetic field and gravitational field In this paper we only consider the first one The solution of Maxwell’s equation depends on several parameters eg on the presence of other charged particles and the spacetime geometry We focus on a particle that even small changes the spacetime geometry One of the known ways to solve the differential equations is the use of Green functions The Green functions that is appropriate for our problem are bitensors Bitensor is a mathematical object that is defined in two distinct points of a manifold The method we specially focus on it using another mathematical object difference of two tensors belonging to two distinct tangent spaces In this manner we come to a new mathematical concept in the formulation of the problem These new mathematical concepts need further research and studies We discuss some aspects of these new concepts in this paper Our paper contains following aspects First we describe the charge particle motion in the curved spacetime and discuss some of its complexities then in Sect 3 the electromagnetic selfforce is discussed by axiomatic approach In Sect 4 the Conservation of energy is considered that is the extension of energy conservation theorem to the curved space Finally in Sect 5 we have discussed the new mathematical concepts that are introduced in the problemTo avoid the complexity of extended charged matters we turned to the point particles We can principally obtain the extended matter’s fields from point’s field even it is not the case for gravitational waves because the Einstein equations are not linear Choice of the point assumption causes a problem the point matters are not compatible with general relativity because it creates black hole a spherical matter distribution with the fixed mass and charge when its radius vanishes becomes a black hole On the other hand we have point particle in the world leptons Thus the general relativity is not a complete theory by itself it needs to join with the quantum theory ie we need quantum gravity So far we have two candidates for quantum gravity string theory and loop quantum gravity In the first one we exclude the point concept and introduce the extended object string In the second one we exclude the continuum structure of spacetime and introduce the spacetime network instead of manifold structure In the present paper we study some small distribution of matter to simplify our considerationsElectromagnetic Axiom 2 flat spacetime axiom If M g ab is Minkowski spacetime the world line is uniformly accelerating and F ab is the halfadvanced halfretarded solution F ab=frac12F ab+F+ ab then fa = 0 at every point on the world lineThe radiation of charged particles in curved space is an important subject in general relativistic kinetic theory In formulating it we see the nonlinear nature of general relativity that comes through two ways We have seen that even the noninteracting particles have some kind of interaction the particle changes the geometry and the new geometry acts on other particles motion The general relativity replaces geodesic with old Newtonian concept ie the force However we can use the force concept as before This issue is based on a fundamental fact the algebraic methods the mechanical approach is more generic than geometric method Where the Einstein’s geodesic method fails the dynamical approach works The radiation of charged particle in curved space like any plasma problem is the simultaneous solving of several equations the set of Maxwell equations the particles motion equations and the geometry equations With some conditions like the fixed background assumption we need to solve the two sets of these three sets However we can solve these sets with the usual methods such as the Green function method and so on we confront a new mathematical concept in the Green function method the bitensor In general the physical quantities are the local concepts that belong to the tangent space of every spacetime points Even the definition of derivative that is the comparison of quantities at two different points must be changed to become a local quantity But the bitensors are not a local quantity A bitensor is not built simply by juxtaposing of two tangent space quantities but these quantities multiplied as the numbers thus this juxtaposing is not the direct product or tensor product These new manipulations of geometrical quantities enter again in the axiomatic approach In this paper we focused on axiomatic approach to obtain the selfelectromagnetic force We have seen in this approach we need to subtract the geometrical quantities belong to two different manifolds In the bitensor approach the product quantities belong to different points of one manifold but in the axiomatic approach these two points belong to different manifolds Thus the operations are not simply juxtaposing the quantities or anything else but the difference of these quantities In physical point of view when we can subtract two quantities which are the same kinds also must have equal dimension Thus here that we are subtracting two things belong to different manifolds we come to a new concept that needs more investigations This is a new geometrical concept We may ask some questions can we make these subtracted quantities a new tensorial quantity by suitable assumptions about each quantity If such extension is present what is the derivative of it We see that there are some technical questions about these new quantities that need some further researchesTo obtain electromagnetic radiation of a charged particle in curved space we must solve the differential equation of electromagnetic potential A mu x with the single particle source term The traditional ways to do it are the method of Green function and the conformal mapping method 10 Since the selffield and the selfforce become infinity we must then regularize them by renormalized mass After that we obtain finite values for the field and force The energy conservation can be obtained with some subtlety In curved space we can reobtain the energy work theorem In a parallel and equivalent manner with some assumption about spacetime manifold we can obtain aforementioned result but with very less calculation This parallel approach is the axiomatic approach In this approach we compare tensorial object belonging to two different manifolds The comparison of tensorial object belonging to a specific manifold are well known But comparison of quantities that belong to two different manifolds is new This new phenomenon is very interesting and valuable for further researchesOpen AccessThis article is distributed under the terms of the Creative Commons Attribution 40 International License http//creativecommonsorg/licenses/by/40/ which permits unrestricted use distribution and reproduction in any medium provided you give appropriate credit to the original authors and the source provide a link to the Creative Commons license and indicate if changes were made

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