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User: As a text script expert, please help me to write a short text script with the topic " EFFECT OF THE MAXIMAL PROPER ACCELERATION IN THE INERTIA 3 Thepresentarticle explores theeffect ofmaximal properacceleration onthemass andkineticenergyofpointchargedparticlesinthecontextofatheoryofhigherorder jet electrodynamics and spacetimes of maximal proper acceleration.Then the notion of 4- momentum and the relation of mass with acceleration in a spacetime with maximal acceleration are discussed, leading to the introduction of the notion of rest mass in our theory.Although this model is an over-simplified description of the bunch, it is enough to show theoretical evidence of observable effects due to the increase of the kinetic energy of the particle with respect to the relativistic kinetic energy due to the effect of maximal proper acceleration, there must be a decrease in the particle population of the bunches.For the LHC, the effect is mainly due to the acceleration in 4 EFFECT OF THE MAXIMAL PROPER ACCELERATION IN THE INERTIA thebendingmagnets, becausethebunchpasses many times throughdipolemagnets compared withthetimeitpasses byradiofrequencycavities andbecausetheproper acceleration in the dipole magnets is relatively higher than in the radio frequency systems.Also note that the proper acceleration reached is several orders of magnitude higher than the reached at LHC, but the bunch is accelerated only one time, which can render the effect more visible for observational purposes.Derivatives with respect to τ are denoted by dot notation: ϑ˙(τ) := d ϑ(τ), etc... By a maximal proper acceleration dτ A we mean that the bound (2.4) for the value of the squared proper acceleration max a2 := η(∇ ϑ˙,∇ ϑ˙) holds good, ϑ˙ ϑ˙ (2.5) 0≤ η(∇ ϑ˙,∇ ϑ˙)< A2 .max c2 q q According to the above re-definition of energy of a point particle, in the instan- taneous Lorentzian coordinate frame where the particle has associated the Lorentz EFFECT OF THE MAXIMAL PROPER ACCELERATION IN THE INERTIA 7 factor γ , the energy of the system is given by the expression (2.10).The model of the classical point charged particle in spacetimes of maximal acceleration In the theory of higher order jet electrodynamics the spacetime structure that determines the proper time is a metric of maximal proper acceleration (2.2).In a general covariant formalism, the equation of motion for a point charged particle of inertial mass m and charge q is expressed as 0 2 (3.1) m ∇ ϑ˙ = qι F − q2g(∇ ϑ˙,∇ ϑ˙)ϑ˙, 0 ϑ˙ ϑ˙ 3 ϑ˙ ϑ˙ where, using the notation from [19], F is the 2-form Faraday form, ι is the interior g ϑ˙ derivative with respect to ϑ˙ and ι F is the dual form determined by the metric η of ϑ˙ the 1-form ι F. ϑ˙ Some remarks related to equagtion (3.1) are in order.Equation (3.1) implies a maximal proper acceleration for particles accelerated by means of electromagnetic fields.If in equation (3.1) we contract the left side with the left side and the right 8 EFFECT OF THE MAXIMAL PROPER ACCELERATION IN THE INERTIA side with the right side using the metric of maximal acceleration g, then it follows that a2 2 2 m2 1− a2 = F2+ q2 (a2)2g(ϑ˙,ϑ˙).Also note that in Bonnor’s theory there is no obvious uniform bound of the acceleration, since the inertial mass m , that could eventually appear in the expression for the maximal 0 acceleration, depends on time.Thus under the influence max 0 of external fields of the accelerator and for acceleration squared a2 small enough compared with A2 , for the model of the bunch as a single particle with charge max and mass (N m ,N q), the bunch will have the same acceleration as an individual 0 point charged particle with charge and mass (m ,q).However, if 0 the bunch being accelerated contains N particles and if the bunch of particles is considered as a sole particle, then the maximal acceleration is decreased by a factor N. Hence the excess in energy with respect to the relativistic energy due to a maximal acceleration is given by the expression 1 a2 ∆El=1/E = N2 , rel 2 A2 (1) max where A2 (1) is the maximal proper acceleration that applies to a sole particle.Thus we have max 2 e2 that for the RF-cavity acceleration, the energy increase between the relativistic theory and the theory of electrodynamics in spacetimes of maximal acceleration that we are discussing is of order ∆El=109Theincrease in the energy of an accelerated bunch with respect to the relativistic energy implies a deficit in the bunch population N. This is because the energy injected in the dynamical system is fixed, but there is a larger rest mass due to the proper acceleration, according to the expression (2.12).Effect of maximal acceleration in plasma wakefield acceleration and laser-electron acceleration.Indeed, current laser electron 12 EFFECT OF THE MAXIMAL PROPER ACCELERATION IN THE INERTIA acceleration systems achieve acceleration of bunches of electrons up to proper accel- erations of order 1022m/s2, which is much larger than the accelerations at LHC.This is lower than in LHC bunches, but the huge acceleration reached and the fact that in our model, the maximal acceleration for an individual electron is of order 10−3 less than for a proton, the discrepancy ∆E leads to figures higher than for Erel RF-acceleration and that potentially can be even higher.However, the effect can be relatively large compared with conventional accelerations because the maximal proper acceleration for an electron is of order 103 smaller than for a proton accelerator.Therefore, as longaslaser-plasmaacceleration canreachmorepopulatedbuncheswiththecurrent top scales of acceleration, the effect discussed in this paper will be increasingly relevant and provide potential observable effect of the equation of motion (3.1) and the maximal proper acceleration (3.3).In our model, the acceleration of one particle is equivalent to the acceleration of the whole bunch of particles considered 14 EFFECT OF THE MAXIMAL PROPER ACCELERATION IN THE INERTIA as a sole particle of mass m N and charge qN.Therefore, as long as s < 1 0 b holds good, the estimates for the acceleration and its effects discussed in this paper provide reasonable results if one is looking for the order of magnitude of the effect of maximal proper acceleration.Fluid models or kinetic models for plasmas in spacetimes of maximal properacceleration could provide significant improvement in the understanding of the dynamics of particle beams in spacetimes with maximal proper acceleration and the effects of maximal proper acceleration on charge beam dynamics.This procedure leads to the expression (3.4), which is a second order ordinary differential equation respect to the proper parameter of η for EFFECT OF THE MAXIMAL PROPER ACCELERATION IN THE INERTIA 19 the point charged particle.This is especially the case of the clock hypothesis in relativity theory, the requirement that the proper time functional associated to the spacetime metric depends formally only on the world line curve and on the tangent vector along the curve where the 1Email: rigato39@gmail.comIn connection with the above generalization of the metric structure of the space- time, a research program aimed to developed a mathematical theory of fields that depend upon higher order jets was developed motivated by the following insight.In our theory, inertial mass and rest mass are different concepts and while in Bonnor’s theory the inertial mass depends on proper time according to a differential equation law, in our model the inertialmassisconstant,buttherestmassdependsontheacceleration andtherefore on the time by means of an algebraic expression.Although this model is an over-simplified description of the bunch, it is enough to show theoretical evidence of observable effects due to the increase of the kinetic energy of the particle with respect to the relativistic kinetic energy due to the effect of maximal proper acceleration, there must be a decrease in the particle population of the bunches.For the LHC, the effect is mainly due to the acceleration in 4 EFFECT OF THE MAXIMAL PROPER ACCELERATION IN THE INERTIA thebendingmagnets, becausethebunchpasses many times throughdipolemagnets compared withthetimeitpasses byradiofrequencycavities andbecausetheproper acceleration in the dipole magnets is relatively higher than in the radio frequency systems.Finally, we have also discussed the notion of higher order fields and the generalization of exterior derivatives and start product operations to higher order jet differential algebra in appendix A, where also the generalized linear electrodynamics that we use has been briefly introduced.We start by considering the construction of the metric of maximal accelera- tion (see appendix B for a discussion of the notion of spacetime of maximal acceler- ation from the point of view of higher order jet geometry).Let us consider a smooth curve ϑ : I → M whose tangent vector field is time like 4 ′ ′ ′ with respect to the limit Lorentzian metric η, η(ϑ(t),ϑ(t)) < 0, where ϑ(t) is the tangent vector of the curve ϑ : I → M parameterized by an arbitrary parameterDerivatives with respect to τ are denoted by dot notation: ϑ˙(τ) := d ϑ(τ), etc... By a maximal proper acceleration dτ A we mean that the bound (2.4) for the value of the squared proper acceleration max a2 := η(∇ ϑ˙,∇ ϑ˙) holds good, ϑ˙ ϑ˙ (2.5) 0≤ η(∇ ϑ˙,∇ ϑ˙)< A2 .ϑ˙ ϑ˙ max Finally, let us remark that τ[ϑ] is equal or larger than the proper time of the metric of maximal acceleration, which is defined by the expression 1/2 η(∇ ϑ˙,∇ ϑ˙) 1/2 (2.6) s[ϑ]:= dτ −g(ϑ˙,ϑ˙) = dτ 1− ϑ˙ ϑ˙ ,The model of the classical point charged particle in spacetimes of maximal acceleration In the theory of higher order jet electrodynamics the spacetime structure that determines the proper time is a metric of maximal proper acceleration (2.2).Under this assumption and assuming compatibility with the covariant Larmor’s radiation formula [18], general arguments lead to a model for the point charged particle de- scribedbyasecondorderdifferentialequation(seeappendixCforaderivationofthe equation of motion in the framework of generalized electrodynamics fields according to the theory developed in [13]).In local Fermi coordinates of η the equation of motion of a point charged particle (3.1) takes the form 2 (3.4) m x¨µ = qFµ x˙ν − q2x¨νx¨ x˙µ, Fµ = gµρF , µ,ρ,ν = 0,1,2,3.Phenomenological consequences of the redefinition of the energy in accelerator physics The difference in the definition of energy E between the theory with maximal properacceleration andtheenergyofthesystemaccordingtothetheoryofrelativity takes the form 1 1 a2 (4.1) ∆E = γ −1 m c2 = γ m c2+ O(a2/A2 )2, ~v  1− a2  0 2 ~v A2 max 0 maxHowever, if 0 the bunch being accelerated contains N particles and if the bunch of particles is considered as a sole particle, then the maximal acceleration is decreased by a factor N. Hence the excess in energy with respect to the relativistic energy due to a maximal acceleration is given by the expression 1 a2 ∆El=1/E = N2 , rel 2 A2 (1) max where A2 (1) is the maximal proper acceleration that applies to a sole particle.Indeed, current laser electron 12 EFFECT OF THE MAXIMAL PROPER ACCELERATION IN THE INERTIA acceleration systems achieve acceleration of bunches of electrons up to proper accel- erations of order 1022m/s2, which is much larger than the accelerations at LHC.Therefore, as longaslaser-plasmaacceleration canreachmorepopulatedbuncheswiththecurrent top scales of acceleration, the effect discussed in this paper will be increasingly relevant and provide potential observable effect of the equation of motion (3.1) and the maximal proper acceleration (3.3).Fluid models or kinetic models for plasmas in spacetimes of maximal properacceleration could provide significant improvement in the understanding of the dynamics of particle beams in spacetimes with maximal proper acceleration and the effects of maximal proper acceleration on charge beam dynamics.Jet bundles and generalized tensors The notion of a generalized higher order field and the generalized higher order electrodynamics is introduced in this appendix.0 0 The following definition provides the fundamental notion of generalized higher order jet tensor and forms, Definition A.1.(eµ1(kx)∧···∧eµp(jkx)) 7→ ǫ gµ1ν1 ···gµpνpeνp+1(jkx)∧···∧eνn(jkx) ν1...νn and then it is extended to an arbitrary generalized form α = α (jkx)eµ1(jkx)∧···∧eµp(jkx) ∈ ΓΛp(M,F(Jk(M))) µ1....µp 0 by the multilineal property, ⋆α(jkx)= α (jkx)ǫ gµ1ν1 ···gµpνpeνp+1(jkx)∧···∧eνn(jkx).2 Since the theory of higher order jet electrodynamics is formally analogous to Maxwell’s theory, then by a similar procedure as in the standard theory, Larmor’s covariant radiation law holds in the form µ dp 2 (C.3) rad = − (1+c ǫα+···) q2(x¨ρx¨σg )(τ)x˙µ(τ).We assume a form of generalized Lorentz equation, where instead of a Maxwell field we have a generalized Maxwell field as a source of a generalized Lorentz force, m (τ)x¨µ = qF¯µx˙ν = qFµ x˙ν + Bµx˙ − x˙µB x˙ν b ν ν ν ν ... ... + Cµx¨ ν − x¨µC ν x˙ν + D(cid:0)µx ν − xµD ν(cid:1)x˙ν +..., with Fµ = gµρF .On the right hand side of the above expression all the contractions that appear in expressions as Bµx˙ − x˙µB x˙ν, etc... are performed with the metric g. ν ν By using an analogous method to Cartan’s moving frame, the generalized Lorentz (cid:0) (cid:1) force equation is cast in the form m (τ)x¨µ = qFµ x˙ν +β x¨µx˙νx˙ − x˙µβ x¨ x˙ν.This procedure leads to the expression (3.4), which is a second order ordinary differential equation respect to the proper parameter of η for EFFECT OF THE MAXIMAL PROPER ACCELERATION IN THE INERTIA 19 the point charged particle.".You have three tasks, which are:\n    1.to summarize the text I provided into a Summary .Please answer within 150-300 characters.\n    2.to summarize the text I provided, using up to seven Highlight.\n    3.to summarize the text I provided, using up to seven Key Insights. Each insight should include a brief in-depth analysis. Key Insight should not include timestamps.\n    Your output should use the following template strictly, provide the results for the three tasks:\n    ## Summary\n    ## Highlights\n    - Highlights\n    ## Key Insights\n    - Key Insights .\n  Importantly your output must use language "English"
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