]>NUPHB114648114648S0550-3213(19)30134-810.1016/j.nuclphysb.2019.114648The AuthorHigh Energy Physics – TheoryParticle production by gravitational perturbations in domain wallsMajdiAmrmajdi.amr77@gmail.comAmman, 11953, JordanAmman11953JordanAmman, 11953, JordanEditor: Stephan StiebergerAbstractIn this work I will discuss the possibility of creating particles due to the interaction of a weak gravitational perturbation with a spin-zero weakly interacting static one-dimensional domain wall in harmonic coordinates up to zero-order approximation in the resulting Neumann solution.1IntroductionThe idea of particle creation by gravitational waves has been considered extensively by many particle physicists where it has been shown in a previous work by F. Sorge [1] that particles can indeed be created by weakly incident gravitational waves in the TT and harmonic gauges [2] due to its interaction with a bounded scalar field after considering methods of quantum field theory in curved spacetimes (namely considering the Klein-Gordon equation in a curved background and a calculation of some coefficients involved in the analysis) where the number of particles created has also been calculated in contrary to the unbounded case where no particle production was shown to exist due to the absence of the β Bogoliubov coefficient (a parameter related to the number of particles created after a time evolution of the field from one vacuum state to another). Although the physics of domain walls is more involved due to its gravitational nature it still can be shown that particles can be produced due to its interaction with a weak gravitational perturbation in the de Donder gauge [3] where the number of particles created has been calculated here for a thin one-dimensional static example using a time dependent Gausian-like gravitational perturbation after the calculation of the Bogoliubov coefficient involved up to zero order approximation of the Neumann solution of the field perturbation of the wall long-after the interaction has been switched off.2Particle production in non-stationary space-timesStarting with the Lagrangian density(1)L(x)=12−g(gμν(x)∇μφ∇νφ−[m2+ξR(x)]ϕ2(x)) for a scalar field φ of particles with mass m in a curved background (where an extra scalar-gravitational coupling term ξR(x)ϕ2 with R(x) being the Ricci scalar has been added) and assuming that(2)limx0⟶±∞gμν=ημν (i.e. asymptotic flatness so that in these limits our field equation reduces to the usual Minkowski Klein-Gordon equation), we can imagine that the space-time considered is a ‘sandwiched’ space-time domain M=M−∪M0∪M+ as shown below:Therefore, it is possible to write the scalar quantum field solution in M− of the Klein-Gordon equation as(3)ϕ(x)=∑i[aiui(x)+ai†ui⁎(x)]inM− The continuation of this solution through M0 will be(4)ϕ(x)=∑i[aiψi(x)+ai†ψi⁎(x)]inM+ As both sets {ui} and {ψi} are complete we can set(5)ψi=∑j[Aijuj+Bijuj⁎] Thus in M+ we have(6)ϕ(x)=∑i[aiψi(x)+ai†ψi⁎(x)]=∑i[ai∑j(Aijuj+Bijuj⁎)+ai†∑j(Aij⁎uj⁎+Bij⁎uj)]=∑i[ai′ui(x)+ai′†ui⁎(x)] where(7)aj′=∑i[aiAij+ai†Bij⁎] which is called a Bogoliubov transformation with A, and B called the α, and β Bogoliubov coefficients respectively.Here the choice of |vac〉 depends on the choice of the basis, where the particle number operator for the ith mode is(8)Ni=ai†aiinM−(9)Ni′=ai′†ai′inM+The expected number of particles in the ith mode in M+ in the in-vacuum is then(10)〈Ni′〉=〈vac|Ni′|vac〉=〈vac|ai′†ai′|vac〉=∑j,k〈vac|(akBki)(aj†Bji⁎)|vac〉=∑j,k〈vac|akaj†|vac〉BkiBij†=(B†B)iiTherefore, the expected total number of particles is given by tr(B†B) and will vanish if B=0, [4,5].3Domain wallsIn simple words a planar domain wall is a sheet-like distribution of matter that is invariant under translations in the two directions in the plane, rotations around the normal, and reflection z→−z through the midplane of the sheet at z=0. Under these conditions the line element can be written as(11)ds2=A(dt2−dz2)+B(dx2+dy2) where A, and B are some functions of t and |z|. Domain walls appear through the breaking of a discrete symmetry of the vacuum which is usually described in terms of a scalar field. In the simplest realization the vacuum has two degenerate states, in which the scalar field φ can take values φ+ and φ−, say. When on one side of two neighboring regions it takes the value φ+ and on the other side φ−, then a domain wall occurs in the separation layer, with the field φ interpolating between these two values. These are the Z2 domain walls, which arise due to a discrete symmetry breaking of two possible states [6].4Particle creation by gravitational perturbation in a 1-D domain wallWe aim now to the study the interaction of a static 1-D domain wall with a gravitational perturbation in harmonic coordinates [7].4.1Background configuration for a domain wallThe action for a scalar field in a curved background is:(12)Sφ=∫−gLφd4x=∫−g(12gμν∂μφ∂νφ−V(φ))d4x where Lφ is the scalar field Lagrangian, and g is the determinant of the metric involved. The action for the gravitational field is(13)Sg=−116πG∫−gRd4x where R is the Ricci scalar. The equations of motion, of the full action Sφ+Sg for the scalar and gravitational fields can be obtained by considering the total Lagrangian density of the joint fields(14)L=−g(12gμν∂μφ∂νφ−V(φ)−116πGR) from which we get by the use of the Euler-Lagrange equations for both fields the two field equations(15)□φ+dVdφ=0 and certainly the Einstein's equation(16)Rμν=8πG(Tμν−12Tααgμν) (where □=1−g∂μ[−ggμν∂ν] considered to be the generalized d'Alembertian) with Rμν being the Ricci tensor, and Tμν is the energy-momentum tensor given by(17)Tμν=∂μφ∂νφ−gμνLφThe domain wall we are considering is specified by the scalar and gravitational field solutions φ‾ and g‾μν respectively, and it is only assumed that φ‾ depends on z only (i.e. φ‾=φ‾(z)), and that the gravitational field depends on both z, and t with g‾μν(t,z)=g‾μν(t,−z).The wall creates a gravitational field with a line element of the form(18)ds‾2=D(z)(dt2−dz2−e−s|t|(dx2+dy2)) where s≅4πGσ and σ is the wall tension. Therefore, the metric of the wall is given by(19)g‾μν=(D0000−De−s|t|0000−De−s|t|0000−D) with the inverse metric(20)g‾μν=(1/D0000−es|t|/D0000−es|t|/D0000−1/D)A somewhat lengthy calculation of the Ricci tensor components (a Mathematica program in appendix C of [8] would be very handy) gives with R00=D″2D,R11=R22=−D″2De−st,andR33=−32(D′D)′ the following three useful equations for our domain wall(21)φ‾″+D′Dφ‾′−DdVdφ‾=0(22)16πG3(φ‾′2+DV)+(D′D)′=0(23)D″+16πGVD2=0 where some extra assumptions on the fields and parameters involved are assumed as in [7] which we will not consider by any way here in this work.Turning off gravity, i.e. using □=ημν∂μ∂ν in all the previously given equations, the scalar field equation of motion admits a solution φ0(z) satisfying:(24)φ0′2=2V(φ0),φ0″=dVdφ(φ0)The gravitational field produced by the wall back reacts producing the scalar field solution φ‾(z) instead of φ0(z).4.2Gravitational perturbation equationsThe fields of the domain wall as we have previously mentioned is specified by φ‾(z) and g‾μν(t,z) and we will consider perturbations around this background configuration,(25)φ=φ‾(z)+δφ(xμ)(26)gμν=g‾μν(t,z)+hμν(xμ) where φ, and gμν obey the Klein-Gordon and Einstein equations (15) and (16) respectively. We now proceed to derive a linearized equation satisfied by δφ.Substituting equation (25) into the field equation (15) and assuming that:1-g‾μν (as given in (19)) differs slightly from ημν (based on the assumption that the wall has a loose tension i.e. s≅0, with D(z)≅1 at least if we demand that D(0)=1),and as a consequence:2-φ‾ differs also slightly from φ0;3-Terms with products of small contributions (such as hμνδφ, (φ‾−φ0)δφ, (φ‾−φ0)hμν, (g‾μν−ημν)δφ, and (g‾μν−ημν)hμν) are neglected;4-∂μ(−ggμν)=0 (i.e. a use of harmonic coordinates [9]) we end up with an equation for the field perturbation δφ(27)ημν∂μ∂νδφ+δφd2Vd2φ(φ0)=hzzdVdφ(φ0) which can be treated as a unique field by its own (a massless Klein-Gordon field with a source term as will be shown in what follows) and assumed to vanish at both sides of the wall (0,l) where l represents the coherence length of the wall related to the local maximum of the potential V(0) (at least in the φ4 theory) by V(0)≅η2l2 with ±η being the VEV's [10]. Before the gravitational perturbation is switched on (at t⟶−∞) we trivially have δφ=0 which corresponds to the no-particle production ground state |0〉in which is expressed in the notation of (3) of the M− region with the (properly normalized) complete set of basis solutions(28){un=1nπe−inπltsin(nπzl)} of a free massless Klein-Gordon field with a vanishing boundary condition at both sides of the wall.All what we need to show for particle creation to take place is a solution of the field perturbation (at t⟶∞) as a new basis solution that mixes both frequency modes of the old basis solutions, i.e. we need to reach the condition(29)ψi=∑j[Aijuj+Bijuj⁎]The solution of (27) now proceeds as follows:First we can rewrite (27) in the form(30)ημν∂μ∂νδφ=hzzdVdφ(φ0)−δφd2Vd2φ(φ0) which can be considered as an equation of a massless scalar field with a source given by the term on the right hand side of (30), and therefore, the formal solution to δφ is(31)δφ=∫d4x′G(x,x′)(hzzdVdφ(φ0)−δφd2Vd2φ(φ0)) which is a Fredholm integral equation of the second kind with a zero-order approximation given by(32)δφ≈∫d4x′G(x,x′)hzzdVdφ(φ0) where G(x,x′) is the Green's function for a z-bounded wave equation given by(33)Gret(x,x′)=θ(t−t′)2lπ2∑n=1∞∫dk‾∥sin[ωk(t−t′)]ωksin(nπzl)sin(nπz′l)eik‾∥.(x‾∥−x‾∥′) [1] where ωk is given by(34)ωk=n2π2l2+k‾∥2What we need now is to look at the time dependence of the solution (32) to see if we have a two frequency-mode solution for particles to exist after the interaction and calculate the number of particles created if it does indeed exist.The solution to (32) under the assumption that the wall is symmetric in both the x, and y directions gives us a convincing justification to consider a gravitational perturbation that depends only on t and z from which we get(35)δφ=2∑n=1∞sin(nπzl)∫dt′dz′θ(t−t′)sin[nπl(t−t′)]nπsin(nπz′l)hzzdVdφ(φ0)=∑n=1∞sin(nπzl)[einπlt{∫dt′dz′θ(t−t′)e−inπlt′nπisin(nπz′l)hzzdVdφ(φ0)}−e−inπlt{∫dt′dz′θ(t−t′)einπlt′nπisin(nπz′l)hzzdVdφ(φ0)}] writing this in terms of the basis solutions of the original free-field equation (28) we get(36)δφ=∑n=1∞[un⁎{∫dt′dz′θ(t−t′)e−inπlt′nπisin(nπz′l)hzzdVdφ(φ0)}−un{∫dt′dz′θ(t−t′)einπlt′nπisin(nπz′l)hzzdVdφ(φ0)}] Comparing (36) with (4), and (5) above (and by assuming that δφ exists in a flat (or a semi-flat) background for a weakly interacting domain wall after the perturbation has been switched off) we can write(37)δφ=∑n=1∞ψn with(38)ψn=−∫dt′dz′θ(t−t′)einπlt′nπisin(nπz′l)hzzdVdφ(φ0)un+∫dt′dz′θ(t−t′)e−inπlt′nπisin(nπz′l)hzzdVdφ(φ0)un⁎ from which we can write(39)ψn=∑j−∫dt′dz′θ(t−t′)eijπlt′jπisin(jπz′l)hzzdVdφ(φ0)δnjuj+∫dt′dz′θ(t−t′)e−ijπlt′jπisin(jπz′l)hzzdVdφ(φ0)δnjuj⁎ and finally consider that the β-Bogoliubov coefficient to be(40)Bnj=∫dt′dz′θ(t−t′)e−ijπlt′jπisin(jπz′l)hzzdVdφ(φ0)δnj which indeed should prove that particles should be produced after such an interaction of the wall with the gravitational perturbation to zero order approximation of the expansion of the field perturbation under all the assumptions given above.4.3A simple calculational example for a thin domain wall caseConsidering a time dependent Gaussian-like perturbation (suitable for the case of a thin domain wall), i.e.,(41)h∼He−α2t2 we get(42)Bnj=H∫dt′dz′e−ijπlt′jπisin(jπz′l)e−α2t2dVdφ(φ0)δnj which gives(43)Bnj=Hαe−j2π24l2α2∫dz′sin(jπz′l)jidVdφ(φ0)δnj Now according to (10) we have for the total number of particles created after the interaction of a thin domain wall with such a gravitational perturbation(44)N=tr(B†B)=H2α2∑je−j2π22l2α2∫dzdz′sin(jπzl)sin(jπz′l)jdVdφ0(z)dVdφ0(z′) representing a possible value for the number of particles being created in a thin static one-dimensional domain wall after its interaction with a Gaussian-like gravitational perturbation to zero order approximation.5ConclusionAs we can see the proof of particle production in a general static one-dimensional domain wall has been shown to zero order approximation and the number of particles created for the case of a thin domain wall considering a Gaussian like perturbation has been calculated which may stimulate the idea for a similar calculation to other systems other the one at hand and give some sort of encouragement to the applicability of the combination of the methods of quantum field theory to classical theory of general relativity (i.e. the subject of quantum field theory in curved backgrounds). 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