application/xmlNote on “Domain wall universe in the Einstein–Born–Infeld theory” [Phys. Lett. B 679 (2009) 160]S. Habib MazharimousaviM. HalilsoyDomain wallBorn–InfeldEinstein–MaxwellNon-linear electromagnetismPhysics Letters B 697 (2011) 497-499. doi:10.1016/j.physletb.2011.02.034journalPhysics Letters BCopyright © 2011 Elsevier B.V. All rights reserved.Elsevier B.V.0370-2693697521 March 20112011-03-21497-49949749910.1016/j.physletb.2011.02.034http://dx.doi.org/10.1016/j.physletb.2011.02.034doi:10.1016/j.physletb.2011.02.034http://vtw.elsevier.com/data/voc/oa/OpenAccessStatus#Full2014-01-01T00:14:32ZSCOAP3 - Sponsoring Consortium for Open Access Publishing in Particle Physicshttp://vtw.elsevier.com/data/voc/oa/SponsorType#FundingBodyhttp://creativecommons.org/licenses/by/3.0/JournalsS300.3PLB27409S0370-2693(11)00179-110.1016/j.physletb.2011.02.034S0370-2693(09)00843-010.1016/j.physletb.2009.07.026Elsevier B.V.TheoryFig. 1The plot of radius a(τ) of the FRW universe for n=4, on the domain wall as a function of proper time. The oscillatory behavior reveals a bounce at a distance greater than the horizon (a>rh). The choice of parameters is: C<0, q=4.5, m=6 and ℓ=0.3. The exact location of the event horizon (rh) is shown in the smaller figure for f(r).NoteNote on “Domain wall universe in the Einstein–Born–Infeld theory” [Phys. Lett. B 679 (2009) 160]S. HabibMazharimousavi⁎habib.mazhari@emu.edu.trM.Halilsoymustafa.halilsoy@emu.edu.trDepartment of Physics, Eastern Mediterranean University, G. Magusa, North Cyprus, Mersin 10, Turkey⁎Corresponding author.Editor: T. YanagidaAbstractThe interaction between bulk and dynamic domain wall in the presence of a linear/non-linear electromagnetism make energy density, tension and pressure on the wall all variables, depending on the wall position. In Lee et al. (2009) [1] this fact seems to be ignored.KeywordsDomain wallBorn–InfeldEinstein–MaxwellNon-linear electromagnetismThe (n+1)-dimensional bulk space time with Z2 symmetry can equivalently be chosen as (i.e. Eq. (4) of Ref. [1])(1)ds2=−f(R)dT2+dR2f(R)+R2dΩn−12 in which dΩn−12 is the line element on Sn−1. The n-dimensional domain wall (DW) in the FRW form is(2)ds2=−dτ2+a(τ)2dΩn−12 with the constraint(3)f(a)T˙2−a˙2f(a)=1 in which a dot implies ddτ.The Israel junction condition(4)[Kμν−gμνK]=−κn+12Sμν leads to (with Z2 symmetry)(5)−2(n−1)af+a˙2=κn+12(ρ+σ),for ττ component,(6)2(n−2)af+a˙2+f′+2a¨f+a˙2=κn+12(p−σ),for θiθi components. As considered in Ref. [1] the DW energy momentum Sμν=diag(−ρ−σ,p−σ,…,p−σ) is given by(7)Sμν=2−gδδgμν∫dnx−g(−σ+Lm) in which (Eq. (22) of Ref. [1])(8)Lm=L0+Can−1A¯τ and C=±q2(n−1)(n−2)κn+12. By using (7) one finds(9)Sμν=−2δLDWδgμν+LDWgμν for LDW=(−σ+Lm). The latter equation implies (see Appendix A)(10)Sττ=2Can−1A¯τ+L0−σ, and(11)Sθiθi=L0−σ(i=1,…,n−1). Comparison with the general form of Sμν implies that the induced electrostatic energy density on the DW is(12)ρ=−2Can−1A¯τ−L0 while the pressure is(13)p=L0. Now, taking into account Eqs. (5) and (6), we get two equations to be satisfied simultaneously, i.e.(14)−2(n−1)af+a˙2=κn+12(−2Can−1A¯τ−L0+σ) and(15)2(n−2)af+a˙2+f′+2a¨f+a˙2=κn+12(L0−σ). Herein A¯τ is given in terms of the bulk potential and metric function by(16)A¯τ=A¯Tf+a˙2f. The angular part of Israel equation admits(17)κn+12(σ−L0)=−(2(n−2)af+a˙2+f′+2a¨f+a˙2), which is clearly not a constant. In Ref. [1] the authors consider a new constant parameter χ2=κn+12(σℓ)2/4(n−1)2 and by setting L0=0 (i.e. zero pressure) they find an equation of motion for the dynamic domain wall, based only on Eq. (14), which reads(18)a˙2+V(a)=0. Plotting rescaled form of V(a) for fixed values of χ (namely χ=1.1) is the last stage of Ref. [1]. Based on our argument on the other hand setting χ to a constant value is equivalent to setting σ=const. which is obviously in contradiction with the form of σ we found in Eq. (17) above. In other words, choosing σ=const. does not satisfy both of the Israel junction conditions at the same time.Unlike this case, if we neglect the interaction between the bulk and domain wall in the form of Nambu–Goto action, i.e.(19)SDW=−σ∘∫Σdnx−g we observe that(20)Sμν=−2δLDWδgμν+LDWgμν=−σ∘gμν. This means from Sμν=diag(−ρ−σ,p−σ,…,p−σ)=diag(−σ∘,−σ∘,…,−σ∘) that −ρ=p=const. (which is set to zero for simplicity). As a result the two Israel junction conditions are consistent, i.e.(21)−2(n−1)af+a˙2=κn+12σ∘,(22)2(n−2)af+a˙2+f′+2a¨f+a˙2=−κn+12σ∘. By differentiating (21) one obtains(23)f′+2a¨f+a˙2=2f+a˙2a, which reduces (22) to (21). Therefore these two equations amount to the single equation (21).Our conclusion to this problem simply implies a more complicated equation of motion for the dynamic domain wall that emerges from the substitution of Eqs. (17) into (14), i.e.,(24)a¨+(G−1)aa˙2+(G−1)fa+f′2=0, in which(25)G=κn+12(Can−2A¯T1f). Given the complexities of f(R) and A¯T for the Einstein–Born–Infeld theory [1], Eq. (24) is a rather difficult differential equation to be solved. To give an idea about its structure yet we resort to the 5-dimensional cosmological Einstein–Maxwell theory (n=4 and β→∞ limit of Ref. [1]). Solution for f(R) and A¯T are given (from Eqs. (12) and (16) of [1] with β→∞) by(26)f(R)=1+R2ℓ2−m2R2+q2R4,(27)A¯T=32qR2. Plugging these expressions with (25) into (24) (for κn+12=1, C=−23q and R=a(τ)) plots the f(R) which in turn determine numerical integrations of (24) for specific parameters. We remark, that depending on the initial conditions and parameters falling into black hole or escaping to infinity and any possibility in between those two extremes are available. We plot, for instance in Fig. 1 the bouncing property of a(τ) with the choice C<0. It should be remarked that with the choice C>0, there is no bounce.Appendix ATo find Sττ and Sθiθi we use the formula (9), and consider(28)LDW=(−σ+Lm), in which(29)Lm=L0+Can−1A¯τ=L0+Can−1A¯μgμτ. Now, in our variational principle we assume σ to be independent of gμν. Variation of LDW with respect to the canonical variable gμν leads accordingly to(30)δLDW=Can−1A¯μδgμτ+δ(Can−1)A¯μgμτ=Can−1A¯μ(12gμτgαβ−gμαgτβ)δgαβ, which, after substitution into (9), it implies(31)Sαβ=−2δLDWδgαβ+LDWgαβ=−2Can−1A¯μ(12gμτgαβ−gμαgτβ)+(−σ+L0+Can−1A¯τ)gαβ.One obtains(32)Sττ=−Can−1A¯τ−(−σ+L0+Can−1A¯τ)=(−2Can−1A¯τ+σ−L0) or equivalently(33)Sττ=2Can−1A¯τ+L0−σ. In the same manner one finds(34)Sθiθi=L0−σ.References[1]B.H LeeW.LeeM.MinamitsujiPhys. Lett. B6792009160