Note on:"Domain wall universe in the Einstein--Born--Infeld theory"Phys. Lett. B 679 (2009) 160

The 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 [1] this fact seems to be ignored.

The (n + 1)−dimensional bulk space time with Z 2 symmetry can equivalently be chosen as (i.e. Eq. (4) of Ref. [1]) in which dΩ 2 n−1 is the line element on S n−1 . The n-dimensional domain wall (DW) in the FRW form is with the constraint in which a dot implies d dτ . The Israel junction condition As considered in Ref.
Comparison with the general form of S ν µ implies that the induced electrostatic energy density on the DW is while the pressure is Now, taking into account Eq.s (5) and (6), we get two equations to be satisfied simultaneously, i.e. and HereinĀ τ is given in terms of the bulk potential and metric function bȳ The angular part of Israel equation admits which is clearly not a constant. In Ref.
[1] the authors consider a new constant parameter χ 2 = κ 2 n+1 (σℓ) 2 /4 (n − 1) 2 and by setting L 0 = 0 (i.e. zero pressure) they find an equation of motion for the dynamic domain wall, based only on Eq. (14), which readsȧ 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 = cons. 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.
we observe that This (which is set to zero for simplicity). As a result the two Israel junction conditions are consistent, i.e.
By differentiating (21) one obtains which reduces (22) to (21). Therefore these two equations amount to the single Eq. (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 Eq.s (17) into (14) i.e., in which Given the complexities of f (R) andĀ 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) andĀ T are given (from Eq. (12) and (16) of Plugging these expressions with (25) into (24) (for κ 2 n+1 = 1, C = −2 √ 3q 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.
In the same manner one finds Figure Caption: Fig. 1: The 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 > r h ). The choice of parameters is : C < 0, q = 4.5, m = 6 and ℓ = 0.3. The exact location of the event horizon (r h ) is shown in the smaller figure for f (r) .