Non-topological gravitating defects in five-dimensional anti-de Sitter space

Massimo Giovannini

Centro `Enrico Fermi', Compendio del Viminale, Via Panisperna 89/A, 00184 Rome, Italy
² Department of Physics, Theory Division, CERN, 1211 Geneva 23, Switzerland

Email: massimo.giovannini@cern.ch

Received 27 July 2006, in final form 5 October 2006
Published 24 October 2006

PACS numbers: 04.50.Th, 11.10.Kk, 11.25.Mj, 11.27. + d

In the presence of infinite extra dimensions [1] (see also [2, 3]) fields of various spin are localized around higher dimensional gravitating defects whose properties determine, at least partially, the features of the localized interactions. Consider, therefore, one of the simplest incarnations of this idea and suppose that there is only one infinite extra dimension that will be denoted, in what follows, by w. The five-dimensional line element can then be written as

where η_μν is the Minkowski metric with signature mostly minus; the Latin (uppercase) indices run over all the five dimensions while the Greek indices run over the (3  +  1) observable dimensions. The coordinate w runs continuously from –∞ to  + ∞. In the situation described by equation (1), five-dimensional domain-wall solutions are known to exist [4–7] (see also [8–10]) and they have the structure of gravitating kinks whose associated geometry is rather similar to the one generated by 3-brane sources supplemented by a negative cosmological constant [11]. Five-dimensional gravitating kinks can arise both in the case of Einstein–Hilbert gravity and in the case of quadratic gravity theories of Euler–Gauss–Bonnet type (see, for instance, [12–15] and references therein).

The scalar–tensor action adopted for illustrating the present considerations will then be given by

leading to the equations^Note1

where . A consistent solution of equations (3) and (4) can be obtained in the form

where b is a parameter (with dimensions of inverse length) related with the thickness of the scalar profile and where

In equation (6), v arises as an integration constant with the same dimensions of β^–1. In equations (5), (6) and (7) ν is a positive integer (i.e. ν ≥ 1). Since the bulk coordinate w may take both positive and negative values, if ν would be rational or even real, the functions defining the solution will may become imaginary^Note2. The curvature invariants pertaining to the solution defined by equations (5), (6) and (7) can be simply computed and they are

where R_ABCD, R_AB and R are, respectively, the Riemann tensor, the Ricci tensor and the Ricci scalar. In the case of the metric (1), the Weyl invariant vanishes. Since ν ≥ 1, equation (5) implies that a(w) → |bw|^–1 in the limit |bw| → ∞, i.e. for values of the bulk coordinate much larger than the thickness of the configuration. Since ν ≥ 1 is a positive integer, the curvature invariants do not have poles for any finite value of w. The quantity w₀  =  b^–1 is the radius of the (asymptotic) AdS₅ space. Consistently with this behaviour, the explicit form of the curvature invariants goes to a constant for |bw| → ∞.

If ν is odd, i.e. ν  =  2m  +  1 with m  =  0, 1, 2, 3,  ..., equation (6) implies that β( – v) varies between –π/2 and π/2. The plus sign in equation (6) corresponds to the kink solution while the minus sign corresponds to the anti-kink solution. In the case of one spatial dimension, spatial infinity consists of two points, i.e. ±∞; a topological charge is then customarily defined for the characterization of (1  +  1)-dimensional defects such as the ones arising in the case of sine–Gordon system [16]. When ν is odd, therefore, we will have that the topological charge does not vanish and is given, in particular, by

where the plus and the minus signs correspond, respectively, to a kink and to an anti-kink solution.

If ν is even, i.e. ν  =  2n with n  =  1, 2, 3,  ..., from equation (6), β( – v) goes, asymptotically, to the same value both for w → –∞ and for w →  + ∞. Therefore, applying the definition reported in the first equality of equation (12), we will have, in this case, that Q_n  =  0 in spite of the sign appearing in equation (6). This second class of solutions seems then to describe more non-topological rather than topological defects. It should be clear that five-dimensional gravity is essential in order to have this type of profiles. In the absence of gravity, non-topological defects in (1  +  1) dimensions are connected with an additive conservation law, so that one should demand that the system contains, at least, a complex scalar field with global U(1) symmetry [16] or, equivalently, two real scalar fields [17]. Here, however, because of the presence of gravity, bell-like profiles can arise even if is not complex as the solution (6) demonstrates explicitly when ν is even.

For graphical illustration, it is practical to rescale through β in such a way that the rescaled field, i.e. , is dimensionless. In the following, when not otherwise stated, we will also fix, without loss of generality, v  =  0. The cases m  =  0 and m  =  1 (i.e. ν  =  1 and ν  =  3) are illustrated, respectively, in the left and in the right plot of figure 1. As m increases an intermediate plateau develops close to w  =  0 (see the right plot in figure 1 and the left plot in figure 2). In the case m  =  0 (full line in the right plot of figure 2) the potential is of a sine–Gordon type and it is, according to equation (7), U()  =  U₀(5cos 2β – 3)/2. The minima of the potential are located, for ν  =  1, in –π/2 and in  + π/2 (see the right plot of figure 2). By looking simultaneously at figure 1 (full line in the left plot) and at figure 2 (full line in the right plot) it appears that the kink solution connects the minimum in –π/2 to the minimum in π/2 and β correctly interpolates between these two values. This situation is a bit reminiscent of the sine–Gordon system in (1  +  1) dimensions [16] where, however, the potential vanishes at the minima while here it is negative due to the gravitating nature of the solution. As m increases the potential develops, at the centre of the interval of periodicity, a second (local) minimum which is located, for the interval chosen in figure 2, in  =  0. Since the minimum is only local (and not global) the field does not settle down and finally reaches the true global minimum in π/2. As m increases further (dot-dashed lines in both plots of figure 2), the local minimum becomes more and more pronounced and the length of the intermediate plateau in gets larger (see figure 2, left plot). According to equation (6), both  ' and  '² are always finite and regular for every value of the bulk radius. In figures 1 and 2 βv  =  0 has been assumed.

Figure 2. In the left plot the behaviour of is illustrated for large (odd) ν (as in figure 1, ν  =  2m  +  1). In the plot at the right the potential U() is reported for three different values of m as specified in the legend.

If ν is even the scalar profile goes, asymptotically, to the same value for w → ±∞. In figure 3 we report the profile of β and its related potential for few values of even ν and for two different values of βv (i.e. βv  =  0 and βv  =  π/2). In the left plot of figure 3 the scalar field is illustrated as a function of the bulk radius for two different values of βv (i.e. βv  =  0 and βv  =  π/2). By increasing the value of n the width of β increases (dashed line in the left plot of figure 3). Given the properties of this second class of solutions, the case of even ν resembles that of a non-topological defect.

As already mentioned the geometry is AdS₅ for |bw| → ∞. This aspect can be clearly appreciated from figure 4 (left plot) where the warp factor is illustrated for different values of ν: for |bw| → ∞, a(w) ≊ 1/|bw|. The curvature invariants, in the same limit, reach a constant value. In figure 4 (right plot) the Riemann invariant is illustrated. The other curvature invariants are qualitatively similar. As a consequence of the features of the geometry the four-dimensional Planck mass is finite since it is simply given by

where the second equality follows by performing explicitly the integral when a(w) is given by equation (5) and when, as assumed throughout, ν ≥ 1. Since the four-dimensional Planck mass is finite, the tensor fluctuations of the geometry are localized on the profile both for even and odd ν. This occurrence is common also to the case where the defect is modelled by a 3-brane [11]. Less obvious is the fact that the scalar fluctuations of the sources are not localized. These results stem from the analysis of the zero modes of the configurations defined by equations (5), (6) and (7) and will now be swiftly addressed. To discuss this problem one can then adopt the formalism developed in [6]. In five dimensions the perturbed geometry leads to 15 independent degrees of freedom which can be classified according to the way they transform under four-dimensional Poincaré transformations. To the fluctuations of the geometry one has also to add the fluctuation of the , i.e. the fluctuation of the profile of the defect. Therefore, the fluctuations of the geometry and of the scalar profile can be written as

where

On top of h_μν which is divergence-less and trace-less (i.e. ∂_μ h^μ_ν=0, h^μ_μ  =  0) there are four scalars (i.e. E, ψ, ξ and C) and two divergence-less vectors (D_μ and f_μ).

The analysis can be conducted in gauge-invariant terms without assuming any specific form of the background geometry. Neglecting the vector modes of the geometry^Note3 the relevant zero modes are the ones associated with the graviton and with the scalar fluctuations. The decoupled evolution equation of the tensor modes can be written as [6]

where μ_μν  =  a^3/2h_μν is the canonical normal mode of the of the action (2) perturbed to second order in the amplitude of tensor fluctuations [6]. The lowest mass eigenstate of equation (16) is μ(w)  =  μ₀ a^3/2(w). Hence, the normalization condition of the tensor zero mode implies

The integral appearing in equation (17) is always convergent if, as assumed throughout the paper, ν is a positive integer. Therefore, the graviton zero mode is always localized on the configurations discussed here. The scalar normal mode of the action is a linear combination χ (defined in equation (14)) and of ψ (defined in equation (15)). The canonical variable is then [6] and it obeys the equation

The lowest mass eigenstate of equation (18) is given by which is normalizable iff

where the right-hand side follows from the definition off z(w) and from the explicit form of the solution. But the integrand in equation (19) is divergent for |bw| → 0 as |bw|^–2ν. As ν increases, the divergence becomes always more severe. We then conclude that the scalar modes of the geometry are not localized on the defect.

The solutions described in the present paper admit a geometry that is, asymptotically, AdS₅. The question one may ask is how general this occurrence is. Put it in other words one could ask if other regular geometries with finite Planck mass may give rise to interesting topological or non-topological solutions. If the warp factor a(w) is not AdS₅ for |w| → ∞ two possible situations may arise: either the warp factor goes to zero faster than in the AdS₅ or it goes to zero more slowly than in the AdS₅. Let us examine these two situations separately.

If the warp factor goes to zero for |w| → ∞ faster than AdS₅, then we can parametrize it as a(w) ≊ |w|^–1–α where α > 0 (if α  =  0 we would be simply in the case of AdS₅). In this case the four-dimensional Planck mass of equation (13) will be finite. The graviton zero mode will also be localized since the integral appearing in equation (17) will be finite. However, this situation will also lead to curvature invariants that will diverge typically as |w|^4α. Therefore, in this situation, the geometry will not be regular for large absolute value of the bulk radius.

Suppose, next, that the warp factor goes to zero for |w| → ∞ more slowly than in the case of AdS₅. In this case the warp factor can be parametrized, for large |w|, as a(w) ≊ |w|^–1 + β with β > 0. Again, the case β  =  0 simply corresponds to the AdS₅ case. In this case the curvature invariants are all finite for |w| → ∞ and converge faster than |w|^–1 provided β > 1/4. The four-dimensional Planck mass is only finite provided β < 2/3. The same condition arises for the normalizability of the graviton zero mode. Therefore, in this second situation and according to the chosen parametrization, 1/4 < β < 2/3.

Consider, as an example, the following choice of the warp factor:

For large absolute value of the bulk coordinate the asymptotic behaviour of the warp factor is controlled by γ and it is, in particular, a(w) ≊ |bw|^–2/γ. Not for every γ and μ exact solutions exist. However, for γ  =  1 and μ  =  1 there is an exact solution that can be written as

From equation (21) it is clear that, for |w| → ∞ a(w) ≊ |bw|^–2, i.e., according to the considerations of the previous paragraphs, the curvature invariants will diverge for |w| → ∞. Moreover, it is clear from equation (22) that the scalar field does not interpolate between two minima, or, even more precisely, it does interpolate between two minima for → ±∞.

In relation to the AdS₅ nature of the obtained solutions, it would be interesting to understand if the solutions described in this paper could be obtained from a supersymmetric background. The answer to this question seems to be negative since close to the core of the defect the spacetime is not AdS₅. However, a more complete analysis is required.

In conclusion, a new class of solutions of five-dimensional warped geometries has been presented and discussed. This class of solution contains, simultaneously, kink-like profiles and bell-like scalar profiles. The regular geometry of the configuration allows the localization of the tensor modes of the geometry. Neither the scalar nor the vector modes are localized. The present findings seem to suggest that not only domain walls but also gravitating non-topological defects in five dimensions may be used to localize gravitational interactions. Furthermore, it is intriguing that these two rather different physical situations may arise in the same class of solutions.

On the basis of the obtained results many questions may arise. For instance, one could wonder what are the cosmological implications of the considerations developed here. It could be interesting to know, for instance, how the static configurations can be generalized to the time-dependent case. In particular, in the non-topological case it is not clear if the known drawbacks of brane-world cosmologies in five dimensions may be overcome. These topics may represent an interesting avenue for future studies.

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Notes

Note1

 In the following, the prime will always denote a derivation with respect to the bulk coordinate w.

Note2

 From a swift inspection of equations (5), (6) and (7) it may seem that the solution can be continued also for negative values of ν. This is not correct since, if ν < 0,  '² becomes negative, or, equivalently, the parameter β defined in equation (8) becomes imaginary.

Note3

 The vector modes are not localized since their corresponding zero mode is not normalizable as it follows from the evolution equations of D_μ; the other vector, i.e. f_μ, can be gauged away by using the freedom of fixing the coordinate system.