Generalised Ellis-Bronnikov wormholes embedded in warped braneworld background and energy conditions

Ellis-Bronnikov (EB) wormholes require violation of null energy conditions at the `throat'. This problem was cured by a simple modification of the `shape function', which introduces a new parameter $m\ge 2$ ($m=2$ corresponds to the EB model). This leads to a generalised (GEB) version. In this work, we consider a model where the GEB wormhole geometry is embedded in a five dimensional warped background. We studied the status of all the energy conditions in detail for both EB and GEB embedding. We present our results analytically (wherever possible) and graphically. Remarkably, the presence of decaying warp factor leads to satisfaction of weak energy conditions even for the EB geometry, while the status of all the other energy conditions are improved compared to the four dimensional scenario. Besides inventing a new way to avoid the presence of exotic matter, in order to form a wormhole passage, our work reveals yet another advantage of having a warped extra dimension.

However, they are not adequate to form macroscopic wormholes. To be precise, they are allowed in the quantum theory, but, the time it takes to travel through the wormhole should be longer than the time it takes to travel between the two mouths on the outside. Therefore only microscopic wormholes were found using standard model matter.
However, it is also known for long that there are classical ways as well to circumvent the problem of exotic matter that violates the energy conditions [18][19][20][21]. One such way is the framework of large class of the so-called modified theories of gravity. There are large number of such models exist in modified gravity that have non-exotic matter [22][23][24][25][26][27][28], though, in some cases, the convergence condition of null geodesics is violated. Models of dynamical wormholes [29][30][31][32][33] also provide ways of restricting the violation of energy conditions. Other popular class modified gravity theories, where detailed analysis of wormhole geometries is done with viable matter source, are the so-called f (R) and higher order gravity theories [34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50]. Recently, successful modelling and analysis of energy condition satisying wormholes are carried out within the framework of Born-Infeld gravity [51][52][53] and torsional gravity [54][55][56]. Note that, all these different scenarios in general have different signatures in different physical phenomenon, particularly in gravitational lensing [57][58][59]. With the advent of gravitational wave astronomy era [60][61][62] and recent blackhole photography [63], search for wormhole signature do not seem unreal. Recently, the possibility of existence of astrophysical wormholes in the dark matter galactic haloes is raised in [64,65]. Wormholes have entered into the catalogue of the so-called blak hole mimickers and their unique signature can be imprinted on during merger phenomena [66,67] or through nature of their quasi-normal modes [68,69] etc. Such signature would also support the case for modified theories of gravity over general relativity. The main purpose of this article is to investigate an yet unexplored modified gravity scenario namely a wormhole embedded in a warped five dimensional thick braneworld. Note that, earlier investigations are done on wormholes embedded in Kaluza-Klein, DGP and Randal-Sundrum thin braneworld scenario [70][71][72][73][74][75] as such. Below we discuss motivation behind this analysis.
Though yet to be detected in experiments, extra dimensions are around in the literature for almost a century now [76,77]. The reason behind the survival of this idea for so long is in the advantages one get in having them. For instance, while making models of unification (such as superstrings [78]), or how the age-old hierarchy problem can be solved with extra dimensions [79,80]. Recently, in the context of reinterpreting the standard model (and what may lye beyond) using octonions, extra dimensions appear naturally [82]- [88]. Thus theories of extra dimensions have strong footing on basis of fundamental physical symmetries and are not mere useful extension of existing theories as such. Perhaps the most popular among these higher dimensional models are the so-called warped braneworld models [81,89,90].
This model assumes a non-factorizable geometry-a curved five dimensional spacetime where the geometry of the four-dimensional part depends on the extra dimension through a warping factor (a feature unique to this class of models).
Motivated by the appearance of extra dimensions in fundamental physics, we ask what new features extra dimensional models may induce on wormhole passages. Here we investigate a straightforward embedding of a four dimensional wormhole in a static five dimensional warped geometry. The family of wormholes we choose for embedding is based on [91] where the well-known Ellis-Bronnikov (EB) spacetime [92,93] had been extended to provide a generalised family (GEB) of spacetimes that satisfies the null energy condition and further detailed studies is done in [69]. For bulk geometry, we choose the so-called thick braneworld scenario [94,95] where the growing or decaying warping factor is a smooth function of the extra dimension and thus represent thick domain wall solutions. A thick brane scenario is preferred over the originally proposed infinitely thin Randall-Sundrumbranes as the former do not introduce Dirac delta functions in the field equations and they naturally appear if one takes into account quantum effects and minimum length scales.
Our intention here is to first figure out whether such models satisfy energy conditions or not, i.e., whether they admit matter sources that satisfy energy conditions. Recently it is reported that energy conditions are satisfied for wormholes embedded in Randall-Sundrum type thin brane models [96]. We on the other hand, want to see if the presence of a smooth warping factor can lead to a viable wormhole geometry with energy condition satisfying matter source. This is an yet unexplored feature of these class of models.
Our program is as follows. In the next section, we briefly review the above mentioned generalised Ellis-Bronnikov wormhole geometry. Then we introduce our five dimensional model and the resulting field equations. In Section III, we review (analytically wherever possible otherwise numerically/graphically) whether the energy conditions are satisfied (locally and/or globally) or not for the four dimensional model. Following this, we investigate how the status of the energy conditions are modified due to the presence of a warped extra dimension (with both decaying and growing warp factor). At the end we conclude with summarising the key results and future plans.

5D WARPED SPACE-TIME
Assuming phantom (negative kinetic energy) scalar field, Ellis and Bronnikov constructed static, spherically symmetric, geodesically complete, horizon-less Lorentzian wormholegeometry connecting two asymptotically flat regions [92,93]. The merric of the Ellis-Bronnikov wormhole is given by where The parameter m takes only even values to make r(l) smooth over the entire domain of the so-called 'tortoise' or 'proper radial distance' coordinate l (where −∞ ≤ l ≤ ∞). Metric (2), in terms of the usual radial coordinate r, can be written as where r and l are related through the shape function b(r) as, It is straightforward to derive the energy-momentum tensor that results in the geometry represented by metric (2) using Einstein tensor and Einstein equations. In the frame basis (denoted by indices with hat), the diagonal components of the energy-momentum-tensor Tμν (μ,ν =0,1,2,3) can be identified as T00 = ρ, T11 = p 1 = τ , T22 = p 2 and T33 = p 3 , where ρ is the energy-density, p 1 = τ is the radial tension, p 2 and p 3 are the principal pressure [1] of the corresponding matter source. Due to spherical symmetry p 2 is equal to p 3 . Thus the non-zero components of Tμν for the GEB wormhole space-time (2) are where a prime denotes derivative with respect to l. In the following, we introduce a 5D warped spacetime where the 4D part is GEB wormhole. However the corresponding energy momentum tensor will be derived later.
A general warped line element in five dimensions is given as where g αβ can, in principle, be any metric and g 44 can be a function of 3-space, time, and the extra spatial dimension-y, not necessarily separable. The line element, representing an embedded wormhole, we choose to work with is as follows: Here the factor, e 2f (y) , is called a warp factor. The domain of y, which is the extra spatial dimension, can be −∞ < y < ∞. In the following, we shall set f (y) = ± log[cosh(y/y 0 )], which correspond to the well-known thick brane models [94,95]. In such models, the brane is dynamically generated as a scalar field domain wall in the bulk. Note that the warp factor in such models is a smooth function of the extra dimension, unlike the Raldall-Sundrum models where f (y) ∼ |y| (i.e. a function with a derivative jump that implies presence of thin branes). Thick brane models do not posses the jumps and delta functions in the connection and curvature. They also appear naturally in multi-dimensional theories. In fact, Quantum fluctuations are expected to create an effective brane thickness. Note that the Ricci Scalar for metric (11), is given by Thus the curvature invariants of the warped model are essentially singularity free unlike some models of black holes in higher dimensions. However, if one considers thin brane models for embedding, then Dirac Delta functions would appear in the curvature to account for those infinitely thin branes.

III. ENERGY CONDITIONS
The energy conditions (EC) are mathematical restriction on solutions of the Einstein equations, to rule out the non-physical solutions. A spacetime geometry may satisfy the one or many or all of the weak, strong, dominant and null energy conditions (WEC, SEC, DEC and NEC) to be physically viable solutions of Einstein equations. These energy conditions lead the following inequalities involving the energy and momentum densities- Violation of these energy conditions would imply the existence of exotic matter (Matter with negative energy density). Note that, as the NEC is implied by the WEC, we would not discuss NEC in the present context. Primarily, Morris and Thorne found that for existence of stable traversable wormholes, violation of WEC is required atleast at the throat [1]. Further studies revealed that presence of the so-called exotic matter is necessary for the stability of all classes of static wormholes. But, there are no observation in support for the presence of exotic matter, which raises doubt on the reality of wormholes.
However, in various modified gravity models, as mentioned in the Introduction, modification of general relativity can serve the purpose of exotic matter. Thus providing stability to wormohole geometry even in presence of energy condition satisfying matter as such. In the following we are going to explore whether the presence of warped extra dimension can lead to similar result. To compare the two models given by Eq. (2) and Eq.
(11), we check analytically (wherever possible) and graphically how the energy conditions behaving throughout the spacetime.

A. Inequalities of WEC for GEB-Space-time
WEC states that the energy density of any matter distribution should be non-negative for any time-like observer in space-time, which implies ρ ≥ 0 and ρ + p i ≥ 0. Using Eqs (6) to (9), we define 'inequality functions', f Here, prime denotes derivative with respect to the 'Tortoise' coordinate l and f comes from the fact that p 2 = p 3 (see Eqs. (8) and (9)). According to the inequalities of WEC, the given functions (f , where µ = 0, 1, 2, 3) should be greater than or equal to zero. Using the expression of r(l) (given by Eq. (5)) and setting b 0 = 1, the inequality functions simplify as given in Table I for three cases with m = 2, m = 4 and m = 8.
Function  Thus the WEC is partially satisfied so to speak. We plot the inequality functions versus l in Fig. 1 is essentially equal to the matter energy density, the first plot in Fig. 1 shows that the negative energy matter accumulates most near the throat for m = 2 and moves away from the throat for m > 2. Thus exotic matter gets localised inside a increasingly narrow region with increasing m. Thus giving a physical understanding of the parameter m. It can be further shown that for increasing m the minima of the energy densities approaches l = ±b 0 . Thus characterising ±b 0 as a length scale where there is uniformly distributed (in l) positive density matter bounded by two infinitely thin (in limit m → ∞) negative energy "walls", beyond which the of the energy density vanishes rapidly.

B. Inequalities of SEC for GEB-Space-time
. Thus, from Eqs (6)-(9) and (18)-(19) we write the 'inequality functions' for SEC, f Eq. (20) implies that the first inequality function of SEC is always zero over the entire domain of coordinate l (−∞ ≤ l ≤ ∞). Eqs. (21) and (22), imply that the behaviour of these inequality functions are same as those analysed in the case of WEC.
DEC essentially implies ρ − |p j | ≥ 0. Again, from Eqs (6-9) we write the corresponding inequality functions as  Table II.       We get the energy-momentum tensor as the Einstein tensor for our metric (11) as before.
Clearly, Eqs. (25)- (29), suggests that the energy inequalities might behave differently due to the appearance of the new terms that depend on the derivatives of the decaying/growing warp factors. In the following, we analyse these inequality functions in detail.

A. Inequalities of WEC
The inequality functions F A=01,2,3,4 for WEC corresponding to our 5D model (11) are given below: (l, y) = ρ(l, y) + p 2 (l, y) It is easy to see that F (W ) 1 and F (W ) 2,3 only gets an overall positive multiplicative factor. So there status would not change compared to the 4D case. However, due to appearance of y-derivative terms, there is possibility that F 0 may satisfy the inequality (at least partially/locally) for decaying warp factor. It is also clear that, for growing warp factor there is no such hope. In the following, we analyse these inequalities as functions of l at various locations along the extra dimension. In Table IV, for decaying warp factor, we present the analytical expressions of the inequality functions, at y = 0 (typical location of the brane). Note that, at y = 0, F , therefore behaviour of this inequalities everywhere in l is similar as the four dimensional case.  Remarkably, in contrast to the four dimensional case (see Table I), here we see that F (does satisfy the inequality for decaying warp factor everywhere) vs l as their behaviour is obvious from their analytic expressions.

B. Inequalities of SEC
As usual, we can write the first inequality, ρ(l, y) + 4 i=1 p i (l, y) ≥ 0, of SEC using Eqs (25) - (29). All remaining inequalities of SEC, ρ(l, y) + p i (l, y) ≥ 0, are already being implied in the WEC section above. The inequality function F In Table V,   Decaying   Table V implies that, for the case of decaying warp factor, at or near y = 0, the inequality is not satisfied anywhere in l for all m. On the other hand, this inequality is always satisfied everywhere in l in the growing warp factor scenario. Let us to look at the parameter space again in the case of decaying warp factor which is presented in Fig. 6. This shows that for m = 2 the inequality is satisfied almost everywhere in l − y plane except the region which essentially represents the location of the thick brane. However, for of m → ∞, the inequality is satisfied at or near of l → ±1 (these are the regions about to pinch for large m in Fig. 6). On the other hand, in presence of the growing warp factor, the inequality is satisfied everywhere in the l − y plane as is obvious from Eq. (34). Taking suggestion from the parameter space plots, in Fig. 7, we present the variation of

C. Inequalities of DEC
The inequality functions for DEC in our 5D model is given below. It is clear that the DEC would not satisfy for a growing warp factor. However, for the decaying warp factor, these conditions may be satisfied in limited domain on the l − y space.
The functional dependence of the inequality functions on l at the location of the brane, for decaying warp factor, is written down in Table VI.  can be negative or positive. In general one can say that the fourth inequality of DEC is always satisfied for m = 4 and partially satisfied for m = 8.
We plot a limited part of the parameter space in l − y plane where the second (Fig. 8) and fourth (Fig. 9) inequalities of DEC are satisfied. Fig. 8 suggests that in the limit m → ∞ and y → 0, the domain of l where the inequality is satisfied increases. On the other hand, Fig. 9 implies that, in the limit m → ∞, the inequality is satisfied only at y → 0 and l → 0, i.e. on the brane and near the throat. After discussing the energy conditions for a GEB wormhole embedded in a 5D warped braneworld background in detail, we summarise in   Table III and Table VII summarises the key results and we discuss them in a systematic manner in the following.
• Table III  • On the other hand, improved significantly compared to the 4D GEB model. One of the key aspects of the 5D model is to resolve the particular drawback mentioned above.
• Physical understanding behind the rise of positive energy density is the following. In thick brane models, it has been shown that [94], a warped geometry with growing warp factor is sourced by bulk phantom or tachyon fields, whereas a decaying warp factor shows up in presence of matter fields with positive energy density, e.g. a scalar field with Sine-Gordon potential. Thus in presence of a decaying warp factor, the negative energy density (needed in 4D EB scenario) is compensated for.
A complete picture of the role played by the warped extra dimension, particularly the one with a decaying warp factor, towards forming a viable wormhole geometry can be realised by looking at the geodetic potentials, particle trajectories and evolution of geodesic congruences as such. One may also ask, to what extent a EB wormhole embedded in 5D braneworld with a decaying warp factor can mimic physical aspects GEB model. These and studies on other astrophysical aspects such as lensing effect, stability of the wormhole will be reported in future communications.