Wormholes in 4D Einstein-Gauss-Bonnet Gravity

Recently Glavan and Lin [Phys. Rev. Lett. 124, 081301 (2020)] formulated the Einstein-Gauss-Bonnet (EGB) gravity by re-scaling GB coupling constant as $\alpha/(D-4)$ and taking limit $D \to 4$ at the level of field equations. The GB contribution, in the resulting novel 4D EGB theory, makes a nontrivial contribution and the theory preserves the number of degrees of freedom thereby free from the Ostrogradsky instability, and also bypasses the Lovelock theorem. We obtain an exact spherically symmetric wormhole solutions in the novel 4D EGB gravity for an isotropic and anisotropic matter sources. In this regard, we consider also a wormhole with a specific radial-dependent shape function, a power-law density profile as well as by imposing a particular equation of state. To this end, we analyze the flare-out conditions, embedding diagrams, energy conditions and the volume integral quantifier. In particular our $-$ve branch results, in the limit $\alpha \rightarrow 0$, reduced exactly to \emph{vis-$\grave{a}$-vis} 4D Moriss-Thorne wormholes of GR.


I. INTRODUCTION
Within the context of general relativity, wormholes are topological bridges connecting two different asymptotically at regions of spacetime [1,2] as well as two different asympotically de Sitter (dS) or anti-de Sitter (AdS) regions [3]. Interest in wormhole space-times dates back to 1916, when Flamm [4] propose 'tunnel structure' in the Schwarzschild solution represents a wormhole. Paging through history one finds, Einstein and Rosen [5] proposed a "bridge structure" that connect two exterior regions of a Schwarzschild black hole space time, and thus forms an inter-universe connection. This was the first attempts to obtain a regular solution without a singularity, namely, 'Einstein-Rosen bridge' (ERB). However, it became soon clear that obtained solution was invalid particle model as mass-energy of such a curved-space topology is the order of Planck mass. The term wormhole for these bridges was first used by J. A. Wheeler [6,7] for microscopic charge-carrying wormholes. They showed that wormholes would collapse instantly upon formation. Moreover, if such a wormhole somehow opened, it would pinch off so quickly even a single photon could be transmitted through it, thereby preserving Einsteinian causality. Despite the dubious possibility of existing a wormhole solution, their study has opened up remarkably fruitful avenues of research. tion in cylindrical symmetry, though separate effort is needed to achieve asymptotic flatness on both sides of the throat, see [17,18].
Subsequently, motivated by Morris Thorne [1] idea, there has been intense activities in the investigation of wormholes in the modified theories of gravity and also in the higher dimensional theories of gravity [19][20][21][22][23][24] including in the Kaluza-Klein gravity [25][26][27]. Advantages in such theories are one can avoid nonstandard fluids and this was the main motivation for extensive research in modified gravity theories. Moreover, specific modifications of Einstein gravity allow additional degrees of freedom in the gravitational sector which can be used, amongst others, to resolve the dark energy and the dark matter problems. Wormholes in f (R) gravity [28] have been constructed without exotic matter. In the same context, Refs. [29][30][31][32][33] and the citations therein, are quite useful to understand wormhole geometries with different inputs and examined the validity of energy conditions. Such wormhole geometries are supported by a non-minimal curvature-matter coupling are obtained also [34] and also the curvature-matter coupled theory, f (R, T) gravity, where exact wormholes solutions were obtained [42][43][44] and also interesting work on wormholes in Born-Infeld theory [38] and scalar-tensor tele-parallel theories [39].
The wormholes geometries also received significant attention in higher curvature Einstein-Gauss-Bonnet (EGB) theory [35][36][37], and also in the Lovelock gravity [40,41]. It may be mentioned that EGB theory, particular case of Lovelock gravity, is a natural generalizations of general relativity, to higher dimensions, introduced by Lanczos [47], and rediscovered by David Lovelock [45]. The EGB gravity has been widely studied, because it can be obtained in the low energy limit of string theory [49,53], is known free from instabilities when expanding about flat spacetime [52], and also leads to the ghost-free nontrivial gravitational self-interactions [50].
However, the EGB theory is topological in 4D as the GB Lagrangian is a total derivative, so it does not contribute to the gravitational dynamics and thereby for non-trivial gravitational dynamics in EGB theory one requires D ≥ 5. This issue of the EGB theory was recently addressed by Glavan & Lin [54] by rescaling the Gauss-Bonnet coupling constant α as α/(D − 4), and taking the limit D → 4 at the level of the field equation and the resulting EGB in gives rise to non-trivial dynamics in 4D. For definiteness we shall call it the novel 4D EGB gravity, which has some in interesting property viz. it bypasses the conclusions of Lovelock's theorem and avoids Ostrogradsky instability. It worth pointing a priori that dimensional regularization of this was consid-ered by Tomozawa [55] with similar consequences.
The novel 4D EGB theory received compelling attention initiated by Glavan and Lin [54] who also proposed a static spherically symmetric vacuum black holes which hold interesting properties, e.g., the black holes are free from the singularity problem, at small distances, the gravitational force is repulsive and an infalling particle fails to reach the singularity [54]. This is in contrast to the analogous HD black holes [52] Schwarzschild like curvature singularity inevitably forms. Other cascades of work includes, Charged version of spherically symmetric black holes [56] in an anti-de Sitter spacetime [56], a Vaidya-like radiating black holes in Ref. [58], generating black holes solution was also addressed in Ref. [64] also regular black holes [59,60] Additionally, black hole solutions and their physical properties, such as rotating black holes using Newman-Janis algorithm [61,62], rotating black hole as particle accelerator [63]. On the other hand, thermodynamical properties of antide Sitter black hole [65], geodesics motion and shadow [66], gravitational lensing [67,68], relativistic stars in 4D EGB [69], and we refer the reader to [70][71][72][73][74][75][76][77][78][79][80] for other contributions and related issues details. Important contributions in the context include objections on the 4D EGB theory raised in Ref. [81] and the derivation of regularized field equations in Refs. [82,83].
However, the wormhole geometries is still unexplored, e.g., the generalization of Moris-Thorne wormhole solutions [1] is still unknown. It is the purpose of this paper to obtain this wormhole in the novel 4D EGB theory of gravity and investigate how the squared curvature terms effect the wormhole geometriy.
The paper is organized as follows: In the next Sec. II, we review the field equations in the novel 4D EGB gravity and show that it makes a nontrivial contribution to gravitational dynamics in 4D as well. In Sec. III, among other things, we obtain the field equations and four wormhole solutions including ans exact isotropic/anisotropic wormhole spacetime in the novel 4D EGB gravity with a constant redshift function. The embedding diagrams of our wormhole metrics is the subject of Sec. IV. The Sec. V is devoted to elaborate the energy conditions and the volume integral quantifier in VI. Finally we comment on our results in Sec. VII.
We have used units which fix the speed of light and the gravitational constant viah = G = c = 1.

II. BASIC EQUATIONS OF EGB GRAVITY
We begin with a short review on the EGB gravity in D-dimensions and also derive the equations of motion. The gravitational action of the EGB theory reads where g denotes the determinant of the metric g µν and α is the Gauss-Bonnet coupling coefficient with dimension [length] 2 . The discussion, in this paper, will be given here corresponding to the case with α ≥ 0. The term L GB is the Lagrangian defined by Here, S matter is the matter fields appearing in the theory. The variation of (1) with respect to metric g µν gives the field equations [58] where δg µν is the energy momentum tensor of matter with the following expression with R the Ricci scalar, R µν the Ricci tensor, R µν is the Lancoz tensor and R µσνρ the Riemann tensor. In general the GB terms is total derivative in 4D space-time, and hence do not contribute to the field equations. However, with re-scaled coupling constant α/(D − 4), and considering maximally symmetric spacetimes with curvature scale K [58], we obtain obviously the variation of the GB action does not vanish in D = 4 because of the re-scaled coupling constant [54].

III. WORMHOLE SOLUTIONS FOR THE NOVEL 4D EGB
To begin discussion on the wormhole in the novel 4D EGB, it is mandatory to consider the general static, spherically symmetric metric D-dimensional metric [35] given by where and Φ(r) is the redshift functions of an infalling body, and it must be finite everywhere to avoid the presence an event horizon. On the other hand, b(r) represents the spatial shape function of the wormhole geometry, it determine the shape of the wormhole in the embedding diagram [35]. Note that b(r) should obey the boundary condition b(r = r 0 ) = r 0 at the throat r 0 where r 0 ≤ r ≤ ∞. Now, to ensure the traversibility of wormhole, the function b(r) must satisfy the flaring-out condition that can be obtained from the embedding calculation, and reads This condition can also be written in a compactified form, namely, b (r 0 ) < 1 at the throat r = r 0 . The condition 1 − b(r)/r ≥ 0 is also imposed.
We consider an anisotropic fluid for the matter source defined by the stress energy tensor where u ν is the D-velocity and χ ν is the unit spacelike vector in the radial direction with ρ(r) is the energy density and P r (r) and P t (r) are the radial and transverse pressures, respectively. On using the metric (6) with stress tensor (8), in the limit D → 4, the components of the field equations (3) can be written as 8πP r (r) = αb(r) where the prime denotes a derivative with respect to the radial coordinate r. In this context, we have five unknown functions of r, i.e., ρ(r), P r , P t (r), b(r) and Φ(r).
We provide below several plan of action for solving the system of equations. Further, we need an additional restriction to close the system and solve the field equations.

A. Isotropic solution
In order to simplify the problem we consider a constant redshift function, namely, a wormhole solution with zero tidal force, i.e. Φ(r) = Φ 0 = const, which simplifies the calculations, and provides interesting exact wormhole solutions. In a recent works [82,83], it was shown that by taking the trace of the field equations (3) one has the simple form where the trace T = T µ ν . Using the metric form (6) for isotropic fluid matter we can use the relation P r (r) = P t (r) = ωρ(r) obtain the following condition 2b (r) Utilizing Eq. (12) and (9), we obtain a differential equation for the shape function as Solving the above equation for b(r) which has the following form and apply the condition b(r = r 0 ) = r 0 , we find the constant of integration Finally, the explicit form of shape function is Note that the solution given by (17) holds for ω = 1/3 only. The ± sign in Eq. (17) refers to two different branches of solution. Boulware and Deser [52] have demonstrated that EGB black holes with +ve branch sign are unstable and the graviton degree of freedom is a ghost, while the branch with −ve sign is stable and is free of ghosts. In our case, in the limit α → 0, the +ve positive branch leads to which is a wormhole solution in a de-Sitter/ anti-de Sitter spacetimes depending on the sign of α. On the other hand, in the limit α → 0, the −ve goes over and the standard Morris-Thorne wormhole is obtained when the cosmological constant vanishes. Henceforth we restrict to the −ve branch in which case the 4D EGB wormhole metric reads with the unit sphere line element is given by dΩ 2 2 = dθ 2 + sin 2 θdφ 2 . Notice that the above analyses is correct for the interval ω ≥ −1/3 along with ω = 1/3. For values ω < −1/3 the sign of the solutions flips.

B. Supporting conditions in anisotropic fluid scenario
We shall argue that there are two ways to find an anisotropic solution. The first way is to use Eq. (12), while the second way is to utilize the conservation of the energy-momentum tensor. As one can easily see that for static and spherical symmetric equations of motion with the anisotropy fluid is in hydrostatic equilibrium. Therefore, covariant derivative of the energy momentum tensor of matter is given by P r (r) = 2(P t (r) − P r (r)) r − (ρ(r) + P r (r))Φ (r). (22) Taking into account Eqs. (10)(11) along with an interesting EoS, leads to the following expression where the EoS parameter ω t and the redshift function Φ(r) are constants. Solving the last differential equation we get the following shape function where C 1 is a constant of integration, which can be fixed using the condition b(r = r 0 ) = r 0 , Then the shape function modifies to b(r) = r 3 2α Similarly for the +ve branch sign in the limit α → 0, we obtain which is a wormhole solution in a de-Sitter/ anti-de Sitter spacetimes depending on the sign of α. On the other hand, in the limit α → 0, the −ve goes over which reduces to the standard Morris-Thorne wormhole. Henceforth, we restrict to the −ve branch for anisotropic fluid in the novel 4D EGB takes the form As we have already pointed out, the second way to obtain our solution (26) is based on the condition (12). Thus, using equations (23) and (12), we will re-derive the expression 2b (r) Substituting all the expressions (9-11) and solve for b(r), one can independently arrive the same solution (25).
One can also consider a specific energy density to obtain the shape function. Here, we shall consider the following energy density profile given in [88]: where β and ρ 0 are constants. Solving the Eq. (9), we obtain the following shape function in which C is a constant of integration. Using the condition b(r = r 0 ) = r 0 , we obtain Substituting the above expression into the Eq. (32), one obtains the two different branches of the solution, which are Note that the branch with −ve sign is stable and we chose as the physical solution, provided β ≥ 4. On the other hand β = 3, there is an apparent singularity. Notice that there is a third case when β = 1, 2, where the sign in the solution flips. The −ve branch solutions for β ≥ 4 are asymptotically flat, for example by choosing β = 4 one can see this by considering the limit α → 0, the −ve gives b(r) On the other hand, in the limit α → 0, the +ve goes over   which corresponds to the wormhole solution in a de-Sitter/ anti-de Sitter spacetimes. Finally one can check that for β = 1, 2 our solutions are not asymptotically flat. Using the shape function (32) the wormhole metric defines by the expression where As we pointed out one should be careful as we already pointed out when β = 1, 2, the sign in the solution flips. As a special case one can consider the limit β → 0, in that case the energy density is a constant quantity i.e. ρ = ρ 0 . As a special example one can consider a wormhole supported by vacuum energy, i.e. ρ 0 = ρ vac , which can be described by the energymomentum tensor written as [3,87] T (vac) µν In fact it quite easy to observe that the effect of an energy-momentum tensor with a cosmological constant is obtained by moving the Λg µν term to the right-hand side in our field equation (3). The vacuum can therefore be thought of as a perfect fluid [3,87] ρ vac = Λ 8π (39) and One can easily get the expression for shape function, inserting the Eq. (39) to Eq. (9), and the b(r) is To this end, we need a specific EoS such that the Eq. (12) will be satisfied. Note that due to the presence of the cosmological constant our solution (41) is not asymptotically flat. Same situation has precisely pointed out in our articles [91,92], where solutions are supported by quantum effects using the Casimir energy and GUP corrected Casimir energy.
With these setups, the radial and transverse pressures in (10) and (11) are leads to and respectively. In the present work we are going to focus on n = 1. From Figs. (1-3), we observe that the geometrical flare-out condition is satisfied at the wormhole throat, i.e., b (r 0 ) < 1. In all plots the wormhole throat is set to one, i.e. r 0 = 1. Again in general we need some EoS in order to satisfy the trace of the field equation that we already discussed.

IV. EMBEDDING DIAGRAM
In this section we will analyze the embedding diagram that helps us to impose the demand of the spacetime metric (6) describe a wormhole. Of particular interests the geometry, we consider an equatorial slice θ = π/2 at some fix moment in time t = const. With this constraint the metric (6) becomes, The reduced metric (46) can be embedded into a 3dimensional Euclidean space, and in cylindrical coordinates r, φ and z has the form The embedded surface z(r) can be obtained by reversing and integrating from the last two equations, we obtain the slope With the illustration of Fig. 4, we explore the geometrical properties of these matrices (21) and (29)   Here also NEC is violated, this implies WEC is also violated. For plotting the constants are α = 0.1 and r 0 = 1 along with ω t = 0.5.

V. ENERGY CONDITIONS
In this section we present a detail description of the energy conditions, that are sets of inequalities depending on energy momentum tensor. To be specific, we start by finding wormhole solutions for weak energy condition (WEC), i.e. T µν U µ U ν , where U µ is a timelike vector. For the given diagonal EM tensor, the WEC implies ρ(r) ≥ 0 and ρ(r) + P i (r) ≥ 0, In next, the null energy condition (NEC) is given by T µν k µ k ν , where k µ is null vector. The NEC for diagonal EM tensor implies that wereas strong energy condition (SEC) asserts that T µν − 1 2 Tg µν U µ U ν ≥ 0 for any timelike vector U µ . The strong energy condition (SEC) asserts that gravity is attractive, ρ(r) + ∑ P i (r) ≥ 0, and ρ(r) + P i (r) ≥ 0. (51) Note that the WEC or SEC imply NEC, but it follows that any violation of the NEC also violates the SEC, WEC, and DEC.   Now we consider, the reduced NEC for master Eqs. (9)(10)(11), when evaluated at the throat is given by (52) On the other hand the strong energy condition (SEC) stipulates that yielding ρ(r) + P t (r)| r=r 0 = b(r) r 3 − 9αb(r) + 4rαb (r) 8πr 6 | r=r 0 .
In Figs. (5-8) we present the energy conditions for all specific cases. In all plots we have chosen a positive value of the Gauss-Bonnet coupling constant α, and obtained results suggest that the energy conditions in general are not satisfied at the wormhole throat. Particularly, one can easily see that for α > 0 the NEC, and consequently the WEC, are violated at the throat, due to the flaring-out condition.

VI. VOLUME INTEGRAL QUANTIFIER
The starting point of this discussion is to evaluate the volume integral quantifier, which provides information about the 'total amount of exotic matter' required for wormhole maintenance. To do this one can compute the definite integrals T µν k µ k ν and T µν U µ U ν , where U µ is the four-velocity [13,84]. The most usual choice is the integral including ρ and P r , with the following definite integrals I V = (ρ(r) + P r (r)) dV, where dV = r 2 sin θdrdθdφ. In this method, the total amount of exotic matter measured by (57) which can be written also as Suppose now that the wormhole extends from the throat, r 0 , with a cutoff of the stress energy tensor at a certain radius a, one deduces where r 0 is the throat radius and hence the minimum value of r. The key point of this discussion is when the limit as a → r + 0 , one verifies that I V → 0. For each wormhole solutions we found from Figs. (9-10) that one may construct wormhole solutions with small quantities of exotic matter, which needs to hold open the wormhole throat.

VII. ENDING COMMENTS
In this paper, we explore Morris-Thorne wormholes, i.e., static and spherically symmetric traversable worm-holes, in the framework of recently formulated novel 4D EGB. Throughout our discussion we consider a constant redshift function i.e., Φ (r) = 0, which simplifies the calculations and provides interesting exact wormhole solutions. Firstly, we have found an exact solution in 4D EGB supported isotropic matter source. Importantly we also found a wormhole solution supported by anisotropic matter source with EoS relating two pressure components. Moreover, we have considered a specific shape function, power law energy density profile.
Here we have shown that the flare-out condition is satisfied for different models with the positive coupling constant α. To this end, we have analyzed the null, weak, and strong conditions at the wormhole throat with a radius r 0 , and shown that in general the classical energy conditions are violated by some small and arbitrary quantities at the wormhole throat for α > 0. The GB quadratic curvature terms made a profound influence on the obtained solutions which revealed interesting features.
Further, the results presented here are a generalization of previous discussions on Moriss-Thorne wormholes of GR which are encompassed as special case in the limit α → 0. The possibility of generalization of wormholes to rotating case and more general Lovelock gravity theories [57] are interesting problems which are being actively considered.