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Traversable wormholes in massive gravity theory

In general relativity (GR), negative pressure plays a role of wormhole tension, forbidding travelling through such a wormhole. The same problem is also found in a wormhole solution for GR with a cosmological constant. Here we consider it in the context of a modified gravitational theory so called massive gravity. Although, the massive gravity is constructed aiming in explaining the cosmic acceleration, the wormhole solution exists in this theory. There are many free parameters in wormhole solution for the de Rham-Gabadadze-Tolley (dRGT) massive gravity. The necessity of the exotic matter can be alleviated. As a result, the wormhole with less size of throat, tension as well as volume of exotic matter can be obtained.


Introduction
Wormhole, such as Einstein-Rosen bridge in the Schwarzschild spacetime, is a topological bridge connecting two spacetimes. It is impossible, even for a photon, to travel through wormhole since the throat pinches off very quickly. In GR, using exotic matter with a negative radial pressure typically solves the problem. The wormhole with the exotic matter is the traversable wormhole (TWH), however such exotic matter violates the null energy condition (NEC) [1]. In modified gravity, constructing stable throat of TWH might be possible due to the new geometrical effects. For example, GR with a cosmological constant still be used the exotic matter but its volume can be reduced [2]. To bound the exotic matter within a finite volume, the thin shell filled with some kind of matter is introduced [3]. In massive gravity, a graviton acquires non-zero mass while it is massless in GR. The theory is motivated from linearized GR with interaction terms. It is found that taking the limit of zero graviton mass, the theory does not reduce to GR, called the vDVZ discontinuity [4]. Promoting the theory to a nonlinear version is proposed. However, theories with large classes of nonlinear interaction contain ghost instability. In 2010, it is cured by de Rham, Gabadadze, and Tolley's ghost-free nonlinear interaction form. The model is known as dRGT massive gravity [5]. Although, there exist the wormhole solutions for both asymptotically de Sitter (dS) and anti-de Sitter (AdS) space in dRGT model, we choose to analyze only dS case because it is consistent with observations. The TWH is also supposed to satisfy weak gravity and weak tidal force conditions [6]. We expect that TWH with less volume of the exotic matter (compared to [2]) can be constructed in dRGT model.

Action and field equation
The action of dRGT massive gravity in four dimensions is [5] where R, m g and κ are the Ricci scalar, graviton mass parameter and a constant related to Newton's constant. U is a potential due to the existence of nonzero-mass graviton which eliminates the BD ghost. Each potential U i is defined by . α 3 and α 4 are free parameters. The square bracket denotes the traces of the matrix f µν is a fiducial metric which is necessary to introduce in order to construct the nontrivial potential U . Varying the action in the equation (1) with respect to the metric tensor g µν , the field equation is where As well this constraint can be obtained by varying the action with respect to f µν . Both are taken in the form of the conservation equation and X µν is an effective momentum tensor.

Traversable wormhole in dRGT massive gravity
Three main regions of wormhole structure are exterior, interior regions and thin shell [3]. Exterior region is empty spacetime outside. Interior region is where the matter field locates. The exterior and interior regions are separated by a surface boundary (filled with some matter) called the thin shell. This shell contains matter inside the wormhole. The existence of the throat in the interior region is one of the conditions for traversability. Consider static and spherically symmetric metric in the form of suitable functions, where Φ is redshift function related to gravitational potential and b is wormhole shape function.
For exterior solution, we solve the field equation in empty space T µν = 0. The (0, 0), (1, 1) and (2, 2)-components of the field equation (2) are, respectively where Λ, γ, and ζ are free parameters which can be written as α and β. Λ < 0 and Λ > 0 correspond to a cosmological constant for de Sitter and anti-de Sitter spacetimes respectively. The prime denotes derivative with respect to r. Moreover, it is noted that the (  3 Schwarzschild case C 1 = 2µ. For the interior region, there is energy momentum tensor of exotic matter, T µν = diag(ρ, −τ, p, p), where ρ, p, and τ are the energy or mass, pressure, and negative radial pressure (known as tension) respectively. With exotic matter, the field equation (2) becomes G µν + m 2 g X µν = κT µν . For traversable wormhole, redshift function Φ(r) must be finite. For simplicity, let Φ be a constant (Φ = 0), i.e. zero gravity. Hence the wormhole possesses weak gravity and weak tidal force. By this setting, the (0, 0), (1, 1) and (2, 2)-components of the field equation become (set κ = 1), respectively, At the throat of the wormhole (r = r 0 ), the shape function must satisfy b(r 0 ) = r 0 . The tension at the throat in the latter expression of equation (7) becomes τ (r 0 ) = 1 To match the interior and exterior solutions at the thin shell with suitable junction conditions, this is to match the metric and to match the equations. For the first matching condition, the metric g µν of interior solution and the exterior solution are specified at the surface boundary to be continuous. The wormhole matter distributes within the range r = r 0 to r = a, where a is a radius of the wormhole mouth which is connected to the thin shell. The metric matching conditions at the mouth a can be obtained as g tt(ext) = g tt(int) and g rr(ext) = g rr(int) . This leads to matching functions Φ ext (a) = Φ int (a) and b ext (a) = b int (a). As a result, the solutions of the redshift and shape functions at the thin shell can be expressed, respectively, as The second matching condition connects the interior and the exterior solutions. However, a fringe of the matter exists, hence thin shell formalism must be used in order to eliminate the fringe. In this formalism, the energy-momentum tensor of the thin shell [6] is t µν = + − G µν dr, wherer is a unit vector in radial direction in the thin shell. The limits of integration − and + are the values ofr at the middle between interior region-thin shell and thin shell-exterior region, respectively. The functions Φ(r) and b(r) must be continuous and smooth in the thin shell. The solutions obtained from integration of Φ and b then vanish. As a result, t 00 = 0, which means that the thin shell has no mass or energy. Similarly, t 11 also equals zero. However, in (2, 2)component of the equation (6), there is a term Φ with non-vanishing t 22 . This component is solved as where P is a tangential or surface pressure. As mentioned, one has Φ − = 0 for interior solution. In the exterior solution, differentiating the first expression of the equation (9), we have Φ + = (1 − b(a)/a) −1 µ a 2 + Λ 3 a + γ 2 . Eventually, the tangential pressure of TWH in dRGT model is Signs of tangential pressure depend on types of matter inside the thin shell. From the above expression, there exists a finite value of a for P = 0. Hence, it is not necessary to use the thin shell of other matter for TWH in dRGT gravity. It is one of the advantages of this model compared to GR.

Results and discussion
Even though, TWH in dRGT massive gravity can be constructed for both asymptotically dS (Λ < 0) and AdS (Λ > 0) spacetimes. In this work, we are interested only in the asymptotically dS case because it is consistent with observations in cosmology. Moreover, in asymptotically dS case, it is more convenient to compare the results to the case of GR with a cosmological constant because TWH in that case exists only for dS spacetime. In the figure 1, we see that TWH with small tension for GR case is vary large (r 0 → ∞ as τ (r 0 ) → 0). However, the tensions at the throat approach to zero for finite r 0 in both GR with a comological constant and dRGT cases. The existence of the throat requires b(r) to satisfy the condition b(r 0 ) = r 0 . The form of b(r) is not unique. The chosen form denoted as b int (r) must be consistent with the equations (7)-(8). By matching the functions such as b int (a) = b ext (a), r 0 can be found in term of a and a is computed from the equation (10) with the simple condition: P = 0 that means form of the matter of TWH is without the thin shell. Therefore, a volume of the of the exotic matter in wormhole can be computed by V min = 4 3 π(a 3 − r 3 0 ). We compute the volume of exotic matter for two different forms of shape functions as b(r) = (r 0 r) 1/2 and b(r) = r 0 /r with choosing Λ = −0.1, µ = 0.5, γ = −1 and ζ = 1. For b(r) = (r 0 r) 1/2 , the volumes of exotic matter in GR with cosmological constant and dRGT model are 59.65 and 3.79 respectively. For b(r) = r 0 /r, volumes in GR with cosmological constant and dRGT model are 33.03 and 3.81 respectively. These mean that, in order to construct TWH, the using of exotic matter in dRGT gravity is less than the using of exotic matter in GR with cosmological constant.

Conclusion
In dRGT model, the volume of exotic matter with negative radial pressure (the tension) can be minimized. Other conditions for TWH are less tension and not too large size. The redshift function in interior region is set as Φ = 0 (zero gravity), which implies the weakness of gravitation field and tidal force. The thin shell is not needed in order to construct TWH in dRGT model (unlike the GR case). We compare volumes of exotic matter in GR with cosmological constant case and the dRGT case. TWH in dRGT model need less amount of exotic matter than in it is needed in GR with cosmological constant case.