Casimir Wormholes in Brans-Dicke Theory

Undoubtedly, one of the most fascinating features of general relativity (GR) that has attracted many researchers so far, is the possible existence of hypothetical geometries which have nontrivial topological structure, known as wormholes. Such structures constitute a short-cut or tunnel between two universes or two separate regions of the same universe [2, 3]. Wormhole physics can originally be traced back to Ludwig Flamm in 1916 who tried to give interpretations of the Schwarzschild solution through 2D embedding diagram of Schwarzschild wormhole [4]. In 1935 specific wormhole-type solutions were considered by Einstein and Rosen [5]. Their motivation was to construct potential models for elementary particles represented by a “bridge” which connect two identical sheets. Such a mathematical description of a physical space that is connected by a wormhole-type solution was subsequently denoted as the “Einstein-Rosen bridge”. After the pioneering work by Einstein and Rosen, studies in the field of wormhole physics were laid dormant for about two decades. It was at the end of 1950s when Misner and Wheeler through their pioneering works [6, 7] invented the concept of wormhole in order to provide a mechanism for having charge without charge. They realized that wormholes connecting two asymptotically flat spacetimes could provide possible existence of non-trivial solutions to the coupled Einstein-Maxwell field equations, where, the lines of electric field flux as observed in one part of the universe could thread the throat and reappear in other part. These objects were denoted as “geons”, i.e., gravitational-electromagnetic entities [8]. However, it was later realized that the geonlike-wormhole structures seem to be a mere curiosity since they were understood as non-traversable [9] wormholes and additionally would develop some type of singularity [10], see also [11] for historical notes.

Over the past decades, there has been a remarkable amount of literature produced in the realm of wormhole physics. This renewed interest in the subject was mainly sparked through the seminal paper by Morris and Thorne [12, 13]. In the framework of GR, they introduced static spherically symmetric metric and discussed the required conditions for physically meaningful Lorentzian traversable wormholes. The Morris-Thorne (MT) wormholes allow a two way communication between two regions of the spacetime by a minimal surface called the wormhole throat, through which, matter and radiation can travel freely. However, traversability of MT wormhole requires inevitably the violation of NEC. In other words, in order that the throat of these wormholes can be kept open some form of “exotic matter” is needed [13]. This type of matter has negative energy density and its energy momentum tensor (EMT) violates the NEC [3, 14]. Although the violation of energy conditions, at least in classical GR, is unacceptable from the common viewpoint of physicists, several researches have tried to address the issue of exotic matter in wormhole physics. Work along this line has been performed in the literature for different matter distributions, for example, traversable wormhole geometries constructed by matter fields with exotic EMT have been studied in [15]. The case of phantom or quintom-type energy as the ingredient to sustain traversable wormholes has been investigated in [16] and wormholes supported by nonminimal interaction between dark matter and dark energy has been explored in [17], see also [18] for a comprehensive review. At quantum level, it has been shown that some effects due to quantum field theory, e.g., Hawking evaporation process [19] and Casimir effect [20] can allow for violation of energy conditions. Negative energy densities in wormhole configurations can also be produced through gravitational squeezing of the vacuum [21], see also [3, 22] for more details. However, due to the problematic nature of exotic matter, many attempts have been made so far in order to minimize its usage with the aim of overcoming the issue of energy conditions in wormhole structures [23]. Also, in order to avoid the usage of exotic matter in traversable MT wormholes, several modifications to GR have been proposed [24]. More precisely, the need for exotic matter in classical MT wormholes can be resolved in modified theories of gravity which for example contain higher curvature corrections or fields other than metric in their gravitational actions. These corrections or additional degrees of freedom that are absent in GR can support traversable wormhole spacetimes. In this regard, several studies have been performed in recent years among which we can quote: traversable wormholes in higher-dimensional GR [25], nonsymmetric gravitational theory [26], Einstein-Gauss-Bonnet gravity [27], brane world scenarios [28], Lovelock [29] and $f({\rm R})$ gravity theories [30], Einstein-Cartan theory [31], modified gravities with curvature-matter coupling [32], Rastall gravity [33] and other theories [34].

Among different matter sources supporting wormhole geometries, Casimir effect is a well-known source that owing to its associated negative energy, it was considered as a good candidate to produce a traversable wormhole, a long time ago by Morris and Thorne [13] and sometime later by Visser [3]. However, only very recently a traversable wormhole solution with Casimir energy as its supporting matter has been found by Garattini [35]. After this innovative work, many researchers have investigated wormhole configurations endowed with Casimir energy in different frameworks, such as, corrections due to generalized uncertainty principle [36], modified gravity theories [37], Casimir source modified by a Yukawa term [38], Casimir wormholes in Yang-Mills theory [39] and other works [40].

The search for exact spacetimes admitting wormhole structures in BD theory has opened up a new field of study in the literature. In particular, the occurrence of classical Lorentzian wormholes in BD theory is quite intriguing since it is a natural theory that emerged as a Machian alternative to GR. The motivation for studying these structures in BD theory is that this theory involves a scalar field that can play the role of classical exotic matter required for construction of traversable Lorentzian wormholes. The search for wormhole spacetimes in BD theory has been initiated by Agnese and La Camera [41] where they obtained a static spherically symmetric vacuum solution that supports a two-way traversable wormhole for $\omega<-2$ and one way for $\omega>-3/2$. Nandi et al. [42] also showed that three of the I-IV classes of BD solutions obtained by Brans [43] support a two-way traversable wormhole for $\omega<-2$ and $0<\omega<\infty$. Analytical wormhole solutions in BD theory in the presence of matter were found by Anchordoqui et al. [44]. The authors obtained some regions of the parameter space in which the BD scalar field may play the role of exotic matter, implying that it might be possible to build a wormhole-like spacetime in the presence of ordinary matter at the throat. Paging through the literature, one finds many interesting articles on BD wormholes today, see for instance [45, 46, 47, 48, 49, 50]. Nevertheless, in the scientific literature, there is no trace of the consequences of Casimir energy on wormholes in BD theory. Motivated by these considerations, in the present work we investigate wormhole solutions in this theory in the presence of Casimir energy. Our report is then is organized as follows: In Sec. (2) we give a brief review on BD theory along with its corresponding field equations. In Sec. (3), using the general static spherically symmetric MT metric we recast the BD field equations, given in tensorial form, into a set of coupled differential equations. In subsection (3.1) we search for exact wormhole solutions with zero tidal force and check the energy conditions for them. Subsection (3.2) is devoted to the solutions with non-vanishing redshift function. At the end of this subsection we examine stability of the obtained solutions through equilibrium condition. Our conclusions are drawn in Sec. (4). We use a system of units so that $\hbar=c=G=1$.

The BD action in Jordan frame is given by [51]

where

is the action describing ordinary matter fields, i.e., any form of matter different from scalar field $\phi$, and $\omega$ is a dimensionless coupling parameter. We note that matter is not directly coupled to the BD scalar field as $\mathcal{L}^{m}$ is independent of $\phi$. However, in the gravitational sector we have a non-minimal coupling between the scalar field and curvature. In BD theory, the gravitational interaction is described by the metric tensor and by the BD scalar field, which together with the matter variables describe the dynamics of the model [52]. Varying action (1) with respect to metric we obtain the BD field equation as

where

is the matter EMT. Variation of the action with respect to $\phi$ gives

Now, taking the trace of Eq. (3) we get the following expression for Ricci scalar

where $T^{(m)}$ is the trace of EMT. Using the above equation to eliminate $R$ from Eq. (5) we finally get the evolution equation for BD scalar field as

In view of the above equation we observe that the BD scalar field is not directly coupled to $T^{(m)}_{\mu\nu}$ or $\mathcal{L}^{m}$, it acts back on ordinary matter only through the metric tensor in the manner determined by Eq. (3). The BD coupling parameter is a free parameter of the theory which represents the strength of coupling between the scalar field and the spacetime metric. In the present work, we exclude the case with $\omega=-3/2$ as for this value of coupling parameter, the BD theory is known to be pathological [53] in the sense that the BD scalar field is non-dynamical and, more importantly the Cauchy problem becomes ill-posed [54].

In the present section we seek for wormhole configurations in BD theory whose supporting energy density originates from the Casimir effect. The associated energy to this effect in various configurations has been calculated for flat and curved boundaries [55]. Studies conducted in this field over the past years have shown that the Casimir energy and its corresponding attractive or repulsive force can assume different forms under various circumstances, namely, $i)$ geometry and distance of the objects involved in a Casimir setup, $ii)$ the type of the quantum field under consideration e.g., electromagnetic, fermionic and scalar field, and $iii)$ the boundary conditions imposed on the objects, e.g., Dirichlet and Neumann boundary conditions for a scalar field [55, 56], conductor boundary conditions for the electromagnetic field [57] and bag boundary conditions for the spinor field [58]. For example, in the case of scalar field confined within a cubic configuration the Casimir energy density is given as $\rho_{\rm c}=-0.015/a^{4}$ where $a$ is the cuboid side length [59]. For the case of wedge-shaped geometries, i.e., the situation where space is divided by two conducting planes making an angle $\varphi_{0}$ with each other, the associated energy density for electromagnetic field [60] and scalar field [61] is given respectively as

where $r$ is the radial distance from the intersection of two planes. For a sphere of radius $R$ the Casimir energy density of the electromagnetic field has been calculated and the result is $\rho_{\rm c}(r)\approx{\rm const}/RD^{3}$ where $D=R-r\ll R$ and $r$ is the distance from the center of the sphere [62, 63]. Work along this line has been carried out for various other geometries with more complicated configurations such as, scalar Casimir energy for two spheres and a sphere and a plane [64], the case of a cylindrical shell [65], two parallel cylinders and a cylinder parallel to a plane [66], see also [55, 57, 67] for more details. In view of the above considerations, one may recognize that the Casimir energy is proportional to inverse powers of distance between the objects involved in a Casimir configuration. We therefore take the energy density associated to Casimir effect as $\rho_{\rm c}(r)=\lambda/r^{m}$ where $\lambda$ and $m>0$ are constant parameters.

Let us now consider the general static and spherically symmetric line element representing a wormhole spacetime which is given by

where $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\varphi^{2}$ is the standard line element on a unit two-sphere and, $\Phi(r)$ and $b(r)$ are denoted as redshift and shape functions, respectively. The radial coordinate ranges from $r_{0}$ to spatial infinity where the surface at $r=r_{0}$ is known as the wormhole’s throat. Although the radial metric component i.e., $g_{rr}$ diverges in the limit $r\rightarrow r_{0}$, the proper radial distance [12, 2]

is required to be finite everywhere. This quantity decreases from $\ell=+\infty$ in the upper universe to $\ell=0$ at the throat, and then from zero to $\ell=-\infty$ in the lower universe. We now define the time-like and space-like vector fields, respectively as $u^{\mu}=[{\rm e}^{-\Phi(r)},0,0,0]$ and $v^{\alpha}=\left[0,\sqrt{1-b(r)/r},0,0\right]$, so that $u^{\mu}u_{\mu}=-1$ and $v^{\alpha}v_{\alpha}=1$. The anisotropic EMT of matter source then takes the form

with $\rho(r)$, $p_{r}(r)$, and $p_{t}(r)$ being the energy density, radial and tangential pressures, respectively. We therefore obtain the components of field equation (3) as

where a prime denotes differentiation with respect to radial coordinate $r$. Also the evolution equation (7) reads

The conservation of EMT, i.e., $\nabla^{\mu}T^{(m)}_{\mu\nu}=0$ leads to the following equation

In view of the above system of differential equations we find that on substitution for energy density and pressure profiles from Eqs. (12)-(14) into equation (16) and after simplification we arrive at evolution equation for BD scalar field, i.e., Eq. (15). This means that, from equations (12)-(16) only four of them are independent. Having this in mind, we proceed to find wormhole configurations assuming matter energy density, Eq. (12), obeys the Casimir energy density for a specified configuration. We therefore set

Substituting for energy density and pressure profiles into Eq. (15) gives

Equations (17) and (18) construct a system of two coupled differential equations to be solved for four unknowns $\{b(r),\phi(r),\Phi(r),V(r)\}$. Obviously this system is under-determined, hence, in order to solve it we have to specify the functionality of two of the unknowns. This is our task in the next section in which we assume physically reasonable forms for the BD scalar field and the redshift function and try to find the scalar field potential and the shape function.