Spherically symmetric wormholes in General Relativity and modified gravity with a Kalb-Ramond background

The field strength of the KR field is denoted by a completely antisymmetric rank three tensor $H_{\mu\nu\lambda}$. The action for the KR field strength with gravity is [28]:

where, $\kappa=8\pi$. The field equations that can be obtained from the action in Eq. (3) are,

where, $G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R$, is the Einstein tensor, and $T_{\mu\nu}$ is the stress-energy tensor given as,

The dual psuedoscalar “axion” $H$ can be defined as,

For the KR field strength, the Bianchi identity is,

In four dimensions, the KR field possess only one degree of freedom [29, 9, 10, 28]. In the background of the spherically symmetric line element described in Eq. (1), the only surviving non-zero component is $H_{023}$, and it depends only on the radial co-ordinate $r$. Therefore, the square of the field strength can be written as $H_{\mu\nu\lambda}H^{\mu\nu\lambda}=6H_{023}H^{023}={[h(r)]}^{2}$ [28, 30]. Using the field equation in Eq. (5), the Bianchi identity in Eq. (8), and the line element in Eq. (1), we get the relation

where, a prime denotes derivative with respect to $r$. Eq. (9) can be solved to obtain

where $B$ is a constant of integration. Moreover, the components of the stress-energy tensor from Eq. (6) can be written as

In the context of spherical symmetry, Eq. (2.1) suggests that the stress-energy tensor of the KR field can be expressed as a perfect fluid, having the diagonal stress-energy tensor [28].
As described previously, the parameter $\Phi(r)$ in Eq. (1), provides a measure of the gravitational redshift and should be well defined through out the space-time. However, there remains the possibility of a simple form of solution known as the zero-tidal force solution, where $\Phi’(r)=0$ everywhere [16]. Stationary observes shall measure precisely zero-tidal forces, and this particular form of the solution is termed in literature as the “tideless” solution. Moreover, setting $\Phi’(r)=0$, simplifies the calculations significantly.

For an anisotropic $T_{\mu\nu}$, with the background line element in Eq. (1), the EFEs are,

The Ricci scalar is given as $R=\frac{2b^{\prime}}{r^{2}}$. From the line element in Eq. (1), we get $h^{2}=\frac{B^{2}}{r^{3}}$ and using Eq. (12), we get,

Solving Eq. (15), we get,

where, $c_{1}$ is a constant of integration. Eq. (16) gives the form of the shape function for the wormhole the minimally coupled scenario, and we use the same approach for the remainder of the cases as well. In order to determine $c_{1}$, we use one of the constraints on the shape function, viz. $b(r_{0})=r_{0}$. Therefore, the shape function becomes

The profiles of the shape function have been shown in Figure 1, and it can be seen that the space-time is asymptotically flat, and flares out for $B=0.01$. With this particular form of the shape function, the various energy conditions and stability are numerically analyzed and presented in the next sections.

In the context of the non-minimal coupling of the KR field to gravity, the well-studied paradigm involves the non-minimal coupling of the KR vacuum expectation value (VEV) with curvature. Although the KR Lagrangian is Lorentz-invariant, coupling the KR-VEV non-minimally yields the peculiar property of spontaneously breaking of Lorentz symmetry locally [31, 32], while preserving the $U(1)$ gauge invariance of the electromagnetic theory in a torsioned background. Wormhole solutions have been analyzed in this framework before [33]. Here, we deviate from this approach, and present a novel action for the KR field strength non-minimally coupled to gravity instead:

where, $\kappa=8\pi$, and $\xi$ is the non-minimal coupling constant. For simplicity, only the first order non-minimal coupling is considered, i.e., coupling with the Ricci scalar $R$. The field equations obtained from the action in Eq. (18) are,

where ${\left[T^{eff}\right]}_{\mu\nu}$ is the effective stress-energy tensor given as

With the components,

Finally, the field equations are obtained as

The Ricci scalar for the line element in Eq. (1) is given as $R=\frac{2b^{\prime}}{r^{2}}$, and $h^{2}=\frac{B^{2}}{r^{3}}$ here. Now, from Eq. (23), we get

Solving Eq. (26), we get,

where, $c_{2}$ is a constant of integration. The constant $c_{2}$ is analysed using the condition at the wormhole throat, $b(r_{0})=r_{0}$. Thus, the shape function then takes the form

It is known that the non-minimal coupling constant should be small. Considering constraints from astrophysical scenarios, where a possible range is $\xi\lesssim 0.01$ [34], we consider the value of the non-minimal coupling parameter as $\xi=0.01$. It can be seen from Figure 2 that the obtained space-time is asymptotically flat, and flares out as desired. The energy conditions and stability conditions for this space-time are presented in the next sections.

It is well-known that several modified gravity theories can account for the shortcomings of GR such as the dark energy problem. Similarly, pathologies of stable and traversable wormholes can also be avoided in several modified gravity proposals. To probe the deviations from GR solutions in our framework, we consider the case of $f(R)$ and $f(R,T)$ modified gravity theories, which serve as suitable alternatives to GR in resolving several issues. Among these, $f(R)$ gravity is perhaps the most widely studied alternative to GR. In this framework, the Ricci scalar $R$ is replaced by an arbitrary function of it in the Lagrangian. Further, $f(R,T)$ gravity serves as a simple framework to non-minimal curvature-matter coupling, and is similar in its essence to the Rastall gravity framework [35].

The energy conditions are sets of inequalities which the stress-energy tensor of matter sources should respect, so that the energy density of matter fields is measured to be positive by any observer traversing a time-like curve [43]. The weak energy condition (WEC) implies $\rho\geq 0$, the NEC implies $\rho+p_{r}\geq 0$, and $\rho+p_{t}\geq 0$, the strong energy condition (SEC) implies $\rho+p_{r}+2p_{t}\geq 0$. These energy conditions are analyzed for all the four cases in this section.
Figure 5 shows the WEC for Cases I and II. As expected and reported in literature [28], the pseudoscalar axion has a positive energy density and satisfies the WEC for both the cases. Figure 6 shows the WEC for Cases III and IV. It is seen that the KR field strength as a source shows a positive energy density, thus satisfying the WEC. In Case III, it can be seen that the energy density exhibits no apparent changes in behavior depending on the $f(R)$ model parameter $\chi$.
Figure 7 shows the NEC term $\rho+p_{r}$ for Cases I and II. It can be observed that the NEC is violated at the throat, which is a characteristic of traversable wormholes. Next, the NEC term $\rho+p_{t}$ is analyzed and shown in Figure 8. It can be observed that the second NEC term $\rho+p_{t}$ is satisfied at the throat. However, since both $\rho+p_{r}\geq 0$ and $\rho+p_{t}\geq 0$ should hold, we infer that the NEC is violated here. Further, the SEC is shown in Figure 11.

Figure 9 shows the first NEC term $\rho+p_{r}$ for Cases III and IV. It can be observed that the first NEC term is violated at the throat for all considered values of $\chi=0.5,1.5,\text{and}\leavevmode\nobreak\ 2$ in Case III, and the first NEC term is also violated at the throat for Case IV with $\lambda=-12$.

Figure 10 shows the second NEC term $\rho+p_{t}$ for Cases III and IV. It is seen that the second NEC term is satisfied at the wormhole throat for both the cases. However, as a whole the NEC is violated for Cases III and IV owing to the violation of the first NEC term $\rho+p_{r}$.

From Figure 11 it can be observed that the SEC in Case I is marginally satisfied ($\rho+p_{r}+2p_{t}=0$) as reported previously [28], while the SEC for the non-minimally coupled scenario is violated at the wormhole throat.

Figure 12 shows the SEC term $\rho+p_{r}+2p_{t}$ for Cases III and IV. It can be observed that the SEC is violated at the wormhole throat for both the cases.

Along with the energy conditions, it is useful to check two more parameters namely, the equation of state (EoS) parameter $\omega=p_{r}/\rho$, and the anisotropy parameter $\Delta=p_{t}-p_{r}$. The EoS parameter provides information regarding the nature of the matter source threading the wormhole geometry, and the anisotropy parameter explains whether the space-time geometry is attractive or repulsive [16]. The profile of the EoS parameter near the throat is shown in Figure 13.

From Figure 13, it can be observed that the EoS parameter $\omega$ has a negative value and $\omega<<-1$, indicating a phantom-like behavior of the source. Moreover, it is important to note that the value of the constant “$B$” plays a crucial role in determining these parameters. Higher values of the constant “$B$” can lead to values of the EoS parameter $-1<\omega<0$, near the wormhole throat. However, with higher values of the constant “$B$” the various conditions required for the viability of the shape function are not satisfied. This highlights how the metric function constraints the various energy conditions, EoS, and anisotropy parameter.

Figure 15 shows the EoS parameter $\omega$ for Cases III and IV. It can be observed that the EoS parameter has a negative value near the wormhole throat for both the cases. For the case of $f(R)$ modified gravity (Case III), the EoS parameter $\omega<-1$ for $\chi=0.5,1.5\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ 2$ near the throat signifying a phantom-like nature of the source, while for the case of $f(R,T)$ modified gravity (Case IV), the EoS parameter is $-1<\omega<0$, near the wormhole throat, indicating a quintessence like behaviour of the source.
Figure 14 shows the anisotropy parameter $\Delta$ for Cases I and Case II. The anisotropy parameter is positive at the wormhole throat, signifying a repulsive geometry, as in the in-falling signals from one asymptotically flat region can cross the wormhole throat and emerge on the other region.
Figure 16 shows the anisotropy parameter $\Delta$ for Cases III and IV. The anisotropy parameter is positive near the wormhole throat for both cases, signifying a repulsive geometry, which is a characteristic feature of traversable wormholes. Table 3.2 below presents a summary of the energy conditions for Cases I-IV.

The stability of stellar structures can be investigated using the Tolman-Oppenheimer-Volkov (TOV) equation, first reported in the context of neutron stars [44, 45]. A more generalized version of the formalism was developed in [46], and this can be leveraged to probe the stability of wormholes in terms of the hydrostatic, gravitational, and anisotropic forces in the space-time. The generalized TOV equation [46, 47, 48] is given as

where $\epsilon(r)=2\Phi(r)$. $F_{h}$ represents the hydrostatic force, $F_{g}$ the gravitational force, and $F_{a}$ the anisotropic force. These three terms of the TOV equation can determine the equilibrium anisotropic mass distribution [48] in that stable stellar structures satisfy Eq. (58).

Figure 17 shows the components of the TOV equation, $F_{\mathrm{h}}$ and $F_{\mathrm{a}}$, for Case I and Case II. It is evident from Eqs. (58) and (59) that as $\Phi^{\prime}(r)=0$, the gravitational force $F_{\mathrm{g}}=0$. Figure 17 shows that the hydrostatic force $F_{\mathrm{h}}$ and anisotropic force $F_{\mathrm{a}}$ cancel each other out, rendering a stable wormhole configuration for both Cases I and II.

Next, Figure 18 shows the corresponding terms of the TOV equations for Cases III and IV. It can be observed from Figure 18 that the wormhole configuration in Case III ($f(R)$ gravity) is not stable, as the hydrostatic force $F_{\mathrm{h}}$ and anisotropic force $F_{\mathrm{a}}$ do not cancel each other. However, in Case IV ($f(R,T)$ gravity), it can be seen that the anisotropic force $F_{\mathrm{a}}$ and hydrostatic force $F_{\mathrm{h}}$ cancel each other out, thus rendering a stable wormhole configuration.

Further, the amount of exotic matter in wormhole space-times can be estimated using the averaged null energy condition, $\int_{\lambda_{1}}^{\lambda_{2}}T_{ij}k^{i}k^{j}d\lambda\geq 0$, evaluated along the radial coordinate $r$. However, instead of this line integral, a more generalized description of the amount of energy condition violating matter can be estimated by using a volume integral, namely the volume integral quantifier (VIQ) [49, 50, 51], which is defined as,

The VIQ gives an estimate of the amount of NEC violating matter required for wormhole configurations after an exterior metric is matched with the wormhole space-time with the stress-energy tensor cutting off at some $r=a$ away from the throat. The requirement of arbitrarily small quantities of NEC violating matter is satisfied provided $I_{v}\rightarrow 0$ as $a\rightarrow r_{0}$ [49, 50]. It can be observed from the Figure 19 that as $a\rightarrow r_{0}$, then, $I_{v}\rightarrow 0$, signifying that the wormhole configurations may be possible with arbitrarily small amounts of exotic matter for all the four cases.