Quasinormal Modes and Hawking Radiation Sparsity of GUP corrected Black Holes in Bumblebee Gravity with Topological Defects

Although the recent discovery of Gravitational Waves (GWs) has provided strong support to the General Relativity (GR), GR is not renormalizable at UV scale, and hence it describes the gravitation at classical level only. On the other hand, the Standard Model (SM) of particle physics describes particles and other three fundamental interactions at the quantum level. As both GR and SM are the most successful field theories in describing nature, the unification of these two theories is one of the prime pursuits of physicists to describe all fundamental interactions of nature at the quantum level so that we will understand nature to the most deeper level. In the quest of this unification goal, some theories of Quantum Gravity (QG) have already been introduced, whose direct test would be possible only at the Planck scale ($\approx 10^{19}$ GeV). As this energy scale is far more beyond the reach of current experiments, so direct tests of those QG models are not possible at present and nor will be in the near future. However, there are some effects of QG models, such as the breaking of Lorentz symmetry [1], which may be observed at the current low energy scales [2]. The violation of Lorentz symmetry arises as a possibility in the context of loop quantum gravity, noncommutative field theories, string theory, standard-model extension (SME) etc. Especially, SME is a most general field theoretical framework that includes the fields of SM and GR with the terms in the lagrangian containing information about the Lorentz symmetry violation. The effects of Lorentz symmetry breaking in the gravitational sector of SME were studied in Ref.s [3, 4, 5, 6, 7, 8, 9, 10]. In the case of GWs, the Lorentz violation was studied in Ref.s [11, 12]. The simplest extended gravitational field theories with the spontaneous breaking of Lorentz and diffeomorphism are usually referred to as bumblebee models [1] in which breaking of Lorentz symmetry takes place due to the nonzero vacuum expectation value of a single vector field, known as the bumblebee field. It should be noted that the diffeomorphism violation is always associated with the local Lorentz symmetry violation. Recently a spherically symmetric exact vacuum solution to Einstein field equations in the presence of a spontaneous breaking of Lorentz symmetry due to nonzero vacuum expectation value of the bumblebee field has been obtained in Ref. [2]. This study also explored three classical tests viz., the advance of perihelion, bending of light and Shapiro’s time delay, and showed that corrections from Lorentz violation were present even in the absence of a massive gravitational source and the Lorentz violation background deforms the spacetime. In another study, particle motion in Snyder noncommutative spacetime structures was studied in the presence of Lorentz violation [13].

In a recent study, it was seen that the Lorentz symmetry breaking reduces the greybody factor of black holes in Generalized Uncertainty Principle (GUP) modified bumblebee gravity [14]. In a different study, the photon orbits of Kerr-Sen-like black holes in bumblebee gravity have been investigated, where the authors examined the effects of charge, Lorentz violation parameter and plasma as a dispersive medium [15]. The impact of a topological defect on black holes in bumblebee gravity was studied for the first time in Ref. [16]. In that work, the authors studied the black hole horizon, temperature and the photon sphere extensively and found that the radius of the shadow of black holes increases with an increase in the global monopole parameter. Their study also shows an increasing effect of the deflection angle coming from the Lorentz symmetry breaking parameter and the global monopole. The exact traversable wormhole solution in bumblebee gravity has been obtained in Ref. [17]. In this work, the authors studied the energy conditions of the wormhole and deflection angle of light in explicit form. They found that the bumblebee wormhole solutions support the normal matter wormhole geometries under some certain conditions.

In this work, we shall study the quasinormal modes of GUP corrected black holes in the presence of a topological defect in bumblebee gravity. The quasinormal modes are basically some complex numbers that are related to the emission of GWs from the compact and massive perturbed objects in the universe [18, 19, 20]. The real part of the quasinormal modes is related to the emission frequency, while the imaginary part is connected to its damping. In recent times, the properties of GWs and quasinormal modes of black holes were studied extensively in different modified gravity theories [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33]. In Ref. [25], the quasinormal modes of black holes in bumblebee gravity were studied, and it was shown that the Lorentz violation has a significant impact on the quasinormal frequencies. In another recent study, the quasinormal modes of GUP corrected Schwarzschild black holes have been studied and it was seen that the GUP correction could change the quasinormal frequencies and the decay rates of GWs [34]. Being inspired by this study, we have considered the GUP correction in black holes in bumblebee gravity. To have a better comparison, we have considered three different perturbations viz., scalar, electromagnetic and gravitational perturbations to calculate the quasinormal modes of black holes. To obtain the quasinormal modes with higher accuracy, we have implemented Padé averaged WKB approximation method upto sixth order. WKB approximation method provides a good approximation to the quasinormal modes, however, in some cases the method may be deficient to calculate quasinormal modes [35]. For a comparison of our results in time domain, we have studied the time evolution of the perturbation profiles and obtained the quasinormal modes from the time domain analysis. Moreover, we have also studied the aspect of the sparsity of black holes along with greybody factors and Hawking radiation power spectrum. The greybody factor of a black hole is related to the quantum nature of a black hole and it has been widely studied for black holes in different modified theories. In Ref. [36], greybody factors of a black hole in dRGT massive gravity have been studied using rigorous bound and the matching technique. In another work, the nonlinear electrodynamic effects on the black hole shadow, deflection angle, quasinormal modes and greybody factors have been studied in details for a magnetically charged black hole in ‘double-logarithmic’ nonlinear electrodynamics [37]. The greybody factors in black strings have been studied in Ref. [38] in dRGT massive gravity theory where the authors calculated the rigorous bounds on greybody factors. The light rays in a Kazakov-Solodukhin black hole has been studied using Gauss-Bonnet theorem and the greybody bounds have been calculated in details in a recent study [39]. In another important work, quasinormal modes and greybody factors have been also studied in $f(R)$ gravity minimally coupled to cloud of strings in $(2+1)$ dimensions, where the authors found explicit analytical results for decay rate, reflection coefficient, greybody factors and temperature of the black hole [40]. The scalar perturbations of a single-horizon regular black hole has been studied in Ref. [41] where the regular black hole solution was obtained using polymer quantization inspired by loop quantum gravity. The quantum corrections like GUP and non-commutativity are also implemented in black holes to study the thermodynamic behaviours [42]. GUP effects on Hawking temperature of a black hole in warp Dvali-Gabadadze-Porrati (DGP) gravity model have been studied in Ref. [43], where it was found that the mass, angular momentum of vector and scalar particles impact the Hawking temperature of a black hole, and it is connected with the type of the particle emitted from the black hole.

Finally, we perform a study on the area of the black holes with the help of the adiabatic invariance method. Since, the area of black holes is connected with the GW echo [44, 45, 46], entropy spectrum and power spectrum of black holes, we believe the results can be further used to study the impacts of Lorentz violation on different aspects of black holes.

One may note that as the Lorentz symmetry breaking carries the possibilities of beyond standard model physics, it has been an objective of many experimental searches [47, 48]. Apart from bumblebee gravity, such symmetry breaking also arises in different theories including string theory [49] and noncommutative geometry [50]. Such theories with a spontaneously broken symmetry can give rise to different topological defects such as domain wall, cosmic string or monopole solutions etc. Hence, global monopoles are a possibility in bumblebee gravity. In such theories, the topological defects like global monopoles arise during the phase transitions in the early universe via Kibble mechanism [51] and since isolated defects are stable in nature, one may expect that they can still exist in the present universe. The presence of such relics can have some important implications in different aspects of astrophysics and cosmology including structure formation and inflation [52, 53]. In this work, we consider one such type of relics known as global monopole and study its implications in black hole properties such as quasinormal modes, sparsity etc. If one can have some direct or indirect evidence of the existence of such a monopole, they will contribute significantly to the understanding of beyond standard model physics. In other words, such an evidence might represent the first field observed to break Lorentz symmetry, which is not described by the standard model. Apart from the bumblebee field and global monopole, we have considered another ingredient known as GUP. GUP quantum corrections are expected for a black hole in quantum regime or in such length scale. In such an SME theory, GUP has been implemented previously [14]. GUP is based on a momentum-dependent modification in the standard dispersion relation which is conjectured to violate the Lorentz invariance [54]. In another work, it has been shown that the GUP deformation parameter is connected with the violation of Lorentz invariance [55]. Hence for a study of the black holes in bumblebee gravity, inclusion of global monopoles and GUP corrections are necessary for a better understanding of the system.

The work is organized as follows. In the next section 2, we have included a very short review on the field equations of bumblebee gravity and obtained a GUP corrected black hole solution around the global monopole in the bumblebee gravity. The black hole solutions with non-vanishing $\Lambda_{eff}$ are obtained in section 3. Here we have discussed the GUP corrected de Sitter and anti-de Sitter black hole solutions. We have obtained the quasinormal modes of black holes in section 4. In this section, we have tried to show the dependency of the quasinormal modes on different parameters of black holes. We have studied the time evolution of three different perturbations in section 5. The possibility of experimental detection of quasinormal modes has been studied in brief in the section 6. The sparsity of black holes is studied in section 7. Finally, we have summarized our results and findings in section 8. Throughout the paper we have considered $G=c=k_{B}=\hbar=l_{p}=1$, where $l_{p}$ is the Planck length.

Here we shall briefly review the simplest bumblebee gravity model and then derive the GUP corrected black hole solution in this gravity. In general, bumblebee models are vector or tensor theories that include a potential term which provides nonzero vacuum expectation values in the configuration of the fields. This behaviour of the fields affects the dynamics of other fields coupled to them, maintaining the conservation laws and geometric structures as required by pseudo-Riemannian manifold used in GR [1].

The Lagrangian density of the simplest bumblebee gravity model for the bumblebee field coupled to gravity in a torsion-free spacetime in the presence of the global monopole is given by [2]

where $\kappa=8\pi$, $\Lambda$ is the cosmological constant, $\mathcal{B}_{\mu}$ is the bumblebee field with the field strength tensor $\mathcal{B}_{\mu\nu}=\partial_{\mu}\mathcal{B}_{\nu}-\partial_{\nu}\mathcal{B}_{\mu}$. $V\!\!\left(\mathcal{B}^{\mu}\mathcal{B}_{\mu}\pm b^{2}\right)$ is the potential with $b^{2}$ as a real positive number, for which the spontaneous Lorentz violation takes place and $\xi$ is the coupling constant for a non-minimal gravity-bumblebee field interaction. $\mathcal{L}_{M}$ is basically the Lagrangian density of matter. In our case, we shall consider this to be for the global monopole. First, we shall calculate the field equations of the theory for the vanishing cosmological constant i.e. $\Lambda=0$. The corresponding field equations associated with the theory can be obtained from the Lagrangian (2.1) by varying its action with respect to the metric $g_{\mu\nu}$, which are given by

where $G_{\mu\nu}$ is the standard Einstein’s tensor. It is seen the energy-momentum tensor part of these field equations is a sum of two tensors: $T_{\mu\nu}^{B}$ and $T_{\mu\nu}^{M}$. The tensor $T_{\mu\nu}^{B}$ is that part of energy momentum tensor which depends on the bumblebee field and is given by

where the prime denotes the derivative with respect to the argument, and the usual matter part $T_{\mu\nu}^{M}$ for our case is considered to be of the form:

Here, the parameter $\eta$ is a constant quantity which is connected to the global monopole charge. The second set of field equations can be obtained by considering the variation of the Lagrangian (2.1) action with respect to the bumblebee field and it is given by

where $\mathcal{J}_{\nu}=\mathcal{J}_{\nu}^{\mathcal{B}}+\mathcal{J}_{\nu}^{M}$, in which $\mathcal{J}_{\nu}^{M}$ acts as a source term for the bumblebee field, while $\mathcal{J}_{\nu}^{\mathcal{B}}=2V^{\prime}\mathcal{B}_{\nu}-\frac{\xi}{\kappa}\mathcal{B}^{\mu}R_{\mu\nu}$ is the self interaction bumblebee field current.Taking trace of equation (2.2) we get,

Now, using equation (2.6) in equation (2.2) we can have,

As mentioned above, the potential in the Lagrangian (2.1) generates the nonzero vacuum expectation value for the field $\mathcal{B}_{\mu}$ since for $V\!\!\left(\mathcal{B}^{\mu}\mathcal{B}_{\mu}\pm b^{2}\right)=0$ it is required that

One can see that the solution of equation (2.8) gives a nonzero vacuum expectation value $\left\langle\mathcal{B}_{\mu}\right\rangle=b_{\mu},$ where $b_{\mu}$ is a vector, which is a function of spacetime coordinates and hence $b^{\mu}b_{\mu}=\pm\,b^{2}$. This nonzero background vector $b_{\mu}$ spontaneously violates the Lorentz symmetry. For the completeness it needs to be mentioned that $\pm$ signs in front of $b^{2}$ decide whether the background field $b_{\mu}$ is timelike or spacelike respectively [2, 13, 14, 16, 25]. For a static and spherically symmetric black hole solution with the Lorentz symmetry violation in the bumblebee field we consider the Birkhoff metric as an ansatz:

where $\gamma$ and $\rho$ are functions of $r$ and we fix the bumblebee field in its vacuum expectation value [56], i.e.

These lead the radial background field as given by

where $b_{r}(r)$ is the radial component of $b_{\mu}$. Using this component it is possible to write the equation (2.7) as the extended Einstein’s equations in vacuum in the form as

where the trace of the energy-momentum tensor is given by

The components of equation (2.13) can be written explicitly as

where $\lambda=\xi b^{2}$. Solving these equations (2.15), (2.16), (2.17) and (2.18) we obtain,

Thus, with these results the Lorentz symmetry breaking spherically symmetric solution for the field equations can be given as

Here, we have chosen $\rho_{0}=2M,$ where $M$ is the mass of the black hole and $\mu=-\,\eta^{2},$ the global monopole term. The metric (2.21) will give the Lorentz symmetry breaking standard spherically symmetric solution when $\eta=0$ [2], and the usual Schwarzschild metric for both $\eta=0$ and $\lambda\rightarrow 0$.

The prediction of the existence of a minimal length from different approaches of QG such as the black hole physics and the string theory has many important physical implications at very high energy scales. For example, at Planck scale the Schwarzschild radius of the corresponding black hole becomes comparable to the Compton wavelength. Thus the existence of a minimal length demands for the GUP, which is defined as [57, 58, 59, 60, 61]

where $\alpha$ and $\beta$ are dimensionless positive parameters. It is clear from the above relation that for $\alpha=\beta=0$ one can easily recover the Heisenberg’s uncertainty principle, i.e. $\Delta x\Delta p\geq\frac{1}{2}$. Following Ref. [57], we can obtain a bound for the massless particles as

which changes the equation (2.22) to

where $\cal E$ denotes the GUP corrected energy. Assuming $\Delta p\sim E\sim M$ and $\Delta x\sim r_{h}=\frac{2M}{(1-\mu)}$, we can write the above equation (2.24) in terms of mass as the following mass relation:

This mass relation leads to the GUP corrected event horizon as given by

Finally, the metric for a GUP corrected black hole in the bumblebee gravity with topological defects can be obtained from the metric (2.21) by replacing the black hole mass $M$ with the corresponding GUP corrected mass $M_{gup}$ as given by

and the event horizon of this black hole would be

In presence of non-vanishing cosmological constant, the metric functions which satisfy the modified field equations in absence of magnetic monopole are given by [62]

and

Here $\Lambda_{eff}=\dfrac{\kappa\lambda}{\xi},$ the effective cosmological constant. Now, in presence of global monopole, these metric functions are found to be of the form:

and

Depending on the value of $\Lambda_{eff}$, the above solutions can be classified into anti-de Sitter and de Sitter solutions which are discussed below.

We have obtained the black hole solutions in bumblebee gravity with topological defects in the previous section and implemented GUP corrections to the solutions. In this this section, we shall study the quasinormal modes obtained from these black holes for three different types of perturbations viz., massless scalar perturbation, electromagnetic perturbation and gravitational perturbation. In case of the quasinormal modes of a black hole obtained from the perturbation of the test field i.e. of scalar field or electromagnetic field, it is considered that the test field has negligible reaction on the spacetime. The Schrödinger like wave equations are derived from the conservation relations of the test fields on the black hole spacetime. For a scalar field it will be a Klein Gordon type equation and for electromagnetic fields, it will be the Maxwell equations. The Schrödinger like wave equations for gravitational perturbations can be obtained by introducing perturbation to the spacetime metric and the field equations. In this work, we have considered the axial gravitational perturbation and calculated the quasinormal modes associated with it.

Now, considering only the axial perturbations, the perturbed metric can be given by [29]

here the parameters $a_{1},a_{2}$ and $a_{3}$ are the functions of time $t$, radial coordinate $r$ and polar angle $\theta$. These parameters define the perturbation introduced to the metric. $g_{tt}$ and $g_{rr}$ being independent of $t$ and $\theta$, are the zeroth order terms or the background terms.