Bumblebee Gravity Research Articles

Bumblebee Black Holes in Light of Event Horizon Telescope Observations

We report the existence of novel static spherical black hole solutions in a vector–tensor gravitational theory called the bumblebee gravity model, which extends the Einstein–Maxwell theory by allowing the vector to nonminimally couple to the Ricci curvature tensor. A test of the solutions in the strong-field regime is performed for the first time using the recent observations of the supermassive black hole shadows in the galaxy M87 and the Milky Way from the Event Horizon Telescope Collaboration. The parameter space is found largely unexcluded and more experiments are needed to fully bind the theory.

Probing vector hair of black holes with extreme-mass-ratio inspirals

The bumblebee gravity model, with a vector field nonminimally coupled to gravity, is a natural extension of the Einstein-Maxwell theory. In this theory, a black hole can carry a vector hair, making the metric deviate from the Schwarzschild metric. To investigate the detectability of the vector hair, we consider an Extreme Mass Ratio Inspiral (EMRI) system, where a stellar-mass black hole inspiraling into a supermassive black hole. We find that, with a one-year observation of an EMRI by a space-based gravitational-wave detector, we can probe the vector charge as small as $Q\sim 10^{-3}$ in the bumblebee gravity model, which is about three orders of magnitude tighter comparing to current EHT observations.

Gravitational time advancement effect in Bumblebee gravity for Earth bound systems

Gravitational time advancement effect in Bumblebee gravity for Earth bound systems

QNMs of slowly rotating Einstein–Bumblebee black hole

We have studied the quasinormal modes (QNMs) of a slowly rotating black hole with Lorentz-violating parameter in Einstein–Bumblebee gravity. We analyse the slow rotation approximation of the rotating black hole in the Einstein–Bumblebee gravity, and obtain the master equations for scalar perturbation, vector perturbation and axial gravitational perturbation, respectively. Using the matrix method and the continuous fraction method, we numerically calculate the QNM frequencies. In particular, for scalar field, it shows that the QNMs up to the second order of rotation parameter have higher accuracy. The numerical results show that, for both scalar and vector fields, the Lorentz-violating parameter has a significant effect on the imaginary part of the QNM frequencies, while having a relatively smaller impact on the real part of the QNM frequencies. But for axial gravitational perturbation, the effect of increasing the Lorentz-violating parameter ell is similar to that of increasing the rotation parameter {tilde{a}}.

Static spherical vacuum solutions in the bumblebee gravity model

The bumblebee gravity model is a vector-tensor theory of gravitation where the vector field nonminimally couples to the Ricci tensor. By investigating the vacuum field equations with spherical symmetry, we find two families of black-hole (BH) solutions in this model: one has a vanishing radial component of the vector field and the other has a vanishing radial component of the Ricci tensor. When the coupling between the vector field and the Ricci tensor is set to zero, the first family becomes the Reissner-Nordstr\"om solution while the second family degenerates to the Schwarzschild solution with the vector field being zero. General numerical solutions in both families are obtained for nonzero coupling between the vector field and the Ricci tensor. Besides BH solutions, we also reveal the existence of solutions that have a nonvanishing $tt$-component of the metric on the supposed event horizon where the $rr$-component of the metric diverges while the curvature scalars are finite. These solutions are not supported by existing observations but present certain properties that are of academic interests. We conclude the study by putting the BH solutions into tests against the Solar-System observations and the images of supermassive BHs.

Polarizations of gravitational waves in the bumblebee gravity model

Lorentz violation modifies the dispersion relation of gravitational waves (GWs), and induces birefringence and anisotropy in propagation. Our study shows that Lorentz violation can also activate multiple polarizations of GWs. We use the gauge invariants to investigate the polarizations of GWs in the bumblebee gravity model, and obtain the following results. (i) For a vector background ${b}^{\ensuremath{\mu}}$ with only a nonzero temporal component ${b}^{t}$, there are five independent propagating degrees of freedom (d.o.f.), which is similar to the Einstein-\ae{}ther theory. (ii) The presence of a spatial component in the background defines a preferred spatial direction which breaks rotational symmetry. We denote $\stackrel{^}{\mathbit{b}}$ as the direction of the spatial part of the background and ${b}_{s}$ as its length. If GWs propagate along $\stackrel{^}{\mathbit{b}}$, the polarization content is similar to the purely timelike case. (iii) If the propagation direction of GWs is separated by an angle $\ensuremath{\beta}$ to $\stackrel{^}{\mathbit{b}}$, and $\ensuremath{\beta}=\mathrm{arccos}({b}^{t}/{b}_{s})$, there are only two tensor polarizations. (iv) If $\ensuremath{\beta}\ensuremath{\ne}\mathrm{arccos}({b}^{t}/{b}_{s})$, there are only two independent d.o.f., and the vector and scalar modes degenerate with the tensor modes. The tensor perturbations can activate a mixture of all six polarizations simultaneously. Finally, we point out the difference in GWs between the bumblebee gravity model and the minimal Standard-Model Extension framework in the linearized regime. Current observations have placed stringent constraints on the anisotropy induced by the background, while our theoretical study still reveals some novel phenomena and provides more understanding about the interaction between the Lorentz-violating vector field and gravity.

Strong gravitational lensing and shadow constraint from M87* of slowly rotating Kerr-like black hole

Motivated by (i) more and more interest in strong gravitational lensing by supermassive black holes due to the achievement of EHT observations, (ii) the ongoing popular topic on the possibility of Lorentz symmetry being broken in gravitation and its consequences, we will apply the Einstein bumblebee gravity with Lorentz violation (LV) to the study of strong gravitational lensing effect and the black hole shadow of slowly rotating Kerr-like black hole. In the strong gravitational lensing sector, we first calculate the deflection angle; then treating the slowly rotating Kerr-like black hole as supermassive M87* black hole, we evaluate the gravitational lensing observables (position, separation and magnification) and the time delays between the relativistic images. In the black hole shadow sector, we show the effect of LV parameter on the luminosity of the black hole shadow and photon sphere using the infalling spherical accretion. Moreover, we explore the dependence of various shadow observables on the LV parameter, and then give the possible constraint on the LV parameter by M87* black hole of EHT observations. We find that the LV parameter shows significant effect on the strong gravitational lensing effect, the black hole shadow and photon sphere luminosity by accretion material. Our results point out that the future generations of EHT observation may help to distinguish the Einstein bumblebee gravity from GR, and also give a possible constrain on the LV parameter.

Novel features of Schwarzschild-like black hole of Lorentz violating bumblebee gravity

A possible avenue for observing quantum gravity (QG) effects at low energy scales is to introduce spontaneous Lorentz violation (LV) in new models of gravity theories. One such model in the literature is bumblebee gravity yielding Schwarzschild-like black hole and weak field Solar System observations involve LV corrections characterized by the parameter ℓ. Here we first show that these LV corrections have a novel genesis in the conical angle Δ = πb subtended at the origin of the spacetime of massless bumblebee gravity. Exploiting the resultant asymptotic light deflection angle πb −1 as a new input in the exact deflection formula, we next study the strong field lensing properties of the Schwarzschild-like black hole exploring how they differ from those of the Schwarzschild black hole of general relativity. It is shown that the angular image separation and ratio of fluxes could respectively be s ∼ e πℓ and r ∼ e−πℓ times those of the Schwarzschild black hole (ℓ = 0). However, the shadow of the Schwarzschild-like black hole is independent of ℓ suggesting that observations of shadow radii cannot reveal QG effects. Finally, we raise an interesting issue about the measurability of the LV corrections caused by strong field lensing. An appendix briefly outlines lensing by the spinning bumblebee black hole.

Superradiance scattering off Kerr-like black hole and its shadow in the bumblebee gravity with noncommutative spacetime

We consider a bumblebee gravity-based Kerr-like black hole in a noncommutative (NC) background and study the superradiance effect and the shadow cast. We extensively study the different aspects of the black hole associated with a generalized Kerr-like spacetime metric endowed with the corrections jointly linked with Lorentz violation and NC spacetime effect. We examine the deviation of shape, and size of the ergosphere, energy emission rate, in this generalized situation. We also examine the influence of admissible values of Lorentz violating parameter ell and NC parameter b on the superradiance effect and shadow of the black hole. The admissible range has been determined from the observation of the Event Horizon Telescope (EHT) collaboration concerning M87^{*} astronomical black hole. We observe that the superradiance phenomena has a crucial dependence on the parameter ell and b apart from its dependence on a or {hat{a}}= sqrt{ell +1}a which is linked with the spin of the black hole. We also observe that with the increase in Lorentz violating parameter ell , the size of the black hole shadow increases, and with the increase in the NC parameter b the size of the black hole decreases. We have made an attempt to constrain the NC parameters b of this modified Kerr-like black hole by using the data available from the Event Horizon Telescope (EHT) collaboration. This study shows that black holes associated with NC Kerr-like spacetime may be a suitable candidate for the astrophysical black holes.

Probing bumblebee gravity with black hole X-ray data

Bumblebee gravity is one of the simplest gravity theories with spontaneous Lorentz symmetry breaking. Since we know a rotating black hole solution in bumblebee gravity, we can potentially test this model with the available astrophysical observations of black holes. In this work, we construct a reflection model in bumblebee gravity and we use our model to analyze the reflection features of a NuSTAR spectrum of the Galactic black hole EXO 1846-031 in order to constrain the Lorentz-violating parameter ell . We find that the analysis of the reflection features in the spectrum of EXO 1846-031 cannot constrain the parameter ell because of a very strong degeneracy between the estimates of ell and of the black hole spin parameter a_*. Such a degeneracy may be broken by combining other observations.

Spontaneous Lorentz symmetry breaking and one-loop effective action in the metric-affine bumblebee gravity

The metric-affine bumblebee model in the presence of fermionic matter minimally coupled to the connection is studied. We show that the model admits an Einstein frame representation in which the matter sector is described by a non-minimal Dirac action without any analogy in the literature. Such non-minimal terms involve unconventional couplings between the bumblebee and the fermion field. We then rewrite the quadratic fermion action in the Einstein frame in the basis of 16 Dirac matrices in order to identify the coefficients for Lorentz/CPT violation in all orders of the non-minimal coupling ξ. The exact result for the fermionic determinant in the Einstein frame, including all orders in ξ, is also provided. We demonstrate that the axial contributions are at least of second order in the perturbative expansion of ξ. Furthermore, we compute the one-loop effective potential within the weak field approximation.

Quasinormal modes and Hawking radiation sparsity of GUP corrected black holes in bumblebee gravity with topological defects

We have obtained the Generalized Uncertainty Principle (GUP) corrected deSitter and anti-de Sitter black hole solutions in bumblebee gravity with atopological defect. We have calculated the scalar, electromagnetic andgravitational quasinormal modes for the both vanishing and non-vanishingeffective cosmological constant using Padé averaged sixth order WKBapproximation method. Apart from this, the time evolutions for all threeperturbations are studied, and quasinormal modes are calculated using the timedomain profile. We found that the first order and second order GUP parametersα and β, respectively have opposite impacts on the quasinormalmodes. The study also finds that thepresence of a global monopole can decrease the quasinormal frequencies and thedecay rate significantly. On the other hand, Lorentz symmetry violation hasnoticeable impacts on the quasinormal frequencies and the decay rate. Wehave studied the greybody factors, power spectrum and sparsity of the blackhole with the vanishing effective cosmological constant for all the threeperturbations. The presence of Lorentz symmetry breaking and the GUPparameter α decrease, while other GUP parameter β and the presenceof global monopole increase the probability of Hawking radiation to reach thespatial infinity. The presence of Lorentz violation can make the black holesless sparse, while the presence of a global monopole can increase the sparsityof the black holes. Moreover, we have seen that the black hole areaquantization rule is modified by the presence of Lorentz symmetry breaking.

Radiative corrections in metric-affine bumblebee model

We consider the metric-affine formulation of bumblebee gravity, derive the field equations, and show that the connection can be written as Levi-Civita of a disformally related metric in which the bumblebee field determines the disformal part. As a consequence, the bumblebee field gets coupled to all the other matter fields present in the theory, potentially leading to nontrivial phenomenological effects. To explore this issue we compute the weak-field limit and study the resulting effective theory. In this scenario, we couple scalar and spinorial matter to the effective metric which, besides the zeroth-order Minkowskian contribution, also has the vector field contributions of the bumblebee, and show that it is renormalizable at one-loop level. From our analysis it also follows that the non-metricity of this theory is determined by the gradient of the bumblebee field, and that it can acquire a vacuum expectation value due to the contribution of the bumblebee field.

Schwarzschild-like black hole with a topological defect in bumblebee gravity

In this paper, we derive an exact black hole spacetime metric in the Einstein–Hilbert–Bumblebee (EHB) gravity around global monopole field. We study the horizon, temperature, and the photon sphere of the black hole. Using the null geodesics equation, we obtain the shadow cast by the Schwarzschild-like black hole with a topological defect in bumblebee gravity. Interestingly, the radius of shadow of the black hole increases with increase in the global monopole parameter. We also visualize the shadows and energy emission rates for different values of parameters. Moreover, using the Gauss–Bonnet theorem, we calculate the deflection angle in weak field limits and we discuss the possibility of testing the effect of global monopole field and bumblebee field, on a weak deflection angle. We find that the global monopole parameter and also Lorentz symmetry breaking parameter has an increasing effect on the deflection angle.

Quasinormal frequencies for a black hole in a bumblebee gravity

After recent observational events like the LIGO-Virgo detections of gravitational waves and the shadow image of the M87* supermassive black hole by event horizon telescope (EHT), the theoretical study of black holes was significantly improved. Quantities as quasinormal frequencies, shadows, and light deflection become more important to analyze black hole models. In this context, an interesting scenario to study is a black hole in the bumblebee gravity. The bumblebee vector field imposes a spontaneous symmetry breaking that allows the field to acquire a vacuum expectation value that generates Lorentz Violation (LV) into the black hole. In order to compute the quasinormal modes (QNMs) via the WKB method, we obtain the Reege-Wheeler's equation with a bell-shaped potential for this black hole. Both QNMs, the scalar and tensorial modes, are computed for the black hole in the bumblebee scenario. The results obtained in bumblebee gravity are compared to Schwarzschild and Einstein-aether black holes. In general, the LV parameter decreases the real part of frequency for scalar and gravitational perturbations. The modulus of imaginary parts is increased with the LV parameter for the scalar field and decreased by the gravitational field. Moreover, the time-domain perturbations are studied and damping profiles are shown.

The gravitational bending angle by static and spherically symmetric black holes in bumblebee gravity

This work investigates the influence of the Lorentz symmetry breaking in the bending angle of massive particles and light for bumblebee black hole solutions. The solutions analyzed break the Lorentz symmetry due to a non-zero vacuum expectation value of the bumblebee field. We use the Ishihara method, which allows us to study the bending angle of light for finite distances, and it is applicable to non-asymptotically flat spacetimes when considering the receiver viewpoint. In order to analyze the deflection of massive particles, we systematize the Ishihara method for its application in the Jacobi metric. This systematization allows the study of the deflection angle of massive particles using the Gauss-Bonnet theorem. We consider two backgrounds: the first was found by Bertolami et al. and is asymptotically flat. The second was found recently by Maluf et al. and is not asymptotically flat due to an effective cosmological constant.

Greybody radiation and quasinormal modes of Kerr-like black hole in Bumblebee gravity model

In the framework of the Lorentz symmetry breaking (LSB), we investigate the quasinormal modes (QNMs) and the greybody factors (GFs) of the Kerr-like black hole spacetime obtained from the bumblebee gravity model. In particular, we analyze the scalar and fermionic perturbations of the black hole within the framework of both semi-analytic WKB method and the time domain approach. The impacts of the LSB on the bosonic/fermionic QNMs and GFs of the Kerr-like black hole are investigated in detail. The obtained results are graphically depicted and discussed.

Bumblebee gravity with a Kerr\u2013Sen like solution and its Shadow

We have considered the bumblebee gravity model where lorentz-violating (LV) scenario gets involved through a bumblebee field vector field B_mu . A spontaneous symmetry breaking allows the field to acquires a vacuum expectation value that generates LV into the system. A Kerr–Sen-like solution has been found out starting from the generalized form of a radiating stationery axially symmetric black hole metric. We compute the effective potential offered by the null geodesics in the bumblebee rotating black hole spacetime. The shadow has been sketched for different variations of the parameters involved in the system. A careful investigation has been carried out to study how the shadow gets affected when Lorentz violation enters into the picture. The emission rate of radiation has also been studied and how it varies with the LV parameter ell is studied scrupulously.

Bumblebee gravity and particle motion in Snyder noncommutative spacetime structures

A metric with a Lorentz violating parameter is associated with thebumblebee gravity model. We study the motion of a particle in thisbumblebee background where the dynamical variables satisfynon-canonical Snyder algebra along with some critical survey onthe classical observations in the bumblebee background to see howthese would likely differ from Schwarzschild background. It hasbeen found that the perihelion shift acquires a generalizedexpression with two independent parameters. One of these two isconnected with the Lorentz violating factor and the other isinvolved in the Snyder algebraic formulation. We also observe thatthe time period of revolution, in general, acquires a Lorentzviolating factor in the bumblebee background, however, for thecircular orbit, it remains unchanged even in the presence of theLorentz violating factor in the bumblebee background. Theparameters used here can be constrained with the same type ofconjecture used earlier.

Metric-affine bumblebee gravity: classical aspects

We consider the metric-affine formulation of bumblebee gravity, derive the field equations, and show that the connection can be written as Levi-Civita of a disformally related metric in which the bumblebee field determines the disformal part. As a consequence, the bumblebee field gets coupled to all the other matter fields present in the theory, potentially leading to nontrivial phenomenological effects. To explore this issue we compute the post-Minkowskian, weak-field limit and study the resulting effective theory. In this scenario, we couple scalar and spinorial matter to the effective metric, and then we explore the physical properties of the VEV of the bumblebee field, focusing mainly on the dispersion relations and the stability of the resulting effective theory.