Gravitational waves effects in a Lorentz-violating scenario

This paper focuses on how the production and polarization of gravitational waves are affected by spontaneous Lorentz symmetry breaking, which is driven by a self-interacting vector field. Specifically, we examine the impact of a smooth quadratic potential and a non-minimal coupling, discussing the constraints and causality features of the linearized Einstein equation. To analyze the polarization states of a plane wave, we consider a fixed vacuum expectation value (VEV) of the vector field. Remarkably, we verify that a space-like background vector field modifies the polarization plane and introduces a longitudinal degree of freedom. In order to investigate the Lorentz violation effect on the quadrupole formula, we use the modified Green function. Finally, we show that the space-like component of the background field leads to a third-order time derivative of the quadrupole moment, and the bounds for the Lorentz-breaking coefficients are estimated as well.


I. INTRODUCTION
Understanding the spacetime structure at the Planck scale and developing a corresponding quantum theory of gravity is one of the most challenging open questions in modern physics.At such high-energy scales, it is possible that some of the low-energy symmetries present in the standard model of particles and gravity, such as Lorentz and CPT symmetries, may be violated.This phenomenon has been observed in various theories, including some string theory vacuum [1,2], Loop Quantum Gravity [3,4], non-commutative geometry [5][6][7], Horava Gravity [8], and Very Special Relativity [9,10].
Gravitational wave detection was one of the most important events for high-energy physics.Two notable events took place in 2015 and were announced on February 11th, 2016 by the VIRGO and LIGO laboratories.The first event, GW150914, was detected on September 14th, 2015 [31].The second one, GW151226, was detected on December 26th, 2015.Both events were caused by the coalescence of two black holes located at distances of approximately 410 Mpc and 440 Mpc, respectively [32].The detection of gravitational waves from these events represents a major milestone in the study of black holes and the nature of gravity, opening up new avenues for research and providing important insights into the workings of the universe at the most fundamental level [33,34].
One relatively simple field that breaks the Lorentz symmetry is the bumblebee field B µ .The Lorentz symmetry is spontaneously broken by the dynamics of B µ , which acquires a non-zero vacuum expectation value (VEV) [35][36][37].This model was originally considered in the context of string theory, where LSB is triggered by the potential The Einstein-aether gravity, as outlined in [38], represents a general tensor-vector theory at the two-derivative levels.This implies that the considered Lagrangian encompasses generic terms, manifesting quadratically in its derivatives.In the context of bumblebee gravity, the introduction of a vector background denoted as b µ , with exclusively a nonzero temporal component, engenders five independent propagating degrees of freedom.Notably, a parallel occurrence of this characteristic is observed in the Einstein-aether gravity theory, as discussed in [39].
This work focuses on analyzing the modifications in the polarization and production of gravitational waves due to Lorentz symmetry breaking (LSB) in the weak field regime.In particular, we investigate the effects of the spontaneous breaking of Lorentz symmetry, triggered by the bumblebee vector field.Then, we solve the modified wave equation for the perturbation field and compare the polarization tensor of the modified gravitational wave solution with the conventional case.Next we introduce a current J µν to the model and derive a Green function for the timelike b µ = (b 0 , 0) and spacelike b µ = (0, b) configurations for the bumblebee VEV.Subsequently, we obtain a modified quadrupole formula for the graviton, which enables us to compare the perturbation for the modified equation with the usual one and to identify the modifications in the theory.

II. MODIFYING THE GRAVITON WAVE EQUATION
The minimal extension of the gravity theory, including the Lorentz-violating terms is given by the following expression [40] S = S EH + S LV + S matter . ( The first term refers to the usual Einstein-Hilbert action, where R is the curvature scalar and Λ is the cosmological constant, which will not be considered in this analysis.The S LV term consists of the coupling between the bumblebee field and the curvature of spacetime where u, s µν and t αβµν are dynamical fields with zero mass dimension [41].And the last term represents the mattergravity couplings which, in principle, should include all fields of the standard model as well as the possible interactions with the coefficients u, s µν and t αβµν .The dynamics of the bumblebee field B µ is dictated by the action [42,43] where we introduced the field strength B µν = ∂ µ B ν − ∂ ν B µ in analogy with the electromagnetic field tensor F µν .In fact, the bumblebee models are not only used as toy models to investigate the excitation originating from the LSB mechanism.They also provide an alternative to U (1) gauge theory for photons [44].In this theory, they appear as Nambu-Goldstone modes due to spontaneous Lorentz violation [45], rather than fundamental particles.
The gravity-bumblebee coupling can be represented by Eq. (3) defining and the potential V (X) reads It is responsible for triggering spontaneously the Lorentz violation and breakdown of the diffeomorphism.Here, b 2 is a positive constant that stands for the non-zero vacuum expectation value of this field.In our study, we aim to investigate how the coupling between gravity and the bumblebee field affects the behavior of graviton.To do so, we consider the linearized version of the metric g µν with the Minkowski background and the bumblebee field B µ , which is split into the vacuum expectation values b µ and the quantum fluctuations Bµ By multiplying the linearized equation of motion with p µ p ν , b µ b ν , p (µ b ν) , the set of constraints are obtained Despite the violation of the diffeomorphism symmetry, twelve constraints in Eq. ( 8) enables only two propagating degrees of freedom, as for the usual Lorentz-invariant graviton.In other words, no graviton mass is produced due to the interaction with the bumblebee field at leading order.The modified dispersion relation (MDR) associated with the physical pole is given by [46] [ where b • p = η µν b µ p µ , with p µ = (p 0 , p) and b µ = (b 0 , b).It turns out that the MDR preserves both stability and causality [46].Also, the respective thermodynamic behavior based on such a modified dispersion relation was addressed in the literature recently by Araújo Filho [47].Also, it is worth mentioning that there exist a correlation of our model under consideration with the Einstein-aether theory.In general lines, the action of this latter case can be written as [48] where In the Einstein-aether theory, the action is generally understood to be composed of two primary components: the usual Einstein-Hilbert action, which provides the foundational framework, and a Lagrange multiplier denoted as λ.Furthermore, covariant terms involving the vector u µ are incorporated as well, ensuring that they contain at most two derivatives [48].Based on the kinetic terms, we can argue that Eq. ( 4) turns out to be a particular case of Eq. (10).

III. POLARIZATION STATES
Studying the polarization of gravitational waves is crucial to gaining insights into the properties of the source and the nature of spacetime.It provides unique information about the motion of massive objects and can reveal any effects encountered during their journey, making it a critical area of research in modern astrophysics [39,[49][50][51][52][53].In this sense, we obtain the modified polarization tensor to the free perturbation, satisfying the constraints (8) and the dispersion relation (9).The source-free equation has the form Consider a plane-wave h µν = ε µν e i(ωt−kz) propagating along the z axis with momentum p µ = (p 0 , 0, 0, p 3 ).The constraint b µ h µ ν = 0 forbids any state in the direction of the background vector b µ .For a timelike b µ = (b 0 , 0, 0, 0), the polarization tensor reads preserving two Lorentz invariant transversal modes.Nevertheless, the group velocity is modified by Now let us consider a spacelike vector with a spatial component along b µ = (0, 0, 0, b 3 ).The polarization tensor still has the form of Eq. ( 12), albeit the wave propagates with dispersion relation Similar modified dispersion relations were obtained for gauge-invariant containing higher mass dimension operators [54,55].The group velocity Therefore, for a timelike or a spacelike b µ , i.e., with ⃗ b parallel to ⃗ p, the transversal characteristic of the gravitational waves is preserved and the wave velocity is slowed by the Lorentz violation.
On the other hand, considering ⃗ b perpendicular to ⃗ p, e.g., b µ = (0, 0, b 2 , 0), we obtain It is worth mentioning that the polarization tensor (15) has two independent states, ε 11 and ε 13 .The plus state ε 11 corresponds to the transversal polarization in the x direction and it is present in the Lorentz invariant wave.Nonetheless, the presence of the cross-state ε 13 leads to a longitudinal polarization along the z direction -a feature forbidden in the Lorentz invariant theory.In other words, the background vector transforms one transversal into a longitudinal degree of freedom.The dispersion relation, in its turn, is kept invariant Moreover, the line element is modified by In this manner, the Lorentz-violating gravitational waves produce contractions and dilatations in the xz plane, and also time deformations.The effects of these modified gravitational waves on particles can be observed by considering the geodesic deviation equation, which for slowly test particle U µ ≈ (1, 0, 0, 0) reads where the S µ is the displacement vector.From Eq.( 15), the temporal and y components vanish D 2 S 0 dτ 2 = D 2 S 2 dτ 2 = 0.Although h 00 is not zero, no time tilde forces arise in this context.In the x direction, Eq.( 18) gives whereas the tidal force in the longitudinal direction has the form For two test particles close enough, their events can be considered simultaneous and the corresponding timelike component S 0 can be neglected.Note that both plus and cross polarizations contribute to the geodesic deviation in the x direction, while the cross state produces variation along the longitudinal direction only.

IV. GRAVITATIONAL WAVES PRODUCTION IN THE PRESENCE OF THE BUMBLEBEE FIELD
For the analysis of the production of GW, we add a current J µν to the homogeneous equation to the graviton field in Eq. ( 11) resulting in The perturbation is determined by For the modified graviton equation, we can represent the Green function in the momentum space so that to get G in the configuration space, we may use the inverse Fourier transform.The inversion is made for two particular cases; the first is considering that b µ has a timelike configuration b µ = b 0 , 0 , in this case, the Green function resumes to and the inversion results to The second case is a spacelike configuration b µ = (0, b), which will reduce G(p) to where Ψ is the angle between p and b, therefore cos 2 Ψ = cos(θ) cos(θ b ) + sin(θ) sin(θ b ) cos(ϕ b − ϕ), with (θ, ϕ) and (θ b , ϕ b ) the angular coordinates of the trivectors p and b, respectively.The inverse Fourier transform of Eq. ( 26) expanded to the second order of |b| is where G R (x − y) is the retarded Green function and with the second term being given by For the usual Einstein-Hilbert linearized theory, we have the formula for the perturbation where t r = t − r is the retarded time and I ij is the quadrupole momentum defined as With b µ = (b 0 , 0) and considering that J µν = 16πGT µν , the perturbations are written as where the retarded time is modified to t ′ r = t − r 1 + ξb 2 0 , evidencing that the waves propagates slower.When b µ = (0, b) the perturbation is Here, we can see the anisotropy in above solution.Such a feature is well-known within the context of theories modified by the bumblebee vector.The vector b selects a preferential direction for the wave propagation.Another peculiarity is that a term involving third derivative of the I ij gives rise to.In addition, terms of dipole are still prohibited due to the momentum conservation.Now, let us analyze the modifications in the simple binary black hole problem pictured in Fig. 1.It represents the movement of a binary system constituted of two masses m 1 and m 2 in the xy plane.The distances from the center of the reference frame to the respective masses are r 1 and r 2 .
Furthermore, the density of energy T 00 is determined from the formula and the equations of motion of the two particles are with M = m 1 + m 2 , l 0 = r 1 + r 2 and ω is the angular velocity of the binary.The only non-zero components of I ij are with µ = m 1 m 2 /(m 1 + m 2 ).By substituting Eq. ( 35) in Eq. ( 32), we have with the spacial tensors A ij and B ij being defined as From Eq. ( 36), we can infer that the frequency of the GW in the presence of the bumblebee field does not change.Besides that, the second term depends of the projection of the bumblebee field in the position vector.Now, we can extract the transverse-traceless part of this solution for the sake of finding the polarization of the waves.Using the projector defined as with P ij ≡ δ jk − n j n k and n j = (0, 0, 1) is the unitary vector in the z direction.Therefore, we obtain two polarization states which can be rewritten as where the resulting amplitude A is given by and the phase difference has the form Fig. 2 shows the comparison of h T T xx for three cases, without bumblebee field (Einstein GW), with ⃗ b perpendicular to ⃗ r, and with ⃗ b and ⃗ r parallel.It is worth mentioning that the wave frequency is considered low, i.e., before coalescence.Besides that, the parameter ξb 2 was overestimated for the effects to be visible.xx for three cases, without bumblebee field (Einstein GW -blue curve), with ⃗ b perpendicular to ⃗ r (red dashed curve), and with ⃗ b and ⃗ r parallel (green dash-doted curve).Note that when ⃗ b and ⃗ r are perpendicular the amplitude is greater and when ⃗ b and ⃗ r are parallel the amplitude is smaller.Besides, a phase difference is observed.In this plot, the frequency is considered low, i.e. before coalescence, and the parameter ξb 2 was overestimated for the effects to be visible.
The Lorentz violation modifies the GW amplitude and produces a phase difference which forms an elliptic polarization.This occurs on the h xx -h xy plane when ⃗ b and ⃗ r parallel.Note that both amplitude and phase depend on the relative direction with respect to the background vector.At leading order, the correction in the amplitude due to the Lorentz violation has the form Therefore, for ⃗ k parallel (θ b = 0) and anti-parallel (θ b = π) to ⃗ b, the Lorentz violation reduces the amplitude to its minimal, whereas for ⃗ k perpendicular to ⃗ b, the correction attains its maximum value, as shown In Fig. (2).Also, in Fig. 3, we display the polarization analysis of gravitational waves in the h xx -h xy plane reveals distinct behaviors in three scenarios: without the bumblebee field (Einstein gravitational waves denoted by the blue curve), with the vector field ⃗ b perpendicular to ⃗ r (illustrated by the red dashed curve), and with ⃗ b and ⃗ r oriented in parallel (depicted by the green dash-dotted curve).It is noteworthy that in the case where ⃗ b and ⃗ r align, the gravitational wave exhibits elliptical polarization.Conversely, when ⃗ b and ⃗ r are perpendicular, the polarization mirrors the circular pattern characteristic of Einstein gravitational waves, albeit with a heightened amplitude.

V. ESTIMATION OF BOUNDS
In order to set upper bounds for the Lorentz violation parameter, we can compare the group velocity that comes from ( 11) with the one obtained in [56], which is the same used by the LIGO collaboration [57].Therefore, using the group velocity for the massive graviton where m g is the upper bound for the assumed mass of the graviton and k g , is the graviton energy.The LIGO and Virgo collaborations reported that the peak of the gravitational wave signal detected in event GW150914 has a frequency of ν = 150 Hz [57] so that the energy is k g = hν ∼ 6.024 • 10 −13 eV (for h = 4.136 • 10 −15 eV).Besides that, the lower bound for the graviton mass found by them is given by m g < 1.20 • 10 −22 eV so that which is the value to be used to estimate the bounds of the Lorentz violating parameters.This, we can perform the same estimate for the group velocity (44) found in this work.First, let b µ be space-like so that For the case where ⃗ b e ⃗ p are orthogonal, For b µ time-like, we have which is the same value found for the space-like parameter.
Therefore, the modifications in the polarization states determined here can be found by gravitational wave detection experiments.Between LIGO and LISA, the one that has a greater perspective of finding such a modification is LISA, as it will be orbiting the planet Earth so its orientation in relation to a theoretical background vector will have more variations than LIGO.This same idea is used in clock comparison experiments, where better results are found when clocks are orbiting planet Earth (e.g.satellites or international space stations).However, as LISA will only be launched in 2034, we can think of some modifications in the experiments that already exist here on Earth, so that the pendulums that form the test masses can be adjusted so that the polarization modified due to a theoretical vector background can be measured.
Based on the LIGO measurements [57], we can estimate the bound for the amplitude.To do so, we suppose the contribution of the Lorentz violation δA is less than the observational error.Such a procedure allows us to estimate an upper bound at the level of ξb 2 ≤ 2.55 × 10 −19 .

VI. CONCLUSION
In this work, we investigated the effects of the Lorentz symmetry breaking due to the presence of a non-zero VEV for a vector field.We used the simplest model in literature, the so-called bumblebee model.We expanded the density Lagrangian of the theory up to the second order of h µν , and we observed the modified equation of motion to the graviton.Hence, we obtained the modified dispersion relation and the corresponding plane wave solution.We also showed that the free graviton had only two degrees of freedom, and we explicitly determined the new polarization tensor dependent on the background field.Furthermore, despite the coupling with the bumblebee field, no massive mode was found for the graviton.
Therefore, the modified wave equation was solved in terms of the Fourier modes.Taking into account the constrains p µ h µν = 0 and b µ h µν = 0, we got modifications in polarization states.For the case where b µ was timelike or ⃗ b was in the same direction of the wave, no modifications appeared in the polarization tensor.Nevertheless, there existed modifications in the group velocity.In addition, for the case where b µ was spacelike and ⃗ b was orthogonal to the momentum, there existed significant modifications, such that one polarization mode was in a orthogonal plane with respect to ⃗ b and the another one was longitudinal with respect to the wave momentum.Besides that, we obtained the following upper bound for the bumblebee parameter, ξ| ⃗ b| 2 < 4.10 −20 .This was done by comparing the group velocity obtained here with the group velocity for the massive graviton and using experimental values of the GW150914 event.
Thereby, we sought for the effects of the spontaneous Lorentz breaking in the production of the gravitational waves.Assuming the presence of a matter source, we calculated the Green function and the corresponding solution for the graviton field in two special configurations.In the first one, we considered a timelike configuration for b µ .In this case, only the velocity of propagation of the wave was smaller by a factor of 1 + ξb 2 0 .In the second case, we regarded a spacelike configuration for the bumblebee VEV.The corrections to the quadrupole formula showed the existence of anisotropy in the solution, showing an explicit dependency in the relative direction between the background field and the position vector.The frequency of the wave did not change, once the tensor I ij remained the same of the usual Einstein-Hilbert theory.In the circular orbit binary solution, we showed that the polarizations were not circular in contrast with the usual solution.Finally, the amplitude changed by a factor of 1 − ξb 2 /2.

Figure 1 :
Figure 1: The representation of binary black hole problem.

Figure 2 :
Figure 2: Comparison of h T Txx for three cases, without bumblebee field (Einstein GW -blue curve), with ⃗ b perpendicular to ⃗ r (red dashed curve), and with ⃗ b and ⃗ r parallel (green dash-doted curve).Note that when ⃗ b and ⃗ r are perpendicular the amplitude is greater and when ⃗ b and ⃗ r are parallel the amplitude is smaller.Besides, a phase difference is observed.In this plot, the frequency is considered low, i.e. before coalescence, and the parameter ξb 2 was overestimated for the effects to be visible.

Figure 3 :
Figure3: Polarization of the GW in the hxx-hxy plane for three cases, without bumblebee field (Einstein GW -blue curve), with ⃗ b perpendicular to ⃗ r (red dashed curve), and with ⃗ b and ⃗ r parallel (green dash-doted curve).It is observed that when ⃗ b and ⃗ r are parallel the GW is elliptically polarized.When ⃗ b and ⃗ r are the polarization is circular like the Einstein GW, but with greater amplitude.