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Gravitational waves in metric-affine bumblebee gravity

A. A. Araújo Filho

Abstract

We study the propagation and emission of gravitational waves in the metric-affine formulation of the bumblebee model, where spontaneous Lorentz symmetry breaking arises from a vector field acquiring a nonvanishing vacuum expectation value. Working in the geometric-optics limit of the linearized theory, we derive the modified dispersion relation governing the graviton modes and show that it depends on the orientation of the wave vector relative to the background vector. The polarization sector is examined for timelike and spacelike configurations of the Lorentz-violating vacuum. In both cases only two independent tensor modes propagate, although their propagation properties and tensor structure depend on the orientation of the background field. We then construct the retarded Green function associated with the modified wave operator and determine the radiation-zone produced by localized sources. In the timelike configuration the Lorentz-violating effects appear through a modified propagation speed and an overall amplitude renormalization, leading to a shifted retarded time while preserving the quadrupole structure of the waveform. In contrast, the spacelike sector introduces anisotropic corrections to the quadrupole amplitude together with an additional contribution proportional to the third time derivative of the quadrupole moment. As an astrophysical application, the gravitational radiation emitted by a circular binary black hole system is evaluated, allowing observational constraints on the Lorentz-violating combination $ξb^{2}$ to be estimated using multimessenger bounds from GW170817/GRB~170817A and waveform consistency requirements from gravitational wave observations.

Gravitational waves in metric-affine bumblebee gravity

Abstract

We study the propagation and emission of gravitational waves in the metric-affine formulation of the bumblebee model, where spontaneous Lorentz symmetry breaking arises from a vector field acquiring a nonvanishing vacuum expectation value. Working in the geometric-optics limit of the linearized theory, we derive the modified dispersion relation governing the graviton modes and show that it depends on the orientation of the wave vector relative to the background vector. The polarization sector is examined for timelike and spacelike configurations of the Lorentz-violating vacuum. In both cases only two independent tensor modes propagate, although their propagation properties and tensor structure depend on the orientation of the background field. We then construct the retarded Green function associated with the modified wave operator and determine the radiation-zone produced by localized sources. In the timelike configuration the Lorentz-violating effects appear through a modified propagation speed and an overall amplitude renormalization, leading to a shifted retarded time while preserving the quadrupole structure of the waveform. In contrast, the spacelike sector introduces anisotropic corrections to the quadrupole amplitude together with an additional contribution proportional to the third time derivative of the quadrupole moment. As an astrophysical application, the gravitational radiation emitted by a circular binary black hole system is evaluated, allowing observational constraints on the Lorentz-violating combination to be estimated using multimessenger bounds from GW170817/GRB~170817A and waveform consistency requirements from gravitational wave observations.
Paper Structure (27 sections, 73 equations, 5 figures, 1 table)

This paper contains 27 sections, 73 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. The metric--affine bumblebee model
  3. Linearized analysis and graviton dispersion relation
  4. The linearized equations
  5. The modified dispersion relation
  6. The polarization states
  7. Timelike background: $b^\mu=(b_0,0,0,0)$
  8. Spacelike background parallel to propagation: $b^\mu=(0,0,0,b_3)$
  9. Spacelike background orthogonal to propagation: $b^\mu=(0,0,b_2,0)$
  10. Geometric and tidal interpretation for $b^\mu\perp k^\mu$
  11. Emission of gravitational waves
  12. The timelike case
  13. The spacelike case
  14. Gravitational radiation from a binary black hole
  15. Timelike configuration
  16. ...and 12 more sections

Figures (5)

  • Figure 1: Binary system viewed in the center of mass frame, where two compact objects with masses $m_1$ and $m_2$ move in circular orbits on the $xy$ plane with radii $r_1$ and $r_2$.
  • Figure 2: Time evolution of the gravitational wave component $h_{xx}^{(t)}(t,r)$ for the timelike configuration, plotted for $\xi b_0^2={0.0,0.1,0.2,0.3}$ with $\omega=0.5$, $r=20$, $\mu=1$, $l_0=1$, and $G=1$.
  • Figure 3: Time evolution of the gravitational wave component $h_{xx}^{(s)}(t,r)$ for the spacelike configuration of the Lorentz--violating background. The curves correspond to $\xi|\mathbf b|^2={0.0,0.1,0.2,0.3}$, with fixed parameters $\omega=0.5$, $r=20$, $\mu=1$, $l_0=1$, $G=1$, and $\theta_b=\pi/4$.
  • Figure 4: Parametric trajectories of the gravitational wave strains in the $(h_{xx},h_{yx})$ plane for a circular binary system, shown for timelike (on the left panel) and spacelike (on the right panel) configurations of the background vector $b^{\mu}$ and different values of $\xi b^{2}$.
  • Figure 5: Ring deformation induced by a spacelike metric--affine bumblebee gravitational wave. Six snapshots of the initially circular ring are shown at phases $\phi={0,\pi/2,2\pi/3,\pi,3\pi/2,2\pi}$, comparing the Lorentz--violating case ($\xi|\mathbf b|^2=0.2$, $\theta_b=\pi/4$) with the GR baseline ($\xi|\mathbf b|^2=0$); the dashed circle denotes the undeformed ring.