We study the gravitation effects on a static and spherically symmetric spacetime due to the vacuum expectation value (VEV) of Kalb-Ramond field. The Kalb-Ramond VEV is a background tensor field which produces a local Lorentz symmetry breaking (LSB) of spacetime. Considering a non-minimal coupling between the Kalb-Ramond (vev) and the Ricci tensor we obtain an exact parameter-dependent power-law hairy black hole. For s = 1, the Lorentz violation produces a solution similar to the Reissner-Nordstrom, despite the absence of charge. The near horizon geometry is modified by including a new inner horizon and shifting the Schwarzchild horizon. Asymptotically, the usual Minkowski spacetime with a background tensor field is recover. By means of the mercury peherilion test, an upper bound to the local Lorentz violation (LV) is obtained and its corresponding effects on the black hole temperature is investigated.
We investigate the effects of the spacetime curvature and extra dimensions on the excitations of the Lorentz violating bumblebee field $B_\mu$. %The self-interacting quadratic potential leads to a spontaneous breaking of the Lorentz symmetry. By assuming the bumblebee field living in a \revision{five dimensional} $AdS_{5}$ bulk, we found an exponential suppression of the bumblebee self-interacting constant $\lambda$ and of the bumblebee vacuum expectation value (VEV) \revision{$b_{M}$} along the \revision{fifth} extra dimension. The fluctuations of the bumblebee \revision{field} upon the VEV can be decomposed into transverse and longitudinal modes with respect to $b_{M}$. For a spacelike $b_{M}$ along the extra dimension and assuming a \revision{thin} FRW 3-brane embedded in the $AdS_{5}$, the transverse mode is localized on the brane. The bulk geometry leads to a propagating and unbound \revision{longitudinal mode along the extra dimension}. On the brane, the cosmological expansion leads to the decay of the longitudinal mode in time, as $\Delta t \approx H^{-1}$.
We investigated the effects of the spacetime curvature and extra dimensions on the excitations of a self-interacting vector field known as the bumblebee field. The self-interacting quadratic potential breaks the gauge invariance and the vacuum expectation value (VEV) of the bumblebee field b M violates the local particle Lorentz symmetry. By assuming the bumblebee field living in a AdS 5 bulk, we found an exponential suppression of the self-interacting constant λ and the bumblebee VEV along the extra dimension. The fluctuations of the bumblebee upon the VEV can be decomposed into transverse and longitudinal modes with respect to b M . We employed the eikonal approximation to study the propagation of both modes. Despite the curvature, the transverse mode is still a massless Nambu-Goldstone (NG) mode and the longitudinal mode keeps its Lorentz violating mass λb 2 . For a spacelike b M along the extra dimension and assuming a FRW 3-brane embedded in the AdS 5 yields to an additional dissipative term to the longitudinal mode. The cosmological expansion leads to decay of the longitudinal mode in a time ∆t ≈ H −1 , where H = ȧ/a is the Hubble parameter and a(t) is the scale factor. For a timelike b M , the longitudinal mode does not propagate and its amplitude decays in time with a −3 .
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