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This work presents an experimental test of Lorentz invariance violation in the infrared (IR) regime by means of an invariant minimum speed in spacetime and its effects on the time when an atomic clock given by a certain radioactive single-atom (e.g. isotope Na${}^{25}$) is a thermometer for an ultracold gas like the dipolar gas Na${}^{23}$K${}^{40}$. So, according to a Deformed Special Relativity (DSR) so-called Symmetrical Special Relativity (SSR), where there emerges an invariant minimum speed V in the subatomic world, one expects that the proper time of such a clock moving close to V in thermal equilibrium with the ultracold gas is dilated with respect to the improper time given in lab, i.e. the proper time at ultracold systems elapses faster than the improper one for an observer in the lab, thus leading to the so-called proper time dilation so that the atomic decay rate of an ultracold radioactive sample (e.g. Na${}^{25}$) becomes larger than the decay rate of the same sample at room temperature. This means a suppression of the half-life time of a radioactive sample thermalized with an ultracold cloud of dipolar gas to be investigated by NASA in the Cold Atom Lab (CAL).

This research aims to provide the geometrical foundation of the uncertainty principle within a new causal structure of spacetime so-called Symmetrical Special Relativity (SSR), where there emerges a Lorentz violation due to the presence of an invariant minimum speed $V$ related to the vacuum energy. SSR predicts that a dS scenario occurs only for a certain regime of speeds $v$, where $v<v_0=\sqrt{cV}$, which represents the negative gravitational potentials ($\mathrm{\Phi }<0$) connected to the cosmological parameter $\mathrm{\Lambda }>0$. For $v=v_0$, Minkowski (pseudo-Euclidean) space is recovered for representing the flat space ($\mathrm{\Lambda }=0$), and for $v>v_0$ ($\mathrm{\Phi }>0$), Anti-de Sitter (AdS) scenario prevails ($\mathrm{\Lambda }<0$). The fact that the current universe is flat as its average density of matter distribution (${\rho }_m$ given for a slightly negative curvature $R$) coincides with its vacuum energy density (${\rho }_{\mathrm{\Lambda }}$ given for a slightly positive curvature $\mathrm{\Lambda }$), i.e. the cosmic coincidence problem, is now addressed by SSR. SSR provides its energy–momentum tensor of perfect fluid, leading to the EOS of vacuum ($p=-{\rho }_{\mathrm{\Lambda }}$). Einstein equation for vacuum given by such SSR approach allows us to obtain ${\rho }_{\mathrm{\Lambda }}$ associated with a scalar curvature $\mathrm{\Lambda }$, whereas the solution of Einstein equation only in the presence of a homogeneous distribution of matter ${\rho }_m$ for the whole universe presents a scalar curvature $R$, in such a way that the presence of the background field $\mathrm{\Lambda }$ opposes the Riemannian curvature $R$, thus leading to a current effective curvature $R_{\mathrm{\text{eff}}}=R+\mathrm{\Lambda }\approx 0$ according to observations. This corrects the notion of gravity as being only of Riemannian origin as the flat space has connection with a background gravity. In view of the current dS scenario with a quasi-zero $\mathrm{\Lambda }$ slightly larger than $\left|R\right|$, we will just obtain a Generalized Uncertainty Principle (GUP) given in the cases of weak gravity and anti-gravity.

We aim to investigate the theory of Lorentz violation with an invariant minimum speed called Symmetrical Special Relativity (SSR) from the viewpoint of its metric. Thus, we should explore the nature of SSR-metric in order to understand the origin of the conformal factor that appears in the metric by deforming Minkowski metric by means of an invariant minimum speed that breaks down Lorentz symmetry. So, we are able to realize that there is a similarity between SSR and a new space with variable negative curvature ($-\infty <{ℛ}<0$) connected to a set of infinite cosmological constants ($0<\mathrm{\Lambda }<\infty$), working like an extended de Sitter (dS) relativity, so that such extended dS-relativity has curvature and cosmological “constant” varying in time. We obtain a scenario that is more similar to dS-relativity given in the approximation of a slightly negative curvature for representing the current universe having a tiny cosmological constant. Finally, we show that the invariant minimum speed provides the foundation for understanding the kinematics origin of the extra dimension considered in dS-relativity in order to represent the dS-length.

In the standard model of neutrino oscillations, the neutrino flavor states are mixtures of mass-eigenstates, and the phenomena are well described by the neutrino mixing matrix, i.e., the PMNS matrix. I review the recent progress on parametrization of the neutrino mixing matrix. Besides that I also discuss on the possibility to describe the neutrino oscillations by a non-standard model in which the neutrino mixing is caused by the Lorentz violation (LV) contribution in the effective field theory for LV. We assume that neutrinos are massless and that neutrino flavor states are mixing states of energy eigenstates. In our calculation the neutrino mixing parts depend on LV parameters and neutrino energy. The oscillation amplitude varies with the neutrino energy, thus neutrino experiments with energy dependence may test and constrain the Lorentz violation scenario for neutrino oscillation.

I present a brief review on the motivation for the study on Lorentz violation and on some of our studies with phenomenological analysis of Lorentz violation effects. I also discuss three effective field theory frameworks for Lorentz violation: the Coleman-Glashow model, the standard model extension (SME), and the standard model supplement (SMS). The situation of the OPERA "anomaly" is also briefly reviewed, together with some discussion on the superluminality of neutrinos within the effective field theory frameworks.

The Vector-Tensor (VT) theories of gravity are a class of alternative theories to General Relativity (GR) that are characterized by the presence of a dynamical vector field besides the metric. They are studied in attempts to understand spontaneous Lorentz violation, to generate massive gravitons, and as models of dark matter and dark energy. In this article, I outline how the nature of singularities and horizons in VT theories differ greatly from GR even under the same ordinary conditions. This is illustrated with Einstein-aether theory where vacuum black hole solutions have naked singularities and vacuum cosmological solutions have new singularities that are otherwise absent in GR. It would be interesting to explore these deviations using gravitational waves.