Abstract
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 [Formula: see text] related to the vacuum energy. SSR predicts that a dS scenario occurs only for a certain regime of speeds [Formula: see text], where [Formula: see text], which represents the negative gravitational potentials ([Formula: see text]) connected to the cosmological parameter [Formula: see text]. For [Formula: see text], Minkowski (pseudo-Euclidean) space is recovered for representing the flat space ([Formula: see text]), and for [Formula: see text] ([Formula: see text]), Anti-de Sitter (AdS) scenario prevails ([Formula: see text]). The fact that the current universe is flat as its average density of matter distribution ([Formula: see text] given for a slightly negative curvature [Formula: see text]) coincides with its vacuum energy density ([Formula: see text] given for a slightly positive curvature [Formula: see text]), 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 ([Formula: see text]). Einstein equation for vacuum given by such SSR approach allows us to obtain [Formula: see text] associated with a scalar curvature [Formula: see text], whereas the solution of Einstein equation only in the presence of a homogeneous distribution of matter [Formula: see text] for the whole universe presents a scalar curvature [Formula: see text], in such a way that the presence of the background field [Formula: see text] opposes the Riemannian curvature [Formula: see text], thus leading to a current effective curvature [Formula: see text] 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 [Formula: see text] slightly larger than [Formula: see text], we will just obtain a Generalized Uncertainty Principle (GUP) given in the cases of weak gravity and anti-gravity.
Highlights
In the last two decades, the physicists have shown a great interest in the theories that contained the breakdown of Lorentz symmetry in many scenarios[3,4,5,6,7,8,9,10,11,12,13] and the so-called Deformed Special Relativities (DSR),[14,15] in spite of the fact that no relevant experimental fact has demonstrated the existence of a Lorentz violation until the present time
There could be the evidence that the Lorentz symmetry breaking may exist in a very low energy regime due to the presence of a vacuum energy density connected to the well-known cosmological constant Λ related to a universal background field associated to an invariant minimum speed
We have explored the nature of the Symmetrical Special Relativity (SSR)-metric[16] in order to understand the origin of the conformal factor that appears in the metric by deforming Minkowski metric by means of a scale factor Θ(v)[19] depending on the minimum speed that breaks down the Lorentz symmetry, leading to a positive cosmological constant
Summary
Lorentz violations can be observable in nature.[1,2] In the last two decades, the physicists have shown a great interest in the theories that contained the breakdown of Lorentz symmetry in many scenarios[3,4,5,6,7,8,9,10,11,12,13] and the so-called Deformed Special Relativities (DSR),[14,15] in spite of the fact that no relevant experimental fact has demonstrated the existence of a Lorentz violation until the present time. This is exactly equivalent of making p = −ρ, i.e. the equation of state (EOS) of vacuum (the cosmological constant) By comparing such term of vacuum of SSR with the well-known ad hoc term of cosmological constant in dS scenario, we will find ρΛ, which is consistent with the result of the modern cosmology of an accelerated universe, where the cosmic coincidence ρΛ ≈ ρmatter leads to a flat universe as it will be shown in Sec. 4, contrary to a Riemannian (curved) universe predicted by the standard cosmology.
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