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Register a new userCurved spacetimes with local $kappa$-Poincare dispersion relation
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We use our previously developed identification of dispersion relations with Hamilton functions on phase space to locally implement the $kappa$-Poincare dispersion relation in the momentum spaces at each point of a generic curved spacetime. We use this general construction to build the most general Hamiltonian compatible with spherical symmetry and the Plank-scale-deformed one such that in the local frame it reproduces the $kappa$-Poincare dispersion relation. Specializing to Planck-scale-deformed Schwarzschild geometry, we find that the photon sphere around a black hole becomes a thick shell since photons of different energy will orbit the black hole on circular orbits at different altitudes. We also compute the redshift of a photon between different observers at rest, finding that there is a Planck-scale correction to the usual redshift only if the observers detecting the photon have different masses.
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Following the methods developed by Corley and Jacobson, we consider qualitatively the issue of Hawking radiation in the case when the dispersion relation is dictated by quantum kappa-Poincare algebra. This relation corresponds to field equations that are non-local in time, and, depending on the sign of the parameter kappa, to sub- or superluminal signal propagation. We also derive the conserved inner product, that can be used to count modes, and therefore to obtain the spectrum of black hole radiation in this case.
The Riemann Hypothesis states that the Riemann zeta function $zeta(z)$ admits a set of non-trivial zeros that are complex numbers supposed to have real part $1/2$. Their distribution on the complex plane is thought to be the key to determine the number of prime numbers before a given number. We analyze two approaches. In the first approach, suggested by Hilbert and Polya, one has to find a suitable Hermitian or unitary operator whose eigenvalues distribute like the zeros of $zeta(z)$. In the other approach one instead compares the distribution of the zeta zeros and the poles of the scattering matrix $S$ of a system. We apply the infinite-components Majorana equation in a Rindler spacetime to both methods and then focus on the $S$-matrix approach describing the bosonic open string for tachyonic states. In this way we can explain the still unclear point for which the poles and zeros of the $S$-matrix overlaps the zeros of $zeta(z)$ and exist always in pairs and related via complex conjugation. This occurs because of the relationship between the angular momentum and energy/mass eigenvalues of Majorana states and from the analysis of the dynamics of the poles of $S$. As shown in the literature, if this occurs, then the Riemann Hypothesis can in principle be satisfied.
The covariant understanding of dispersion relations as level sets of Hamilton functions on phase space enables us to derive the most general dispersion relation compatible with homogeneous and isotropic spacetimes. We use this concept to present a Planck-scale deformation of the Hamiltonian of a particle in Friedman-Lema^itre-Robertson-Walker (FLRW) geometry that is locally identical to the $kappa$-Poincare dispersion relation, in the same way as the dispersion relation of point particles in general relativity is locally identical to the one valid in special relativity. Studying the motion of particles subject to such Hamiltonian we derive the redshift and lateshift as observable consequences of the Planck-scale deformed FLRW universe.
We study noncommutative deformations of the wave equation in curved backgrounds and discuss the modification of the dispersion relations due to noncommutativity combined with curvature of spacetime. Our noncommutative differential geometry approach is based on Drinfeld twist deformation, and can be implemented for any twist and any curved background. We discuss in detail the Jordanian twist $-$giving $kappa$-Minkowski spacetime in flat space$-$ in the presence of a Friedman-Lema^{i}tre-Robertson-Walker (FLRW) cosmological background. We obtain a new expression for the variation of the speed of light, depending linearly on the ratio $E_{ph}/E_{LV}$ (photon energy / Lorentz violation scale), but also linearly on the cosmological time, the Hubble parameter and inversely proportional to the scale factor.
In this article, we summarize two agnostic approaches in the framework of spatially curved Friedmann-Robertson-Walker (FRW) cosmologies discussed in detail in (Kerachian et al., 2020, 2019). The first case concerns the dynamics of a fluid with an unspecified barotropic equation of state (EoS), for which the only assumption made is the non-negativity of the fluids energy density. The second case concerns the dynamics of a non-minimally coupled real scalar field with unspecified positive potential. For each of these models, we define a new set of dimensionless variables and a new evolution parameter. In the framework of these agnostic setups, we are able to identify several general features, like symmetries, invariant subsets and critical points, and provide their cosmological interpretation.
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