Metric Relativity and the Dynamical Bridge: highlights of Riemannian geometry in physics

We present an overview of recent developments concerning modifications of the geometry of space-time to describe various physical processes of interactions among classical and quantum configurations. We concentrate in two main lines of research: the Metric Relativity and the Dynamical Bridge.Comment: 121 page

Chiral symmetry breaking as a geometrical process

This article expands for spinor fields the recently developed Dynamical Bridge formalism which relates a linear dynamics in a curved space to a nonlinear dynamics in Minkowski space. Astonishingly, this leads to a new geometrical mechanism to generate a chiral symmetry breaking without mass, providing an alternative explanation for the undetected right-handed neutrinos. We consider a spinor field obeying the Dirac equation in an effective curved space constructed by its own currents. This way, both chiralities of the spinor field satisfy the same dynamics in the curved space. Subsequently, the dynamical equation is re-expressed in terms of the flat Minkowski space and then each chiral component behaves differently. The left-handed part of the spinor field satisfies the Dirac equation while the right-handed part is trapped by a Nambu-Jona-Lasinio (NJL) type potential.Comment: 6 page

Massless Dirac particles in the vacuum C-metric

We study the behavior of massless Dirac particles in the vacuum C-metric spacetime, representing the nonlinear superposition of the Schwarzschild black hole solution and the Rindler flat spacetime associated with uniformly accelerated observers. Under certain conditions, the C-metric can be considered as a unique laboratory to test the coupling between intrinsic properties of particles and fields with the background acceleration in the full (exact) strong-field regime. The Dirac equation is separable by using, e.g., a spherical-like coordinate system, reducing the problem to one-dimensional radial and angular parts. Both radial and angular equations can be solved exactly in terms of general Heun functions. We also provide perturbative solutions to first-order in a suitably defined acceleration parameter, and compute the acceleration-induced corrections to the particle absorption rate as well as to the angle-averaged cross section of the associated scattering problem in the low-frequency limit. Furthermore, we show that the angular eigenvalue problem can be put in one-to-one correspondence with the analogous problem for a Kerr spacetime, by identifying a map between these "acceleration" harmonics and Kerr spheroidal harmonics. Finally, in this respect we discuss the nature of the coupling between intrinsic spin and spacetime acceleration in comparison with the well known Kerr spin-rotation coupling.Comment: 18 pages, 2 figures; accepted for publication in Classical and Quantum Gravit

Gauss map and the topology of constant mean curvature hypersurfaces of S7\mathbb{S}^{7} and CP3\mathbb{CP}^{3}

We define a Gauss map $\gamma:M\rightarrow\mathbb{S}^{6}$ of an oriented hypersurface $M$ of the unit sphere $\mathbb{S}^{7}$ and prove that $\gamma$ is harmonic if and only if $M$ has CMC. Results on the geometry and topology of CMC hypersurfaces of $\mathbb{S}^{7}$, under hypothesis on the image of $\gamma$, are then obtained. By a Hopf symmetrization process we define a Gauss map for hypersurfaces of $\mathbb{CP}^{3}$ and obtain similar results for CMC hypersurfaces of this space.Comment: 20 page

On the disformal invariance of the Dirac equation

In this paper we analyze the invariance of the Dirac equation under disformal transformations depending on the propagating spinor field. Using the Weyl-Cartan formalism, we construct a large class of disformal maps between different metric tensors, respecting the order of differentiability of the Dirac operator and satisfying the Clifford algebra in both metrics. Then, we have shown that there is a subclass of solutions of the Dirac equation, provided by Inomata's condition, which keeps the Dirac operator invariant under the action of the disformal group.Comment: 12 pages; This matches the version to be published in CQ

Extended disformal approach in the scenario of Rainbow Gravity

We investigate all feasible mathematical representations of disformal transformations on a space-time metric according to the action of a linear operator upon the manifold's tangent and cotangent bundles. The geometric, algebraic and group structures of this operator and their interfaces are analyzed in detail. Then, we scrutinize a possible physical application, providing a new covariant formalism for a phenomenological approach to quantum gravity known as Rainbow Gravity.Comment: 8 pages, 2 figure

Magnetic fields and the Weyl tensor in the early universe

We have solved the Einstein-Maxwell equations for a class of isotropic metrics with constant spatial curvature in the presence of magnetic fields. We consider a slight modification of the Tolman averaging relations so that the energy-momentum tensor of the electromagnetic field possesses an anisotropic pressure component. This inhomogeneous magnetic universe is isotropic and its time evolution is guided by the usual Friedmann equations. In the case of flat universe, the space-time metric is free of singularities (except the well-known initial singularity at t = 0). It is shown that the anisotropic pressure of our model has a straightforward relation to the Weyl tensor. We also analyze the effect of this new ingredient on the motion of test particles and on the geodesic deviation of the cosmic fluid.Comment: 10 pages, 2 figures, new conceptual treatment of the early phenomenological work arXiv:1301.3079. Matches published versio

Slicing black hole spacetimes

A general framework is developed to investigate the properties of useful choices of stationary spacelike slicings of stationary spacetimes whose congruences of timelike orthogonal trajectories are interpreted as the world lines of an associated family of observers, the kinematical properties of which in turn may be used to geometrically characterize the original slicings. On the other hand properties of the slicings themselves can directly characterize their utility motivated instead by other considerations like the initial value and evolution problems in the 3-plus-1 approach to general relativity. An attempt is made to categorize the various slicing conditions or "time gauges" used in the literature for the most familiar stationary spacetimes: black holes and their flat spacetime limit.Comment: 30 pages, 6 figures; published versio

The flexibility of optical metrics

We firstly revisit the importance, naturalness and limitations of the so-called optical metrics for describing the propagation of light rays in the limit of geometric optics. We then exemplify their flexibility and nontriviality in some nonlinear material media and in the context of nonlinear theories of the electromagnetism, both underlain by curved backgrounds, where optical metrics could be flat and impermeable membranes only to photons could be conceived, respectively. Finally, we underline and discuss the relevance and potential applications of our analyses in a broad sense, ranging from material media to compact astrophysical systems.Comment: 8 pages, some improvements in the physical content. Accepted for publication in Classical and Quantum Gravit

Space and time ambiguities in vacuum electrodynamics

It is shown that every regular electromagnetic field in vacuum identically satisfy Maxwell equations in a new manifold where the roles of space and time have been exchanged. The new metric is Lorentzian, depends on the particular solution and forces the flow of time to tilt somehow in the direction of the field lines. We give a detailed description of the transformation and discuss several of its properties, both in the algebraic and differential settings. Examples are given where the new metrics are explicitly computed and carefully analyzed. We conclude with possible applications of the transformation as well as future perspectives.Comment: 22 page