arXiv subjects

Anna Valeria Germinario

Publications and source records attributed to Anna Valeria Germinario.

5 recordsLinked to original sources

arXiv2026

A unified framework for photon and massive particle hypersurfaces in stationary spacetimes

Massive particle surfaces (MPS), introduced by Kobialko, Bogush and Gal'tsov, are timelike hypersurfaces confining the trajectories of charged massive particles with fixed Killing energy and charge-to-mass ratio, in analogy with the photon surfaces of Claudel, Virbhadra and Ellis. We study both classes in standard stationary spacetimes within a unified Finslerian framework. For Killing-invariant hypersurfaces $\mathbb{R}\times S_0$, with $S_0$ a hypersurface of a spacelike slice $S$, we prove that photon hypersurfaces and MPS are exactly those for which $S_0$ is totally geodesic with respect to the Fermat metric and to a Jacobi--Randers metric on $S$, respectively. For MPS we also give a gauge-invariant version, in terms of a Riemannian Jacobi metric and a tangency condition on the reduced magnetic field. For arbitrary timelike hypersurfaces, we give a coordinate-free formulation of the partial umbilicity characterization of MPS and, in dimension at least four and under a natural energy condition, we characterize when an MPS is quasi-umbilical in the sense of B.-Y. Chen. In the Som--Raychaudhuri spacetime of a rigidly rotating charged dust we exhibit MPS which are neither photon hypersurfaces nor quasi-umbilical. Finally, under completeness assumptions on the Jacobi--Randers metric, we prove existence and multiplicity results for proper-time parametrized solutions of the Lorentz force equation connecting a point to a flow line of the Killing field and, under compactness assumptions, existence of solutions with periodic spatial projection, possibly confined to an MPS.

gr-qc↗

arXiv2026

Multiplicity of Finsler geodesics and electromagnetic trajectories via localized deformations

Given a Finsler manifold $(M,F)$ and suitable nested regions $B_1\subset B_2\subset B_3$, with $M\setminus B_1$ connected and non-contractible, as in the model case of concentric balls in $\mathbb R^n$, $n\geq2$, we consider conformal deformations $φ_λF$ which can be chosen so as to leave $F$ unchanged outside $B_3$ and grow on $B_2\setminus B_1$. This growth creates an energy barrier, diverging with $λ$, that low-energy curves cannot cross. The minimax families can therefore be chosen in the path space of $M\setminus B_1$, which contains compact subsets of arbitrarily large category. Consequently, for every $m\in\mathbb N$ and all sufficiently large $λ$, there exist at least $m$ geodesics joining two prescribed points, all avoiding $B_1$. This extends to the Finsler setting an idea of Capozzi, Fortunato and Greco. The same scheme applies to Randers metrics $α+β$, with $α$ deformed and $β$ unchanged, as well as to fixed-energy trajectories of non-relativistic electromagnetic systems and relativistic Lorentz-force trajectories, and light rays in spacetimes endowed with a causal Killing field. In the non-relativistic case only the scalar potential is perturbed, while the magnetic field is preserved. When the Killing field is not everywhere timelike, the Fermat principle yields a possibly singular Randers--Kropina metric on a spacelike hypersurface. Under a natural admissible category condition, combining localized barriers with Randers approximation gives $m$ Randers--Kropina geodesics with pairwise distinct lengths, equivalently $m$ future-pointing light rays with pairwise distinct arrival times.

math.DG↗

arXiv2025

Lorentzian-Euclidean black holes and Lorentzian to Riemannian metric transitions

In recent papers on spacetimes with a signature-changing metric, the concept of a Lorentzian-Euclidean black hole and new elements for Lorentzian-Riemannian signature change have been introduced. A Lorentzian-Euclidean black hole is a signature-changing modification of the Schwarzschild spacetime satisfying the vacuum Einstein equations in a weak sense. Here the event horizon serves as a boundary beyond which time becomes imaginary. We demonstrate that the proper time needed to reach the horizon remains finite, consistently with the classical Schwarzschild solution. About Lorentzian to Riemannian metric transitions, we stress that the hypersurface where the metric signature changes is naturally a spacelike hypersurface which can be identified with the future or past causal boundary of the Lorentzian sector. Moreover, a number of geometric interpretations appear, as the degeneracy of the metric corresponds to the collapse of the causal cones into a line, the degeneracy of the dual metric corresponds to collapsing into a hyperplane, and additional geometric structures on the transition hypersurface (Galilean and dual Galilean) might be explored.

gr-qc↗

arXiv2012

Convex regions of stationary spacetimes and Randers spaces. Applications to lensing and asymptotic flatness

By using Stationary-to-Randers correspondence (SRC), a characterization of light and time-convexity of the boundary of a region of a standard stationary (n+1)-spacetime is obtained, in terms of the convexity of the boundary of a domain in a Finsler n or (n+1)-space of Randers type. The latter convexity is analyzed in depth and, as a consequence, the causal simplicity and the existence of causal geodesics confined in the region and connecting a point to a stationary line are characterized. Applications to asymptotically flat spacetimes include the light-convexity of stationary hypersurfaces which project in a spacelike section of an end onto a sphere of large radius, as well as the characterization of their time-convexity with natural physical interpretations. The lens effect of both light rays and freely falling massive particles with a finite lifetime, (i.e. the multiplicity of such connecting curves) is characterized in terms of the focalization of the geodesics in the underlying Randers manifolds.

math.DG↗

arXiv2010

Convex domains of Finsler and Riemannian manifolds

A detailed study of the notions of convexity for a hypersurface in a Finsler manifold is carried out. In particular, the infinitesimal and local notions of convexity are shown to be equivalent. Our approach differs from Bishop's one in his classical result (Bishop, Indiana Univ Math J 24:169-172, 1974) for the Riemannian case. Ours not only can be extended to the Finsler setting but it also reduces the typical requirements of differentiability for the metric and it yields consequences on the multiplicity of connecting geodesics in the convex domain defined by the hypersurface.

math.DG↗