Andrey A. Shoom

The Schläfli function $f_{n}(x)$ allows to compute volume of a regular $(n-1)$ -dimensional spherical simplex of the dihedral angle $2\alpha =$ arcsec(x) and it has many applications. For example, it defines conjectured upper bounds on the sphere packing problem and the kissing number problem, and a lower bound on the mean-squared error in the quantizing problem. The function is defined recursively via a first-order non-linear differential relation, that makes it difficult to compute, especially for large values of n. Here we present a method for an accurate numerical computation of the Schläfli function $f_{n}(x)$ for $n\geq 4$ in the frequently used in applications interval $x\in [n-1,n+1]$ . The computation is based on the Chebyshev approximation of the function $q_{n}(x)$ , which is related to the Schläfli function via a simple factor of an algebraic expression and regular in the interval. We also present the computation algorithm based on the method.

In this paper we discuss propagation of the weak high-frequency gravitational waves in a curved spacetime background. We develop a so-called spinoptics approximation which takes into account interaction of the spin of the field with the curvature of the background metric. This is achieved by modifying the standard geometric optics approximation by including the helicity sensitive terms of the order 1/ω in the eikonal equation. The novelty of the approach developed in this paper is that instead of study of the high-frequency expansion of the equations for the gravitational field perturbations we construct the effective action for the gravitational spinoptics. The gravitational spinoptics equations derived by variation of the effective action correctly reproduce the earlier obtained results. However, the proposed effective action approach is technically more simple and transparent. It allows one to reduce the study of the high-frequency gravitational waves to study classical dynamics of massless particles with internal discrete degree of freedom (helicity). The formalism is covariant and it can be applied for arbitrary vacuum space-time background.

A gravitational field can cause a rotation of the polarization plane of light. This phenomenon is known as the gravitational Faraday effect. It arises due to different spin-orbit interactions of left- and right-handed circularly polarized components of light. Such an interaction also causes transverse displacement in the light trajectory, in opposite directions for each component. This phenomenon is known as the gravitational spin-Hall effect of light. We study these effects in a local inertial frame in arbitrary vacuum spacetime and show that they are observer dependent and arise due to interaction of light polarization with a local gravitomagnetic field measured by the observer. Thus, to address the effects to a gravitational field alone, one has to consider zero angular momentum observers. Published by the American Physical Society 2024

A gravitational field can cause a rotation of the polarisation plane of light. This phenomenon is known as the gravitational Faraday effect. It arises due to different spin-orbit interaction of left- and right-handed circularly polarised components of light. Such an interaction also causes transverse displacement in the light trajectory, in opposite directions for each component. This phenomenon is known as the gravitational spin-Hall effect of light. We study these effects in a local inertial frame in arbitrary vacuum space-time and show that they are observer dependent and arise due to interaction of light polarisation with a local gravitomagnetic field measured by observer. Thus, to address the effects to a gravitational field alone, one has to consider zero angular momentum observers.

The gravitational Faraday and its dual spin-Hall effects of light arise in space-times of nonzero angular momentum. These effects were studied in stationary, asymptotically flat space-times. Here we study these effects in arbitrary, nonstationary, asymptotically flat space-times. These effects arise from the interaction between light polarization and space-time angular momentum. As a result of such interaction, the phase velocity of left- and right-handed circularly polarized light becomes different, that results in the gravitational Faraday effect. This difference implies different dynamics of these components, that begin to propagate along different paths—the gravitational spin-Hall effect of light. Due to this effect, the gravitational field splits a multicomponent beam of unpolarized light and produces polarized gravitational rainbow. The component separation is an accumulative effect observed in long range asymptotics. To study this effect, we construct uniform eikonal expansion and derive dynamical equation describing this effect. To analyze the dynamical equation, we present it in the local space and time decomposition form. The spatial part of the equation presented in the related optical metric is analogous to the dynamical equation of a charged particle moving in magnetic field under the influence of the Coriolis force.

We study the Gregory-Laflamme instability of 5D compactified, electrically charged black strings. We consider static, linear perturbations of these solutions and derive master equation for these perturbations. Using numerical analysis we search for the threshold unstable modes and derive the critical wave-length. The results are illustrated in the black string mass v.s. electric charge phase diagram. Similar diagram is constructed using global thermodynamic equilibrium argument applied to the charged string and 5D electrically charged black hole. The results derived illustrate that electric charge makes the black string less stable.Comment: 8 pages, 2 figures; Some minor changes in the text. Added reference