Covariant hydrodynamics of Hamiltonian systems

The theory of hydrodynamic reduction of non-autonomous Hamiltonian mechanics (V. Kozlov, 1983) is presented in the geometric formalism of bundles over the time axis R. In this formalism, time is one of the coordinates, not a parameter; the connections describe reference frames and velocity fields of mechanical systems. The equations of the theory are presented in a form that is invariant with respect to time-dependent coordinate transformations and the choice of reference frames.

Redefining of “EPSG” and “PROJ” for current State Coordinate Reference System of the Republic of North Macedonia

The state Coordinate Reference System (CRS) of the Republic of North Macedonia (RNM) has been established a century ago, by the Military Geographic Institute of the Yugoslavia Kingdom. It is in official usage entire period up to day. In international public EPSG registry of geodetic datums, spatial reference systems, Earth ellipsoids, coordinate transformations and related units of measurement, CRS for RNM is recognizable within 3 EPSG codes 6204, 6316 and 8679.First code EPSG 6204 represents current state CRS for the entire country area, based on current law, however unfortunately this CRS is official by the law but it is not used for developing the official spatial data published in geoportals of Agency for Real Estate Cadastre (AREC) and NSDI geoportal of RNM. The second code EPSG 6316 is defined to be used for 6 countries of former Yugoslavia that covers area between 19.5°E up to 22.5°E longitude, which does not correspond with the practical and official usage of CRS for working with spatial data in RNM and CRS law definition in RNM. Third code EPSG 8679 has never been used in RNM, which covers eastern part of RNM and Serbia beginning from 22.5°E.Beside of problems with EPSG codes, default transformation parameters of EPSG 6316 have low accuracy and can not be used for data overlapping with open layers. Therefore, redefined new EPSG codes for state CRS of RNM are proposed in this paper.

Kottler spacetime in isotropic static coordinates

The Kottler spacetime in isotropic coordinates is known where the metric is time-dependent. In this paper, the Kottler spacetime is given in isotropic static coordinates (i.e., the metric components are time-independent). The metric is found in terms of the Jacobian elliptic functions through coordinate transformations from the Schwarzschild-(anti-)de Sitter metric. In canonical coordinates, it is known that the unparameterized spatially projected null geodesics of the Kottler and Schwarzschild spacetimes coincide. We show that in isotropic static coordinates, the refractive indices of Kottler and Schwarzschild are not proportional, yielding spatially projected null geodesics that are different.

Manifolds

We now embark on the full theory, beginning with the concept of a manifold in differential geometry. The meaning of coordinates and coordinate transformations is carefully explained. The metric and its transformation between coordinate frames is discussed. Riemann normal coordinates are described. The concepts of a tangent space and local flatness are discussed and derived. It is shown how to use the metric to calculate distances, areas and volumes, and to describe submanifolds.

Electrodynamics in noninertial frames

The electromagnetic theory is considered in the framework of the generally covariant approach, that is applied to the analysis of electromagnetism in noninertial coordinate and frame systems. The special-relativistic formulation of Maxwell’s electrodynamics arises in the flat Minkowski spacetime when the general coordinate transformations are restricted to a class of transformations preserving the Minkowski line element. The particular attention is paid to the analysis of the electromagnetism in the noninertial rotating reference system. For the latter case, the general stationary solution of the Maxwell equations in the absence of the electric current is constructed in terms of the two scalar functions satisfying the Poisson and the biharmonic equations with an arbitrary charge density as a matter source. The classic problem of Schiff is critically revisited.

Non-Rindler horizons and radiating black holes

In this work we construct metrics corresponding to radiating black holes whose near horizon regions cannot be approximated by Rindler space–time. We first construct infinite parameter coordinate transformations from Minkowski coordinates, such that an observer using these coordinates to describe space–time events measures the Minkowski vacuum to be Planckian. Utilizing these results, we construct a family of black holes that radiate at spatial infinity. As an illustration, we study a subset of the black hole solutions and show that they satisfy the null energy condition.

Bending the Bruhat-Tits tree. Part II. The p-adic BTZ black hole and local diffeomorphism on the Bruhat-Tits tree

In this sequel to [1], we take up a second approach in bending the Bruhat-Tits tree. Inspired by the BTZ black hole connection, we demonstrate that one can transplant it to the Bruhat-Tits tree, at the cost of defining a novel “exponential function” on the p-adic numbers that is hinted by the BT tree. We demonstrate that the PGL(2, Qp) Wilson lines [2] evaluated on this analogue BTZ connection is indeed consistent with correlation functions of a CFT at finite temperatures. We demonstrate that these results match up with the tensor network reconstruction of the p-adic AdS/CFT with a different cutoff surface at the asymptotic boundary, and give explicit coordinate transformations that relate the analogue p-adic BTZ background and the “pure” Bruhat-Tits tree background. This is an interesting demonstration that despite the purported lack of descendents in p-adic CFTs, there exists non-trivial local Weyl transformations in the CFT corresponding to diffeomorphism in the Bruhat-Tits tree.

The Hybrid FEM-DBCI for the Solution of Open-Boundary Low-Frequency Problems

This paper describes a particular use of the hybrid FEM-DBCI, for the computation of low-frequency electromagnetic fields in open-boundary domains. Once the unbounded free space enclosing the system has been truncated, the FEM is applied to the bounded domain thus obtained, assuming an unknown Dirichlet condition on the truncation boundary. An integral equation is used to express this boundary condition in which the integration surface is selected in the middle of the most external layer of finite elements, very close to the truncation boundary, so that the integral equation becomes quasi-singular. The method is described for the computation of electrostatic fields in 3D and of eddy currents in 2D, but it is also applicable to the solution of other kinds of electromagnetic problems. Comparisons are made with other methods, concluding that FEM-DBCI is competitive with the well-known FEM-BEM and coordinate transformations for what concerns accuracy and computing time.