GEMS Embeddings of Schwarzschild and RN Black Holes in Painlevé-Gullstrand Spacetimes

Making use of the higher dimensional global embedding Minkowski spacetime (GEMS), we embed (3 + 1)-dimensional Schwarzschild and Reissner-Nordström (RN) black holes written by the Painlevé-Gullstrand (PG) spacetimes, which have off-diagonal components in metrics, into (5 + 1)- and (5 + 2)-dimensional flat ones, respectively. As a result, we have shown the equivalence of the GEMS embeddings of the spacetimes with the diagonal and off-diagonal terms in metrics. Moreover, with the aid of their geodesic equations satisfying various boundary conditions in the flat embedded spacetimes, we directly obtain freely falling temperatures. We also show that freely falling temperatures in the PG spacetimes are well-defined beyond the event horizons, while they are equivalent to the Hawking temperatures, which are obtained in the original curved ones in the ranges between the horizon and the infinity. These will be helpful to study GEMS embeddings of more realistic Kerr, or rotating BTZ black holes.

Effective Potentials and Orbits in Weyl Conformastatic Slender Disk

By using geodesic equations to obtain a gravitational potential generated from a point-like source, we end up in the concept of a nearly Newtonian gravity to analyse effective potentials of quasi-circular orbits. By means of an approximate solution from an axially static and symmetric Weyl metric, we study an effective gravitational potential to obtain its related rotation curves, orbital planes and orbits. Moreover, using as initial condition a Plummer sphere, some prospects on star cluster disruption are also discussed in this framework.

Conformal geodesics on gravitational instantons

We study the integrability of the conformal geodesic flow (also known as the conformal circle flow) on the SO(3)–invariant gravitational instantons. On a hyper–Kähler four–manifold the conformal geodesic equations reduce to geodesic equations of a charged particle moving in a constant self–dual magnetic field. In the case of the anti–self–dual Taub NUT instanton we integrate these equations completely by separating the Hamilton–Jacobi equations, and finding a commuting set of first integrals. This gives the first example of an integrable conformal geodesic flow on a four–manifold which is not a symmetric space. In the case of the Eguchi–Hanson we find all conformal geodesics which lie on the three–dimensional orbits of the isometry group. In the non–hyper–Kähler case of the Fubini–Study metric on $\mathbb{CP}^2$ we use the first integrals arising from the conformal Killing–Yano tensors to recover the known complete integrability of conformal geodesics.

Projective Collineations of Decomposable Spacetimes Generated by the Lie Point Symmetries of Geodesic Equations

We investigate the relation of the Lie point symmetries for the geodesic equations with the collineations of decomposable spacetimes. We review previous results in the literature on the Lie point symmetries of the geodesic equations and we follow a previous proposed geometric construction approach for the symmetries of differential equations. In this study, we prove that the projective collineations of a n+1-dimensional decomposable Riemannian space are the Lie point symmetries for geodesic equations of the n-dimensional subspace. We demonstrate the application of our results with the presentation of applications.

Existence of conserved quantities and their algebra in curved spacetime

In general relativity, finding out the geodesics of a given spacetime manifold is an important task because it determines which classical processes are dynamically forbidden. Conserved quantities play an important role in solving the geodesic equations of a general spacetime manifold. Furthermore, knowing all possible conserved quantities of a system gives information about the hidden symmetries of that system since conserved quantities are deeply connected with the symmetries of the system. These are very important in their own right. Conserved quantities are also useful to capture certain features of spacetime manifold for an asymptotic observer. In this article, we show the existence of these conserved charges and their algebra in a generic curved spacetime for a class of dynamical systems with the Hamiltonians quadratic and linear in momentum and spin.

Simplification of a System of Geodesic Equations by Reference to Conservation Laws

This paper is purposed to exploit prevalent premises for determining analytical solutions to di erential equations formulated from the calculus of variations. We realize this premises from the statement of Emmy Noether’s theorem; that every system in which a conservation law is observed also admits a symmetry of invariance (Olver, 1993, pp.242; Dresner, 1999, pp.60-62 ). As an illustration, the infinitesimal symmetries for Ordinary Di erential Equations (O.D.E’s) of geodesics of the glome are explicitly computed and engaged following identification of a relevant conservation law in action. Further prospects for analysis of this concept over the same manifold are then presented summarily in conclusion.

Conserved quantities in the presence of torsion: A generalization of Killing theorem

When spacetime torsion is present, geodesics and autoparallels generically do not coincide. In this work, the well-known method that uses Killing vectors to solve the geodesic equations is generalized for autoparallels. The main definition is that of T-Killing vectors: vector fields such that, when their index is lowered with the metric, have vanishing symmetric derivative when acted upon with a torsionful and metric-compatible derivative. The main property of T-Killing vectors is that their contraction with the autoparallels’ tangents are constant along these curves. As an example, in a static and spherically symmetric situation, the autoparallel equations are reduced to an effective one-dimensional problem. Other interesting properties and extensions of T-Killing vectors are discussed.