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Register a new userAnalytic solutions of the geodesic equation in axially symmetric space-times
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The complete sets of analytic solutions of the geodesic equation in Taub--NUT--(anti-)de Sitter, Kerr--(anti-)de Sitter and also in general Plebanski--Demianski space--times without acceleration are presented. The solutions are given in terms of the Kleinian sigma functions.
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The complete analytical solutions of the geodesic equation of massive test particles in higher dimensional Schwarzschild, Schwarzschild-(anti)de Sitter, Reissner-Nordstroem and Reissner-Nordstroem-(anti)de Sitter space--times are presented. Using the Jacobi inversion problem restricted to the theta divisor the explicit solution is given in terms of Kleinian sigma functions. The derived orbits depend on the structure of the roots of the characteristic polynomials which depend on the particles energy and angular momentum, on the mass and the charge of the gravitational source, and the cosmological constant. We discuss the general structure of the orbits and show that due to the specific dimension-independent form of the angular momentum and the cosmological force a rich variety of orbits can emerge only in four and five dimensions. We present explicit analytical solutions for orbits up to 11 dimensions. A particular feature of Reissner--Nordstroem space-times is that bound and escape orbits traverse through different universes.
The numerical evolution of Einsteins field equations in a generic background has the potential to answer a variety of important questions in physics: from applications to the gauge-gravity duality, to modelling black hole production in TeV gravity scenarios, analysis of the stability of exact solutions and tests of Cosmic Censorship. In order to investigate these questions, we extend numerical relativity to more general space-times than those investigated hitherto, by developing a framework to study the numerical evolution of D dimensional vacuum space-times with an SO(D-2) isometry group for Dge 5, or SO(D-3) for Dge 6. Performing a dimensional reduction on a (D-4)-sphere, the D dimensional vacuum Einstein equations are rewritten as a 3+1 dimensional system with source terms, and presented in the Baumgarte, Shapiro, Shibata and Nakamura (BSSN) formulation. This allows the use of existing 3+1 dimensional numerical codes with small adaptations. Brill-Lindquist initial data are constructed in D dimensions and a procedure to match them to our 3+1 dimensional evolution equations is given. We have implemented our framework by adapting the LEAN code and perform a variety of simulations of non-spinning black hole space-times. Specifically, we present a modified moving puncture gauge which facilitates long term stable simulations in D=5. We further demonstrate the internal consistency of the code by studying convergence and comparing numerical versus analytic results in the case of geodesic slicing for D=5,6.
The Lounesto classification splits spinors in six classes: I, II, III are those for which at least one among scalar and pseudo-scalar bi-linear spinor quantities is non-zero, its spinors are called regular, and among them we find the usual Dirac spinor. IV, V, VI are those for which the scalar and pseudo-scalar bi-linear spinor quantities are identically zero, its spinors are called singular, and they are split in further sub-classes: IV has no further restrictions, its spinors are called flag-dipole; V is the one for which the spin axial-vector vanishes, its spinors are called flagpole, and among them we find the Majorana spinor; VI is the one for which the momentum antisymmetric-tensor vanishes, its spinors are called dipole, and among them we find the Weyl spinor. In the quest for exact solutions of fully-coupled systems of spinor fields in their own gravity, we have already given examples in the case of Dirac fields and Weyl fields but never in the case of Majorana or more generally flagpole spinor fields. Flagpole spinor fields in interaction with their own gravitational field, in the case of axial symmetry, will be considered. Exact solutions of the field equations will be given.
In the present work we analyze all the possible spherically symmetric exterior vacuum solutions allowed by the Einstein-Aether theory with static aether. We show that there are four classes of solutions corresponding to different values of a combination of the free parameters, $c_{14}=c_1+c_4$, which are: $ 0 < c_{14}<2$, $c_{14} < 0$, $c_{14}=2$ and $c_{14}=0$. We present explicit analytical solutions for $c_{14}=3/2, 16/9, 48/25, -16, 2$ and $0$. The first case has some pathological behavior, while the rest have all singularities at $r=0$ and are asymptotically flat spacetimes. For the solutions $c_{14}=16/9, 48/25, mathrm{, and ,}, -16$ we show that there exist no horizons, neither Killing nor universal horizon, thus we have naked singularities. Finally, the solution for $c_{14}=2$ has a metric component as an arbitrary function of radial coordinate, when it is chosen to be the same as in the Schwarzschild case, we have a physical singularity at finite radius, besides the one at $r=0$. This characteristic is completely different from General Relativity.
We revisit a little known theorem due to Beltrami, through which the integration of the geodesic equations of a curved manifold is accomplished by a method which, even if inspired by the Hamilton-Jacobi method, is purely geometric. The application of this theorem to the Schwarzschild and Kerr metrics leads straightforwardly to the general solution of their geodesic equations. This way of dealing with the problem is, in our opinion, very much in keeping with the geometric spirit of general relativity. In fact, thanks to this theorem we can integrate the geodesic equations by a geometrical method and then verify that the classical conservation laws follow from these equations.
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