A numerical examination of certain null geodesies of a high angular momentum kerr field

An observer situated anywhere but in the equatorial plane of a high angular momentum Kerr field cannot see the ring singularity. In the visual field of such an observer, what demarcates his own universe from that through the ring? The projections onto a certain submanifold of the null geodesics which pass through a point on the symmetry axis of a specific Kerr field are examined numerically. All the distinct projections are obtained by varying one parameter, essentially the quadratic Killing tensor constant. Various interesting features of the geodesics emerge. Through the ring is a region in which there exist closed time-like curves and which can be used to construct closed time-like curves through any non-singular point of the manifold. Only geodesies of negative angular momentum can enter this region.     

A numerical examination of certain null geodesies of a high angular momentum kerr field

An observer situated anywhere but in the equatorial plane of a high angular momentum Kerr field cannot see the ring singularity. In the visual field of such an observer, what demarcates his own universe from that through the ring?The projections onto a certain submanifold of the null geodesics which pass through a point on the symmetry axis of a specific Kerr field are examined numerically. All the distinct projections are obtained by varying one parameter, essentially the quadratic Killing tensor constant. Various interesting features of the geodesics emerge.Through the ring is a region in which there exist closed time-like curves and which can be used to construct closed time-like curves through any non-singular point of the manifold. Only geodesies of negative angular momentum can enter this region.     

Damour-Ruffini and Unruh theories of the Hawking effect

Internal relations between the Damour-Ruffini approach and the Unruh approach to dealing with the Hawking effects are shown. The Unruh-type analytical wave functions can be obtained by means of the analytical continuation method suggested by Damour and Ruffini. In fact, Unruh-type analytical wave functions are complex conjugate functions of Damour-Ruffini type. Normalizing each of them, or making use of them to construct the Bogoliubov transformation, one can get the same Hawking thermal spectrum. The pure state wave function defined on the connected complexr space-time manifold is a mixture showing thermal properties in the realr space-time manifold, which is divided into two parts by the event horizon.     

NULL GEODESICS IN STATIONARY ERNST-WILD SPACE-TIME

Null geodesics in equatorial plane of stationary Ernst-Wild space-time representing a Kerr black hole immersed in axlsymmetric magnetic field are investigated.As the case of static Ernst space-time there are two types of geodesics which depend on a dimensionless parameter β of the magnetic field.Differing from the static Ernst space-time,however,the critical β_C now depends on a dimensionless parameter q of the angular momentum of the Kerr black hole,and the relation between β_C and q is qualitatively different in two cases:L>O and L<0 where L is a dimensionless parameter of angular momentum of test particles moving along nu11 geodesics.     

Spinor structure of space-time

The spinor structure on space-time manifold is investigated in the frame of Crumeyrolle's approach. Some of his theorems are simplified. The equivalence of this approach to the Milnor and Lichnerowicz one is shown using topological properties of the group space of ₀. The equivalence of any two spinor structures on simply connected space-time is established.Partly supported by the Polish Government under the Research Program MR I.7.     

Dimension spectrum of axiom a diffeomorphisms. II. Gibbs measures

We compute the dimension spectrumf() of the singularity sets of a Gibbs measure defined on a two-dimensional compact manifold and invariant with respect to aC ² Axiom A diffeomorphism. This case is the generalization of the case where the measure studied is the Bowen-Margulis measure—the one that realizes the topological entropy. We obtain similar results; for example, the functionf is the Legendre-Fenchel transform of a free energy function which is real analytic (linear in the degenerate case). The functionf is also real analytic on its definition domain (defined in one point in the degenerate case) and is related to the Hausdorff dimensions of Gibbs measures singular with respect to each other and whose supports are the singularity sets, and we finally decompose these sets.     

Circular orbits and relative strains in Schwarzschild space-time

The equation of the relative strain is analyzed in tetrad form with respect to a family of observers moving on spatially circular orbits, in the Schwarzschild space-time. We select a field of tetrads, which we term phase locking frames, and explicitly calculate how, in the equatorial plane, the orbital acceleration, its gradient and the Fermi drag add together to compensate the curvature and assure equilibrium among a set of comoving neighbouring particles. While equilibrium is achieved in the radial and azimuthal directions, in the direction orthogonal to the equatorial plane there is a residue of acceleration which pulls a particle towards that plane leading to a harmonic oscillation with a frequency equal to the proper frequency of the orbital revolution. This measurement, combined with those of the frequency shift of an incoming photon and the frequency of precession of the local compass of inertia, enables one to determine the relativistic ratio 2M/r, whereM is the gravitational mass of the source andr the coordinate radius of the circular orbits.     

Geometrothermodynamics for black holes and de Sitter space

A general method to extract thermodynamic quantities from solutions of the Einstein equation is developed. In 1994, Wald established that the entropy of a black hole could be identified as a Noether charge associated with a Killing vector of a global space-time (pseudo-Riemann) manifold. We reconstruct Wald’s method using geometrical language, e.g., via differential forms defined on the local space-time (Minkowski) manifold. Concurrently, the abstract thermodynamics are also reconstructed using geometrical terminology, which is parallel to general relativity. The correspondence between the thermodynamics and general relativity can be seen clearly by comparing the two expressions. This comparison requires a modification of Wald’s method. The new method is applied to Schwarzschild, Kerr, and Kerr–Newman black holes and de Sitter space. The results are consistent with previous results obtained using various independent methods. This strongly supports the validity of the area theorem for black holes.     

Second order Kerr deflection

We perform a full second order calculation of the deflection of light along the equatorial plane in the Kerr metric. Previous Kerr deflection calculations were interested in obtaining the correction due to rotation to the Einstein deflection. By expanding to first order in the rotational parameter a, they obtain the Einstein deflection of 4M/r _o and the second order deflection due to rotation of , (where r _o is the point of closest approach). In this paper, we are interested in going beyond the rotational contribution for the purpose of astrophysical applications. We therefore keep all terms up to second order in our final weak field expansion. Besides the rotational contribution, we also obtain an extra second order term of . Since M > a, this extra term is greater than the rotational contribution of in astrophysical applications. When a/M is close to unity the terms are of the same order of magnitude.     

Time machine and geodesic motion in Kerr metric

The existence of nontrivial causality violation is a peculiar property of the space-time around a naked singularity. In the case of Kerr metric witha > m we have found that for a particular class of geodesies that could in principle violate causality, the conditions for causality violation are never satisfied.On leave of absence from Institute of Astronomy, Polish Academy of Sciences, Warsaw, Poland.     

The effects of running gravitational coupling on rotating black holes

In this work we investigate the consequences of running gravitational coupling on the properties of rotating black holes. Apart from the changes induced in the space-time structure of such black holes, we also study the implications to Penrose process and geodetic precession. We are motivated by the functional form of gravitational coupling previously investigated in the context of infra-red limit of asymptotic safe gravity theory. In this approach, the involvement of a new parameter \({\tilde{\xi }}\) in this solution makes it different from Schwarzschild black hole. The Killing horizon, event horizon and singularity of the computed metric is then discussed. It is noticed that the ergosphere is increased as \({\tilde{\xi }}\) increases. Considering the black hole solution in equatorial plane, the geodesics of particles, both null and time like cases, are explored. The effective potential is computed and graphically analyzed for different values of parameter \({\tilde{\xi }}\). The energy extraction from black hole is investigated via Penrose process. For the same values of spin parameter, the numerical results suggest that the efficiency of Penrose process is greater in quantum corrected gravity than in Kerr Black Hole. At the end, a brief discussion on Lense–Thirring frequency is also done.     

Chaos and dynamics of spinning particles in Kerr spacetime

We study chaos dynamics of spinning particles in Kerr spacetime of rotating black holes use the Papapetrou equations by numerical integration. Because of spin, this system exists many chaos solutions, and exhibits some exceptional dynamic character. We investigate the relations between the orbits chaos and the spin magnitude S, pericenter, polar angle and Kerr rotation parameter a by means of a kind of brand new Fast Lyapulov Indicator (FLI) which is defined in general relativity. The classical definition of Lyapulov exponent (LE) perhaps fails in curve spacetime. And we emphasize that the Poincaré sections cannot be used to detect chaos for this case. Via calculations, some new interesting conclusions are found: though chaos is easier to emerge with bigger S, but not always depends on S monotonically; the Kerr parameter a has a contrary action on the chaos occurrence. Furthermore, the spin of particles can destroy the symmetry of the orbits about the equatorial plane. And for some special initial conditions, the orbits have equilibrium points.     

A new approach to the theory of elementary domains

It is shown that Yukawa's theory of elementary domains can be formulated in a general framework. A quantized measure space structure of a space-time manifold is introduced so as to represent faithfully the elementary-domain structure. For a realization of elementary domains, an operator valued measure is defined such that it represents the spatiotemporal distribution of elementary domains. Effects of such a quantized topology are illustrated in the expressions ofS matrices.     

The structure of the b-completion of space-time

Structured space, as a natural generalization of the manifold concept, is defined to be a topological space with a sheaf of real function algebras which are suitably localized and closed with respect to composition with smooth Euclidean functions. Vector fields, differential forms, linear connection and curvature are introduced on structured spaces. It is shown that structured spaces correctly model space-times with singularities. Schmidt's b-boundary of space-time is constructed in the category of structured spaces, and well known difficulties with the b-boundaries of the closed Friedman and Schwarzschild space-times are disentangled. It is argued that the b-boundary of space-time, when considered in the category of structured spaces, can serve as a good definition of classical singularities.     

Some exact solutions of F(R) gravity with charged (a)dS black hole interpretation

In this paper we obtain topological static solutions of some kind of pure F(R) gravity. The present solutions are two kind: first type is uncharged solution which corresponds with the topological (a)dS Schwarzschild solution and second type has electric charge and is equivalent to the Einstein-Λ-conformally invariant Maxwell solution. In other word, starting from pure gravity leads to (charged) Einstein-Λ solutions which we interpreted them as (charged) (a)dS black hole solutions of pure F(R) gravity. Calculating the Ricci and Kreschmann scalars show that there is a curvature singularity at r = 0. We should note that the Kreschmann scalar of charged solutions goes to infinity as r → 0, but with a rate slower than that of uncharged solutions.     

Fuzzy Space-Time Geometry and Particle’s Dynamics

The quantum space-time with Dodson-Zeeman topological structure is studied. In its framework, the states of massive particle m correspond to the elements of fuzzy ordered set (Foset), i.e. the fuzzy points. Due to their partial ordering, m space coordinate x acquires principal uncertainty σ _x . Schroedinger formalism of Quantum Mechanics is derived from consideration of m evolution in fuzzy phase space with minimal number of additional axioms. The possible particle’s interactions on fuzzy manifold are studied and shown to be gauge invariant.     

Potential surfaces for time-like geodesics in the Curzon metric

In this paper we deduce the general pattern of the potential surfaces for time-like geodesics in the Curzon metric. We find that for fairly small energies and orbital angular momenta, the time-like geodesics group into two sets; the geodesics of one set tend to thez-axis asR=(r²+z²)^1/2 0,R=0 being a directional singularity, the others tend to ther-axis. At low energies these two sets are detached but they merge together as the energy increases. Stable circular motion is confined to thez = 0-plane and an energy threshold for stationary motion exists and is equal (per unit of rest-mass energy) to 0.945, a value almost indistinguishable from that in the Schwarzschild space-time.     

A connection between fermionic strings and quantum gravity states: a loop space approach

We present physical arguments-based on loop space representations for Dirac/Klein Gordon determinants—that some suitable Fermionic String Ising Models at the critical point and defined on the space-time base manifold M ì R³{M \subset R^3} are formal quantum states of the gravitational field when quantized in the Ashtekar-Sen connection canonical formalism. These results complements the author previous string-loop space studies on the subject.