Killing Trajectories Research Articles

Fermi–Walker magnetic curves and Killing trajectories in 3D Riemannian manifolds

This paper deals with obtaining some new characterizations on the dynamics of the moving charged particle acting under particular forces in the three‐dimensional space. In this perspective, we have enlarged the research of magnetic curves and associated Killing magnetic fields by adding the Fermi–Walker parallelism and derivative. By doing this, we investigate the Hall–Killing–Fermi effect on the charged particle, then we define new types of Fermi–Walker magnetic curves corresponding to the Killing magnetic field. We carefully construct the Fermi–Walker character of the magnetic field to prove the parallelism of the magnetic trajectories and uniformly accelerated motion of the charged particle. It leads to defining a very special family of magnetic fields, that is, Fermi–Walker uniform magnetic fields. Moreover, we use the Killing feature of the magnetic field to show that the Frenet–Serret scalars and quasislopes of the magnetic curves must meet particular differential equations that lead to obtaining some geometric interpretations of the magnetic trajectories. Last but not the least, both of these two characters allow us to define Killing and Fermi–Walker electric vector fields influencing the motion of the charged particle experiencing the Lorentz force in the three‐dimensional setting.

Hawking radiation, quantum fields, and tunneling

I consider the problem of reconciling "tunneling" approaches to black-hole radiation with the treatment by quantum field theory in curved space-time. It is not possible to do this completely, but using what appears to be the most direct and natural correspondence, the simplest case of such an approach does not describe a tunneling process but rather a WKB approximation to a two-point function. What it computes is a sort of rescaled Unruh temperature associated with world-lines tracing Killing trajectories near the horizon. This temperature is numerically equal to the Hawking temperature, but at this point no argument is known which identifies it with the Hawking effect; indeed, the same temperature can exist when no curvature is present.

The complete solution of the conformastat electrovacuum problem

We find the complete solution of the Einstein-Maxwell field equations without sources for static spacetimes in which the space of Killing trajectories is conformally flat. The result is used to present an improved local characterisation of the Majumdar-Papapetrou class of solutions.

Stationary strings and branes in the higher-dimensional Kerr-NUT-(A)dS spacetimes

We demonstrate complete integrability of the Nambu-Goto equations for a stationary string in the general Kerr-NUT-(A)dS spacetime describing the higher-dimensional rotating black hole. The stationary string in D dimensions is generated by a 1-parameter family of Killing trajectories and the problem of finding a string configuration reduces to a problem of finding a geodesic line in an effective (D-1)-dimensional space. Resulting integrability of this geodesic problem is connected with the existence of hidden symmetries which are inherited from the black hole background. In a spacetime with p mutually commuting Killing vectors it is possible to introduce a concept of a $\xi$-brane, that is a p-brane with the worldvolume generated by these fields and a 1-dimensional curve. We discuss integrability of such $\xi$-branes in the Kerr-NUT-(A)dS spacetime.

Periodic standing-wave approximation: Eigenspectral computations for linear gravity and nonlinear toy models

The periodic standing-wave approach to binary inspiral assumes rigid rotation of gravitational fields and hence helically symmetric solutions. To exploit the symmetry, numerical computations must solve for ``helical scalars,'' fields that are functions only of corotating coordinates, the labels on the helical Killing trajectories. Here we present the formalism for describing linearized general relativity in terms of helical scalars and we present solutions to the mixed partial differential equations of the linearized gravity problem (and to a toy nonlinear problem) using the adapted coordinates and numerical techniques previously developed for scalar periodic standing-wave computations. We argue that the formalism developed may suffice for periodic standing-wave computations for post-Minkowskian computations and for full general relativity.

Conformal Killing horizons

For time dependent black hole space–times the event horizon cannot be described by a Killing horizon. In the case when the space–time admits a timelike conformal Killing field, which becomes null on a boundary called the conformal stationary limit surface, one can locally describe the expanding event horizon by using this boundary, provided that it is a null geodesic hypersurface. In this case the boundary is called a conformal Killing horizon and is shown to be null and geodesic if and only if the twist of the conformal Killing trajectories on the hypersurface vanishes. Moreover if the space–time is conformally related to a stationary asymptotically flat black hole space–time, it is shown that this hypersurface is globally equivalent to the event horizon, provided that the conformal factor goes to a constant at null infinity. When the conformal stationary limit surface does not coincide with the conformal Killing horizon, a generalization of the weak rigidity theorem which establishes the conformal Killing property of the event horizon and the rigidity of its rotation is obtained. A physical definition of surface gravity for conformal Killing horizons is given, which is then used to formulate a generalized zeroth law of black hole physics.

Circular holonomy in the Taub–NUT spacetime

Parallel transport around closed circular orbits in the equatorial plane of the Taub–NUT spacetime is analysed to reveal the effect of the gravitomagnetic monopole parameter on circular holonomy transformations. Investigating the boost/rotation decomposition of the connection 1-form matrix evaluated along these orbits, one finds a situation that reflects the behaviour of the general orthogonally transitive stationary axisymmetric case and indeed along Killing trajectories in general.

ON THE UNIQUENESS OF STATIONARY AXISYMMETRIC ELLIPSOIDAL VACUUM CONFIGURATIONS

Stationary axisymmetric vacuum space–times in general relativity are considered. The Riemannian metric induced on the space of the time-like Killing trajectories is assumed to be ellipsoidal. It is demonstrated that the only ellipsoidal solutions of the vacuum Einstein's equations are the Kerr–NUT family with vanishing electric and magnetic charges.

Stationary rotating matter in general relativity

Stationary rotating matter configurations in general relativity are considered. A formalism for general stationary space times is developed. Axisymmetric systems are discussed by the use of a nonholonomic and nonrigid frame in the three-space of the time-like Killing trajectories. Two symmetric and trace-free tensors are constructed. They characterize a class of matter states in which both the interior Schwarzschild and the Kerr solution are contained. Consistency relations for this class of perfect fluids are derived. As an example of the application of our procedure, we obtain a solution under the assumption of rigid rotation.

Increasing efficiency in Monte Carlo simulations by a killing-and-splitting strategy

We have derived a procedure which increases the efficiency of Monte Carlo simulations. It aims at special cases in ion-solid simulations where the probability of success becomes particularly small. Efficiency results from killing and splitting trajectories according to a certain importance figure assigned to a number of sites along the ion trajectory. A much simple case, however, is studied which, on the one hand, retains the essentials of ion-solid simulations, without, on the other hand, going into unnecessary complications.

The Frenet-Serret formalism and black holes in higher dimensions

The Frenet-Serret formalism is applied to the study of Killing trajectories in higher dimensions (five and six). The curvature and the higher torsions are expressed in terms of an effective field strength Fab and their mutual relations established by the use of the Cayley-Hamilton theorem. This is used to show that the surface on which the timelike Killing vector becomes null is a null surface only if the 'rotation' is also null on it.

Stationary vacuum fields with a conformally flat three-space I. General theory

A generalized notion of conformastat space-times is introduced in relativity theory. In this sense, the conformastat space-time is stationary with the three-space of timelike Killing trajectories being conformally flat. A 3+1 decomposition of the field equations is given, and two classes of nonstatic conformastat vacuum fields are exhaustively investigated. The resulting three metrics form a NUT-type extension of the solution of the static conformastat vacuum problem. We conjecture that all conformastat vacuum space-times are axially symmetric.

Geometry of homothetic Killing trajectories and stationary limit surfaces

We investigate the intrinsic geometry of timelike and null homothetic Killing trajectories. We do this through the use of two different Frenet–Serret formalisms developed by Synge and Bonnor. The curvature of such a nongeodesic timelike curve passing through a spacetime point is a direct measure of the expansion of the congruence at that point. Moreover, the rotation of this congruence is directly related to the torsions of its individual curves. For both classes of curves, timelike and null, the intrinsic scalars can be expressed as exponential functions of suitable parameters. We prove the following theorem twice, first by using the timelike formalism, and then through the null one. A homothetic stationary limit surface (where the homothetic Killing vector becomes null) is a null geodesic hypersurface if and only if the rotation of the homothetic Killing congruence vanishes on that hypersurface. The second, ’’null’’ approach allows us to derive this theorem directly on the hypersurface. Finally, we note that by setting a parameter equal to zero we recover all of the corresponding results for ordinary Killing vectors.

Intrinsic geometry of Killing trajectories

We use the Frenet–Serret formalism to study the intrinsic geometry of Killing trajectories that are admitted by an arbitrary n-dimensional Riemannian space. The intrinsic quantities associated with these curves, i.e. their curvatures, are found to be constants of the motion that can be evaluated in terms of Hankel determinants. The results are then applied to curves in real quantum mechanics.

Classification of stationary space-times

A systematic approach to the geometric structure of stationary gravitational fields is presented. The algebraic type of the trace-free Ricci tensor together with the propagation properties of the eigenrays in the background 3-space defined by the Killing trajectories serve as a basis for classifying the solutions of the stationary field equations. The eigenrays are the integral curves belonging to the solutions ξA of the eigenvalue problemGABξB=μξA,GABspinor representing the gravitational field in the background space. Many of the already known stationary metrics can be derived in the present scheme but new solutions of the field equations are also obtained. The possible types of the vacuum and electrovac fields are discussed in their connection with the corresponding exact solutions.

Motion of charged particles in homogeneous electromagnetic fields

An invariant geometrical description of the world lines of charged particles in arbitrary homogeneous electromagnetic fields is presented. This is accomplished through the combined use of the Frenet-Serret equations and the Lorentz equation. The results apply to flat as well as Riemannian space-time. The intrinsic scalars associated with these curves (i.e., their curvatures and first and second torsions) are found to be constants of the motion when they are well defined. Moreover, they form simple relationships with the field invariants as well as with the energy and momentum densities of the rest frame fields. When they are evaluated in the instantaneous rest frame of the particle, the Frenet vectors lend themselves to simple physical interpretation. It is shown that one cannot distinguish in an intrinsic geometrical manner between the curves of positive and negative charges. The same is true for positive and negative magnetic monopoles if they exist. In such a case, however, one would be able to distinguish intrinsically between ordinary and magnetic charges. The effect of duality rotations of the field tensor on the Frenet scalars is studied. A physical realization of the Frenet frame is obtained by considering the classical description of spin precession. Finally the Frenet formalism is applied to timelike Killing trajectories. These are shown to closely resemble the world lines of charged particles in homogeneous electromagnetic fields.

Invariant Approach to a Space-Time Symmetry

A study is made of Killing vector fields in vacuum Einstein spaces with a restriction primarily to those fields whose associated bivector is nonnull. However, a well-known theorem of Robinson is modified slightly to show that if such a space admits a null bivector associated with a Killing vector, the space must be algebraically special. Consequently, all algebraically general spaces admit a nonnull Killing bivector (KBV) if they admit a symmetry at all. Furthermore, it is shown that if a vacuum Einstein space admits a spacelike or timelike Killing vector field whose associated KBV is nonnull, the Killing trajectories are not geodesics. Computation of invariants from the curvature tensor and the KBV allows an approach which gives a general classification to such spaces which admit at least one hypersurface orthogonal Killing vector field. A few geometrical properties involving the principal null directions of the KBV are also derived for the hypersurface orthogonal cases. In addition, a topological result follows immediately from the behavior of the invariant J.

Almost Symmetric Spaces and Gravitational Radiation

A generalization of the idea of Killing fields to spaces which are not symmetric is given. The field so defined specifies coordinate lines along which the variation of the metric tensor is the slowest possible in a global sense. Thus it generalizes the Killing fields in spaces with a symmetry where the metric tensor does not change along Killing trajectories. Several examples are given, and the method is then applied to spaces containing gravitational radiation of the type considered by Issacson. For spaces containing radiation, it is shown that a real functional λ[ξ], associated with every vector field ξ, measures some parameters associated with the radiation. In the simplest case this parameter is the ``energy density'' of the radiation, but, if a sufficient number of vector fields can be invariantly defined in the background, the average gravitational ``stress'' associated with the wave may also be measured. We conclude with some conjectures about further application of these ideas to the theory of gravitational radiation.