The Generalized Jacobi Equation

The Jacobi equation in pseudo-Riemannian geometry determines the linearized geodesic flow. The linearization ignores the relative velocity of the geodesics. The generalized Jacobi equation takes the relative velocity into account; that is, when the geodesics are neighboring but their relative velocity is arbitrary the corresponding geodesic deviation equation is the generalized Jacobi equation. The Hamiltonian structure of this nonlinear equation is analyzed in this paper. The tidal accelerations for test particles in the field of a plane gravitational wave and the exterior field of a rotating mass are investigated. In the latter case, the existence of an attractor of uniform relative radial motion with speed $2^{-1/2}c\approx 0.7 c$ is pointed out. The astrophysical implications of this result for the terminal speed of a relativistic jet is briefly explored.Comment: LaTeX file, 4 PS figures, 28 pages, revised version, accepted for publication in Classical and Quantum Gravit

On the Clock Paradox in the case of circular motion of the moving clock

In this paper we deal analytically with a version of the so called clock paradox in which the moving clock performs a circular motion of constant radius. The rest clock is denoted as (1), the rotating clock is (2), the inertial frame in which (1) is at rest and (2) moves is I and, finally, the accelerated frame in which (2) is at rest and (1) rotates is A. By using the General Theory of Relativity in order to describe the motion of (1) as seen in A we will show the following features. I) A differential aging between (1) and (2) occurs at their reunion and it has an absolute character, i.e. the proper time interval measured by a given clock is the same both in I and in A. II) From a quantitative point of view, the magnitude of the differential aging between (1) and (2) does depend on the kind of rotational motion performed by A. Indeed, if it is uniform there is no any tangential force in the direction of motion of (2) but only normal to it. In this case, the proper time interval reckoned by (2) does depend only on its constant velocity v=romega. On the contrary, if the rotational motion is uniformly accelerated, i.e. a constant force acts tangentially along the direction of motion, the proper time intervals $do depend$ on the angular acceleration alpha. III) Finally, in regard to the sign of the aging, the moving clock (2) measures always a $shorter$ interval of proper time with respect to (1).Comment: LaTex2e, 9 pages, no figures, no tables. It is the follow-on of the paper physics/040503

Relativistic Equilibrium Distribution by Relative Entropy Maximization

The equilibrium state of a relativistic gas has been calculated based on the maximum entropy principle. Though the relativistic equilibrium state was long believed to be the Juttner distribution, a number of papers have been published in recent years proposing alternative equilibrium states. However, some of these papers do not pay enough attention to the covariance of distribution functions, resulting confusion in equilibrium states. Starting from a fully covariant expression to avoid this confusion, it has been shown in the present paper that the Juttner distribution is the maximum entropy state if we assume the Lorentz symmetry.Comment: Six pages, no figure

Wave and Particle Scattering Properties of High Speed Black Holes

The light-like limit of the Kerr gravitational field relative to a distant observer moving rectilinearly in an arbitrary direction is an impulsive plane gravitational wave with a singular point on its wave front. By colliding particles with this wave we show that they have the same focussing properties as high speed particles scattered by the original black hole. By colliding photons with the gravitational wave we show that there is a circular disk, centered on the singular point on the wave front, having the property that photons colliding with the wave within this disk are reflected back and travel with the wave. This result is approximate in the sense that there are observers who can see a dim (as opposed to opaque) circular disk on their sky. By colliding plane electromagnetic waves with the gravitational wave we show that the reflected electromagnetic waves are the high frequency waves.Comment: Latex file, 22 pages, 1 figure, accepted for publication in Classical and Quantum Gravit

Geometric transport along circular orbits in stationary axisymmetric spacetimes

Parallel transport along circular orbits in orthogonally transitive stationary axisymmetric spacetimes is described explicitly relative to Lie transport in terms of the electric and magnetic parts of the induced connection. The influence of both the gravitoelectromagnetic fields associated with the zero angular momentum observers and of the Frenet-Serret parameters of these orbits as a function of their angular velocity is seen on the behavior of parallel transport through its representation as a parameter-dependent Lorentz transformation between these two inner-product preserving transports which is generated by the induced connection. This extends the analysis of parallel transport in the equatorial plane of the Kerr spacetime to the entire spacetime outside the black hole horizon, and helps give an intuitive picture of how competing "central attraction forces" and centripetal accelerations contribute with gravitomagnetic effects to explain the behavior of the 4-acceleration of circular orbits in that spacetime.Comment: 33 pages ijmpd latex article with 24 eps figure

Markov Chain Monte Carlo Method without Detailed Balance

We present a specific algorithm that generally satisfies the balance condition without imposing the detailed balance in the Markov chain Monte Carlo. In our algorithm, the average rejection rate is minimized, and even reduced to zero in many relevant cases. The absence of the detailed balance also introduces a net stochastic flow in a configuration space, which further boosts up the convergence. We demonstrate that the autocorrelation time of the Potts model becomes more than 6 times shorter than that by the conventional Metropolis algorithm. Based on the same concept, a bounce-free worm algorithm for generic quantum spin models is formulated as well.Comment: 5 pages, 5 figure

A Snapshot of J. L. Synge

A brief description is given of the life and influence on relativity theory of Professor J. L. Synge accompanied by some technical examples to illustrate his style of work

Light-cone coordinates based at a geodesic world line

Continuing work initiated in an earlier publication [Phys. Rev. D 69, 084007 (2004)], we construct a system of light-cone coordinates based at a geodesic world line of an arbitrary curved spacetime. The construction involves (i) an advanced-time or a retarded-time coordinate that labels past or future light cones centered on the world line, (ii) a radial coordinate that is an affine parameter on the null generators of these light cones, and (iii) angular coordinates that are constant on each generator. The spacetime metric is calculated in the light-cone coordinates, and it is expressed as an expansion in powers of the radial coordinate in terms of the irreducible components of the Riemann tensor evaluated on the world line. The formalism is illustrated in two simple applications, the first involving a comoving world line of a spatially-flat cosmology, the other featuring an observer placed on the axis of symmetry of Melvin's magnetic universe.Comment: 11 pages, 1 figur

Potential flows in a core-dipole-shell system: numerical results

Numerical solutions for: the integral curves of the velocity field (streamlines), the density contours, and the accretion rate of a steady-state flow of an ideal fluid with p=K n^(gamma) equation of state orbiting in a core-dipole-shell system are presented. For 1 < gamma < 2, we found that the non-linear contribution appearing in the partial differential equation for the velocity potential has little effect in the form of the streamlines and density contour lines, but can be noticed in the density values. The study of several cases indicates that this appears to be the general situation. The accretion rate was found to increase when the constant gamma decreases.Comment: RevTex, 8 pages, 5 eps figures, CQG to appea

Linear waves in sheared flows. Lower bound of the vorticity growth and propagation discontinuities in the parameters space

This study provides sufficient conditions for the temporal monotonic decay of enstrophy for two-dimensional perturbations traveling in the incompressible, viscous, plane Poiseuille and Couette flows. Extension of J. L. Synge's procedure (1938) to the initial-value problem allowed us to find the region of the wavenumber-Reynolds number map where the enstrophy of any initial disturbance cannot grow. This region is wider than the kinetic energy's one. We also show that the parameters space is split in two regions with clearly distinct propagation and dispersion properties