Gyula Fodor

Multipole moments of axisymmetric systems in relativity

An algorithm is developed for computing the nth gravitational multipole moment of an asymptotically flat, empty, stationary axisymmetric space‐time. The moments are expressed in terms of the expansion coefficients of the Ernst potential on the axis of symmetry. The values of the first ten multipole moments are given.

The Wahlquist metric cannot describe an isolated rotating body

It is proven that the Wahlquist perfect fluid spacetime cannot be smoothly joined to an exterior asymptotically flat vacuum region. The proof uses a power-series expansion in the angular velocity, to a precision of the second order. In this approximation, the Wahlquist metric is a special case of the rotating Whittaker spacetime. The exterior vacuum domain is treated in a like manner. We compute the conditions of matching at the possible boundary surface in both the interior and the vacuum domain. The conditions for matching the induced metrics and the extrinsic curvatures are mutually contradictory.

Small amplitude quasibreathers and oscillons

Quasibreathers (QB) are time-periodic solutions with weak spatial localization introduced in G. Fodor et al. in [Phys. Rev. D 74, 124003 (2006)]. QBs provide a simple description of oscillons (very long-living spatially localized time dependent solutions). The small amplitude limit of QBs is worked out in a large class of scalar theories with a general self-interaction potential, in D spatial dimensions. It is shown that the problem of small amplitude QBs is reduced to a universal elliptic partial differential equation. It is also found that there is the critical dimension, D{sub crit}=4, above which no small amplitude QBs exist. The QBs obtained this way are shown to provide very good initial data for oscillons. Thus these QBs provide the solution of the complicated, nonlinear time dependent problem of small amplitude oscillons in scalar theories.

Radiation of scalar oscillons in 2 and 3 dimensions

Abstract The radiation loss of small-amplitude radially symmetric oscillons (long-living, spatially localized, time-dependent solutions) in two- and three-dimensional scalar field theories is computed analytically in the small-amplitude expansion. The amplitude of the radiation is beyond all orders in perturbation theory and it is determined using matched asymptotic series expansions and Borel summation. The general results are illustrated on the case of the two- and three-dimensional sine-Gordon theory and a two-dimensional ϕ 6 model. The analytic predictions are found to be in good agreement with the results of numerical simulations of oscillons.

Computation of the radiation amplitude of oscillons

The radiation loss of small-amplitude oscillons (very long-living, spatially localized, time-dependent solutions) in one-dimensional scalar field theories is computed in the small-amplitude expansion analytically using matched asymptotic series expansions and Borel summation. The amplitude of the radiation is beyond all orders in perturbation theory and the method used has been developed by Segur and Kruskal in Phys. Rev. Lett. 58, 747 (1987). Our results are in good agreement with those of long-time numerical simulations of oscillons.

Surface gravity in dynamical spherically symmetric spacetimes

A definition of surface gravity at the apparent horizon of dynamical spherically symmetric spacetimes is proposed. It is based on a unique foliation by ingoing null hypersurfaces. The function parametrizing the hypersurfaces can be interpreted as the phase of a light wave uniformly emitted by some far-away static observer. The definition gives back the accepted value of surface gravity in the static case by virtue of its nonlocal character. Although the definition is motivated by the behavior of outgoing null rays, it turns out that there is a simple connection between the surface gravity, the acceleration of any radially moving observer, and the observed frequency change of the infalling light signal. In particular, this gives a practical and simple method of how any geodesic observer can determine surface gravity by measuring only the redshift of the infalling light wave. The surface gravity can be expressed as an integral of matter field quantities along an ingoing null line, which shows that it is a continuous function along the apparent horizon. A formula for the area change of the apparent horizon is presented, and the possibility of thermodynamical interpretation is discussed. Finally, concrete expressions of surface gravity are given for a number of four-dimensional and two-dimensional dynamical black hole solutions.

Perfect fluid spheres with cosmological constant

We examine static perfect fluid spheres in the presence of a cosmological constant. Because of the cosmological constant, new classes of exact matter solutions are found. One class of solutions requires the Nariai metric in the vacuum region. Another class generalizes the Einstein static universe such that neither its energy density nor its pressure is constant throughout the spacetime. Using analytical techniques we derive conditions depending on the equation of state to locate the vanishing pressure surface. This surface can, in general, be located in regions where, going outwards, the area of the spheres associated with the group of spherical symmetry is decreasing. We use numerical methods to integrate the field equations for realistic equations of state and find consistent results.

Numerical simulation of oscillatons: extracting the radiating tail

Spherically symmetric, time-periodic oscillatons -- solutions of the Einstein-Klein-Gordon system (a massive scalar field coupled to gravity) with a spatially localized core -- are investigated by very precise numerical techniques based on spectral methods. In particular the amplitude of their standing-wave tail is determined. It is found that the amplitude of the oscillating tail is very small, but non-vanishing for the range of frequencies considered. It follows that exactly time-periodic oscillatons are not truly localized, and they can be pictured loosely as consisting of a well (exponentially) localized nonsingular core and an oscillating tail making the total mass infinite. Finite mass physical oscillatons with a well localized core -- solutions of the Cauchy-problem with suitable initial conditions -- are only approximately time-periodic. They are continuously losing their mass because the scalar field radiates to infinity. Their core and radiative tail is well approximated by that of time-periodic oscillatons. Moreover the mass loss rate of physical oscillatons is estimated from the numerical data and a semi-empirical formula is deduced. The numerical results are in agreement with those obtained analytically in the limit of small amplitude time-periodic oscillatons.

Axistationary perfect fluids - A tetrad approach

Stationary axisymmetric perfect fluid spacetimes are investigated using the curvature description of geometries. We formulate the equations in terms of components of the Riemann tensor and the Ricci rotation coefficients in a comoving Lorentz tetrad. It is shown that the only incompressible axistationary magnetic perfect fluid is the interior Schwarzschild solution. Further, we find that all rigidly rotating axistationary fluids with magnetic Weyl tensor have local rotational symmetry. Rigidly rotating fluid spacetimes with purely electric or purely magnetic Weyl tensor are shown to be of Petrov type D.

Quadrupole moment of slowly rotating fluid balls

In this paper we use the second order formalism of Hartle to study slowly and rigidly rotating stars with focus on the quadrupole moment of the object. The second order field equations for the interior fluid are solved numerically for different classes of possible equations of state and these solutions are then matched to a vacuum solution that includes the general asymptotically flat axisymmetric metric to second order, using the Darmois-Israel procedure. For these solutions we find that the quadrupole moment differs from that of the Kerr metric, as has also been found for some equations of state in other studies. Further we consider the post-Minkowskian limit analytically. In the paper we also illustrate how the relativistic multipole moments can be calculated from a complex gravitational potential.