Andreas Kleinwächter

Highly accurate calculation of rotating neutron stars

A new spectral code for constructing general–relativistic models of rapidly rotating stars with an unprecedented accuracy is presented. As a first application, we reexamine uniformly rotating homogeneous stars and compare our results with those obtained by several previous codes. Moreover, representative relativistic examples corresponding to highly flattened rotating bodies are given.

Uniformly rotating axisymmetric fluid configurations bifurcating from highly flattened Maclaurin spheroids

We present a thorough investigation of sequences of uniformly rotating, homogeneous axisymmetric Newtonian equilibrium configurations that bifurcate from highly flattened Maclaurin spheroids. Each one of these sequences possesses a mass-shedding limit. Starting at this point, the sequences proceed towards the Maclaurin sequence and beyond. The first sequence leads to the well-known Dyson rings, whereas the end-points of the higher sequences are characterized by the formation of a two-body system, either a core‐ring system (for the second, the fourth, etc., sequence) or a two-ring system (for the third, the fifth, etc., sequence). Although the general qualitative picture drawn by Eriguchi and Hachisu in the 1980s has been confirmed, slight differences arise in the interpretation of the origin of the first two-ring sequence and in the general appearance of fluid bodies belonging to higher sequences.

RELATIVISTIC DYSON RINGS AND THEIR BLACK HOLE LIMIT

In this Letter, we investigate uniformly rotating, homogeneous, and axisymmetric relativistic fluid bodies with a toroidal shape. The corresponding field equations are solved by means of a multidomain spectral method, which yields highly accurate numerical solutions. For a prescribed, sufficiently large ratio of inner to outer coordinate radius, the toroids exhibit a continuous transition to the extreme Kerr black hole. Otherwise, the most relativistic configuration rotates at the mass-shedding limit. For a given mass density, there seems to be no bound to the gravitational mass as one approaches the black hole limit and a radius ratio of unity.

Dirichlet boundary value problems of the Ernst equation

We demonstrate how the solution to an exterior Dirichlet boundary value problem of the axisymmetric, stationary Einstein equations can be found in terms of generalized solutions of the Backlund type. The proof that this generalization procedure is valid is given, which also proves conjectures about earlier representations of the gravitational field corresponding to rotating disks of dust in terms of Backlund-type solutions. As a further result, we find that, in contrast with the Laplace equation, arbitrary boundary values may not be prescribed.

Equilibrium configurations of homogeneous fluids in general relativity

By means of a highly accurate, multi-domain, pseudo-spectral method, we investigate the solution space of uniformly rotating, homogeneous and axisymmetric relativistic fluid bodies. It turns out that this space can be divided up into classes of solutions. In this paper, we present two new classes including relativistic core‐ring and two-ring solutions. Combining our knowledge of the first four classes with post-Newtonian results and the Newtonian portion of the first ten classes, we present the qualitative behaviour of the entire relativistic solution space. The Newtonian disc limit can only be reached by going through infinitely many of the aforementioned classes. Only once this limiting process has been consummated can one proceed again into the relativistic regime and arrive at the analytically known relativistic disc of dust. Ke yw ords: gravitation ‐ relativity ‐ methods: numerical ‐ stars: rotation.

On the black hole limit of rotating discs and rings

Solutions to Einstein’s field equations describing rotating fluid bodies in equilibrium permit parametric (i.e. quasi-stationary) transitions to the extreme Kerr solution (outside the horizon). This has been shown analytically for discs of dust and numerically for ring solutions with various equations of state. From the exterior point of view, this transition can be interpreted as a (quasi) black hole limit. All gravitational multipole moments assume precisely the values of an extremal Kerr black hole in the limit. In the present paper, the way in which the black hole limit is approached is investigated in more detail by means of a parametric Taylor series expansion of the exact solution describing a rigidly rotating disc of dust. Combined with numerical calculations for ring solutions our results indicate an interesting universal behaviour of the multipole moments near the black hole limit.

The multipole moments of the rigidly rotating disk of dust in general relativity

Abstract The gravitational multipole moments of the asymptotically flat global solution to Einsteins equations describing a rigidly rotating disk of dust are calculated and discussed. The so-called rotational multipole moments (higher than the angular momentum) do not vanish. Hence the conjecture that these moments appear only for differentially rotating bodies proves to be false.

On the multipole moments of a rigidly rotating fluid body

Based on numerical simulations and analytical calculations we formulate a new conjecture concerning the multipole moments of a rigidly rotating fluid body in equilibrium. The conjecture implies that the exterior region of such a fluid is not described by the Kerr metric.

Rotating fluid bodies in equilibrium: fundamental notions and equations

An important and successful approach to solving problems throughout physics is to split the world into a system to be considered, its ‘surroundings’ and the ‘rest of the universe’, where the influence of the latter on the system being considered is neglected. The applicability of this concept to general relativity is not a trivial matter, since the spacetime structure at every point depends on the overall energymomentum distribution. Our aim is to find a description of a single fluid body (modelling a celestial body, e.g. a neutron star) under the influence of its own gravitational field. Fortunately, one often encounters such a body surrounded by a vacuum, where the closest other bodies are so far away that an intermediate region with a weak gravitational field exists. In such a situation (see Fig. 1.1) one can discuss the far field of the body. If the distant outside world (the ‘rest of the universe’) is isotropic, which it is according to astronomical observations and the standard cosmological models, then the line element corresponding to the far field of an arbitrary stationary body can be written as follows (see Stephani 2004):

Discussion of the Theta Formula for the Ernst Potential of the Rigidly Rotating Disk of Dust

The exact global solution of the rigidly rotating disk of dust[1] is given in terms of ultraelliptic functions. Here we discuss the “theta formula” for the Ernst potential[2]. The space-time coordinates of the problem enter the arguments of these functions via ultraelliptic line integrals which are related to a Riemann surface.