Reinhard Meinel

Progress in relativistic gravitational theory using the inverse scattering method

The increasing interest in compact astrophysical objects (neutron stars, binaries, galactic black holes) has stimulated the search for rigorous methods, which allow a systematic general relativistic description of such objects. This article is meant to demonstrate the use of the inverse scattering method, which allows, in particular cases, the treatment of rotating body problems. The idea is to replace the investigation of the matter region of a rotating body by the formulation of boundary values along the surface of the body. In this way we construct solutions describing rotating black holes and disks of dust (“galaxies”). Physical properties of the solutions and consequences of the approach are discussed. Among other things, the balance problem for two black holes can be tackled.

The Einsteinian gravitational field of the rigidly rotating disk of dust

This paper presents the gravitational field of a uniformly rotating stationary and axisymmetric disk consisting of dust particles as a rigorous global solution to the Einstein equations. The problem is formulated as a boundary value problem of the Ernst equation and solved by means of inverse methods. The solution is given in terms of linear integral equations and depends on two parameters: the angular velocity Ω and the relative redshift z from the center of the disk. The Newtonian limit z<<1 represents the MacLaurin solution of a rotating fluid in the disk limit. For z→∞ the exterior solution is given by the extreme Kerr solution. This proves a conjecture of Bardeen & Wagoner (1969, 1971)

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.

Constructive proof of the Kerr-Newman black hole uniqueness including the extreme case

A new proof of the uniqueness of the Kerr–Newman black hole solutions amongst asymptotically flat, stationary and axisymmetric electrovacuum spacetimes surrounding a connected Killing horizon is given by means of an explicit construction of the corresponding complex Ernst potentials on the axis of symmetry. This construction, which makes use of the inverse scattering method, also works in the case of a degenerate horizon.

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.

A Physical Derivation of the Kerr–Newman Black Hole Solution

According to the no-hair theorem, the Kerr–Newman black hole solution represents the most general asymptotically flat, stationary (electro-) vacuum black hole solution in general relativity. The procedure described here shows how this solution can indeed be constructed as the unique solution to the corresponding boundary value problem of the axially symmetric Einstein–Maxwell equations in a straightforward manner.

Vandermonde-like determinants and N-fold Darboux/Bäcklund transformations

We define Vandermonde-like determinants and analyze their structure. The resulting scheme is well-suited to achieve a remarkable compactness and transparency in N-soliton formulas or, more generally, in formulas for N-fold Darboux or Backlund transformations.

Asymptotically flat solutions to the Ernst equation with reflection symmetry

It is shown that the class of asymptotically flat solutions to the axisymmetric and stationary vacuum Einstein equations with reflection symmetry of the metric is uniquely characterized by a simple relation for the Ernst potential on the upper part of the symmetry axis ( axis): . This result generalizes a well-known fact from potential theory: axisymmetric solutions to the Laplace equation that vanish at infinity and have reflection symmetry with respect to the plane are characterized by a potential that is an odd function of on the upper part of the axis.

On the black hole limit of rotating fluid bodies in equilibrium

Recently, it was shown that the extreme Kerr black hole is the only candidate for a (Kerr) black hole limit of stationary and axisymmetric, uniformly rotating perfect fluid bodies with a zero temperature equation of state. In this paper, necessary and sufficient conditions for reaching the black hole limit are presented.