Physically realistic solutions to the Ernst equation on hyperelliptic Riemann surfaces

Klein, Christian; Richter, O.

doi:10.1103/physrevd.58.124018

Cited by 24 publications

(54 citation statements)

References 35 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The differential dω ∞ + ∞ − is given up to holomorphic differentials by −K g dK/µ. In [19,20] a physically interesting subclass of Korotkin's solution was identified which can be written in the form…”

Section: P E1mentioning

confidence: 99%

“…In the limit ρ → 0, there is a non-trivial contribution from the quotient of theta functions in (99) which diverges as 1/ρ. Repeating the considerations of [20] in the calculation of a 0 , one finds that the axis potentials can be expressed in terms of theta functions on the surfaceΣ of genus g − 1 given byμ…”

Section: Metric and Harrison Transformationmentioning

confidence: 99%

“…Since these two-sheeted surfaces are a generalization of the well-known elliptic surfaces, a powerful theory is at hand to treat hyperelliptic functions. The main advantages of this class are that it is very rich ('solitonic' solutions as the Kerr solution for a rotating black hole are contained as limiting cases), and that a whole subclass with physically interesting solutions could be identified in [17,18] by studying the analyticity properties of the solution. The corresponding solutions for the electro-vacuum are given on three-sheeted surfaces [14] which are mathematically less well understood.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On explicit solutions to the stationary axisymmetric Einstein-Maxwell equations describing dust disks

Klein¹

2003

Ann. Phys.

Self Cite

View full text Add to dashboard Cite

We review explicit solutions to the stationary axisymmetric Einstein-Maxwell equations which can be interpreted as disks of charged dust. The disks of finite or infinite extension are infinitesimally thin and constitute a surface layer at the boundary of an electro-vacuum. The Einstein-Maxwell equations in the presence of one Killing vector are obtained by using a projection formalism. This leads to equations for three-dimensional gravity where the matter is given by a SU (2, 1)/S[U (1, 1) × U (1)] nonlinear sigma model. The SU (2, 1) invariance of the stationary Einstein-Maxwell equations can be used to construct solutions for the electrovacuum from solutions to the pure vacuum case via a so-called Harrison transformation. It is shown that the corresponding solutions will always have a non-vanishing total charge and a gyromagnetic ratio of 2. Since the vacuum and the electro-vacuum equations in the stationary axisymmetric case are completely integrable, large classes of solutions can be constructed with techniques from the theory of solitons. The richest class of physically interesting solutions to the pure vacuum case due to Korotkin is given in terms of hyperelliptic theta functions. Harrison transformed hyperelliptic solutions are discussed. As a concrete example we study the transformation of a family of counter-rotating dust disks. To obtain algebro-geometric solutions with vanishing total charge which are of astrophysical relevance, three-sheeted surfaces have to be considered. The matter in the disk is discussed following Bičák et al. We review the 'cut and glue' technique where a strip is removed from an explicitly known spacetime and where the remainder is glued together after displacement. The discontinuities of the normal derivatives of the metric at the glueing hypersurface lead to infinite disks. If the energy conditions are satisfied and if the pressure is positive, the disks can be interpreted in the vacuum case as made up of two components of counter-rotating dust moving on geodesics. In electro-vacuum the condition of geodesic movement is replaced by electro-geodesic movement. As an example we discuss a class of Harrison-transformed hyperelliptic solutions. The range of parameters is identified where an interpretation of the matter in the disk in terms of electro-dust can be given.

show abstract

Section: P E1mentioning

confidence: 99%

Section: Metric and Harrison Transformationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On explicit solutions to the stationary axisymmetric Einstein-Maxwell equations describing dust disks

Klein¹

2003

Ann. Phys.

Self Cite

View full text Add to dashboard Cite

show abstract

“…We can also give the metric function a (see [11] and [13]) and k (see [14], and references therein) explicitly. Notice that solutions of the form (6) are characterized by one real valued function G and two real numbers, e.g., a and b defined by E 2 1 : a 1 ib where b has to be positive.…”

mentioning

confidence: 98%

“…Introducing the standard quantities associated with a Riemann surface (for the notation and the cut system, see [13]), one can show that…”

mentioning

confidence: 99%

Exact Relativistic Gravitational Field of a Stationary Counterrotating Dust Disk

Klein

Richter²

1999

Phys. Rev. Lett.

Self Cite

View full text Add to dashboard Cite

We present a solution to the Ernst equation which represents an infinitesimally thin dust disk consisting of two streams of particles circulating with constant angular velocity in opposite directions. These streams have the same density distribution but their relative density may vary continuously. In the limit of only one component of dust, we get the solution for the rigidly rotating dust disk of Neugebauer and Meinel; in the limit of identical densities, the static disk of Morgan and Morgan is obtained. We discuss the Newtonian and the ultrarelativistic limit, the occurrence of ergospheres, and the regularity of the solution.PACS numbers: 04.20.JbIn astrophysics thin disks of collisionless matter, socalled dust, are discussed as models for certain galaxies (see, e.g., [1]) or for accretion disks. A fully relativistic treatment of these models is necessary if a black hole is present since black holes are genuinely relativistic objects. But the exact treatment of dust disks without a central object would provide deep insight both in the mathematical structure of the field equations and in the physics of rapidly rotating relativistic bodies, since dust disks can be viewed as a limiting case for extended matter sources. Corresponding exact solutions hold globallyin the vacuum and in the matter region-and can thus provide physically realistic test beds for numerical codes. Since Newtonian dust disks are known to be unstable and since there are hints by numerical work (see, e.g., [2]) that the same holds in the relativistic case, such solutions could be taken as exact initial data for numerical collapse calculations. Whereas the Newtonian theory of such disks is well established (see [1], and references therein), the same holds in the relativistic case only for static disks which can be interpreted as consisting of two counterrotating streams of matter with vanishing total angular momentum. The first disk of this type was considered by Morgan and Morgan [3]. Infinitely extended dust disks with finite mass were studied by Bičák, and Katz [4] in the static case and by Bičák and Ledvinka [5] in the stationary case. The first to construct the exact solution for a finite stationary dust disk were Neugebauer and Meinel [6] who gave the solution for the rigidly rotating dust disk which was first treated numerically by Bardeen and Wagoner [2]. They solved the corresponding boundary value problem for the Einstein equations with the help of a corotating coordinate system.In this Letter we present a class of new disk solutions to the Ernst equation where such a coordinate system cannot be used. The disks of finite radius r 0 consist of two counterrotating components of dust with respective density s 6 ͑r͒. The angular velocity of both streams of particles is of the same constant absolute value V but of different sign. The relative density g ͑s 1 2 s 2 ͒͑͞s 1 1 s 2 ͒ is a constant with respect to r and z which varies between one, the rigidly rotating dust disk [6], and zero, the static Morgan and Morgan disk [3]. Interesting...

show abstract

On explicit solutions to the stationary axisymmetric Einstein‐Maxwell equations describing dust disks

Klein

2003

Annalen der Physik

Self Cite

View full text Add to dashboard Cite

We review explicit solutions to the stationary axisymmetric Einstein‐Maxwell equations which can be interpreted as disks of charged dust. The disks of finite or infinite extension are infinitesimally thin and constitute a surface layer at the boundary of an electro‐vacuum. The Einstein‐Maxwell equations in the presence of one Killing vector are obtained by using a projection formalism. This leads to equations for three‐dimensional gravity where the matter is given by a SU(2,1)/S[U(1,1)× U(1)] nonlinear sigma model. The SU(2,1) invariance of the stationary Einstein‐Maxwell equations can be used to construct solutions for the electro‐vacuum from solutions to the pure vacuum case via a so‐called Harrison transformation. It is shown that the corresponding solutions will always have a non‐vanishing total charge and a gyromagnetic ratio of 2. Since the vacuum and the electro‐vacuum equations in the stationary axisymmetric case are completely integrable, large classes of solutions can be constructed with techniques from the theory of solitons. The richest class of physically interesting solutions to the pure vacuum case due to Korotkin is given in terms of hyperelliptic theta functions. Harrison transformed hyperelliptic solutions are discussed. As a concrete example we study the transformation of a family of counter‐rotating dust disks. To obtain algebro‐geometric solutions with vanishing total charge which are of astrophysical relevance, three‐sheeted surfaces have to be considered. The matter in the disk is discussed following Bičák et al. We review the ‘cut and glue’ technique where a strip is removed from an explicitly known spacetime and where the remainder is glued together after displacement. The discontinuities of the normal derivatives of the metric at the glueing hypersurface lead to infinite disks. If the energy conditions are satisfied and if the pressure is positive, the disks can be interpreted in the vacuum case as made up of two components of counter‐rotating dust moving on geodesics. In electro‐vacuum the condition of geodesic movement is replaced by electro‐geodesic movement. As an example we discuss a class of Harrison‐transformed hyperelliptic solutions. The range of parameters is identified where an interpretation of the matter in the disk in terms of electro‐dust can be given.

show abstract

Physically realistic solutions to the Ernst equation on hyperelliptic Riemann surfaces

Cited by 24 publications

References 35 publications

On explicit solutions to the stationary axisymmetric Einstein-Maxwell equations describing dust disks

On explicit solutions to the stationary axisymmetric Einstein-Maxwell equations describing dust disks

Exact Relativistic Gravitational Field of a Stationary Counterrotating Dust Disk

On explicit solutions to the stationary axisymmetric Einstein‐Maxwell equations describing dust disks

Contact Info

Product

Resources

About