Axisymmetric Einstein Equations Research Articles

Fully constrained, high-resolution shock-capturing, formulation of the Einstein-fluid equations in 2+1 dimensions

Four components of the axisymmetric Einstein equations in $2+1$ dimensions with negative cosmological constant can be written as ${\ensuremath{\nabla}}_{a}M=\dots{}$ and ${\ensuremath{\nabla}}_{a}J=\dots{}$, where the dots stand for stress-energy terms, and $M$ and $J$ are scalars. In vacuum, they reduce to the constant mass and angular momentum parameters of the BTZ solution of the same name. The integrability conditions for the Einstein equations give rise to two conserved stress-energy currents ${\ensuremath{\nabla}}_{a}{j}_{(M)}^{a}=0$ and ${\ensuremath{\nabla}}_{a}{j}_{(J)}^{a}=0$. The angular momentum current is just the Noether current due to axisymmetry, but the mass current is unexpected in the presence of rotation. The conserved quantity $M$ exists in all dimensions in spherical symmetry, known as the Misner-Sharp, Hawking or Kodama mass, but in $2+1$ dimensions $M$ exists also in axisymmetry, even with rotation. We use $M$ and $J$ to give a fully constrained formulation of the axisymmetric Einstein equations in $2+1$ dimensions, where the Einstein equations are solved by explicit integration from the center along time slices. We use the two conserved matter currents in the construction of a high-resolution shock-capturing formulation of the Einstein-perfect fluid system, in which $M$ and $J$ momentum are then exactly conserved by construction. We demonstrate convergence of the code in the test cases of generic dispersion and collapse and stable and unstable rotating stars.

Exact Solution of a Neumann Boundary Value Problem for the Stationary Axisymmetric Einstein Equations

For a stationary and axisymmetric spacetime, the vacuum Einstein field equations reduce to a single nonlinear PDE in two dimensions called the Ernst equation. By solving this equation with a Dirichlet boundary condition imposed along the disk, Neugebauer and Meinel in the 1990s famously derived an explicit expression for the spacetime metric corresponding to the Bardeen–Wagoner uniformly rotating disk of dust. In this paper, we consider a similar boundary value problem for a rotating disk in which a Neumann boundary condition is imposed along the disk instead of a Dirichlet condition. Using the integrable structure of the Ernst equation, we are able to reduce the problem to a Riemann–Hilbert problem on a genus one Riemann surface. By solving this Riemann–Hilbert problem in terms of theta functions, we obtain an explicit expression for the Ernst potential. Finally, a Riemann surface degeneration argument leads to an expression for the associated spacetime metric.

Static axisymmetric Einstein equations in vacuum: Symmetry, new solutions, and Ricci solitons

An explicit one-parameter Lie point symmetry of the four-dimensional vacuum Einstein equations with two commuting hypersurface-orthogonal Killing vector fields is presented. The parameter takes values over all of the real line and the action of the group can be effected algebraically on any solution of the system. This enables one to construct particular one-parameter extended families of axisymmetric static solutions and cylindrical gravitational wave solutions from old ones, in a simpler way than most solution-generation techniques, including the prescription given by Ernst for this system. As examples, we obtain the families that generalize the Schwarzschild solution and the $C$-metric. These in effect superpose a Levi-Civita cylindrical solution on the seeds. Exploiting a correspondence between static solutions of Einstein's equations and Ricci solitons (self-similar solutions of the Ricci flow), this also enables us to construct new steady Ricci solitons.

Boundary Value Problems for the Stationary Axisymmetric Einstein Equations: A Disk Rotating Around a Black Hole

We solve a class of boundary value problems for the stationary axisymmetric Einstein equations corresponding to a disk of dust rotating uniformly around a central black hole. The solutions are given explicitly in terms of theta functions on a family of hyperelliptic Riemann surfaces of genus 4. In the absence of a disk, they reduce to the Kerr black hole. In the absence of a black hole, they reduce to the Neugebauer-Meinel disk.

Linear Perturbations for the Vacuum Axisymmetric Einstein Equations

In axial symmetry, there is a gauge for Einstein equations such that the total mass of the spacetime can be written as a conserved, positive definite, integral on the spacelike slices. This property is expected to play an important role in the global evolution. In this gauge the equations reduce to a coupled hyperbolic–elliptic system which is formally singular at the axis. Due to the rather peculiar properties of the system, the local in time existence has proved to resist analysis by standard methods. To analyze the principal part of the equations, which may represent the main source of the difficulties, we study linear perturbation around the flat Minkowski solution in this gauge. In this article we solve this linearized system explicitly in terms of integral transformations in a remarkable simple form. This representation is well suited to obtain useful estimates to apply in the non-linear case.

Boundary-value problems for the stationary axisymmetric Einstein equations: a rotating disc

The stationary, axisymmetric reduction of the vacuum Einstein equations, the so-called Ernst equation, is an integrable nonlinear PDE in two dimensions. There now exists a general method for analysing boundary-value problems (BVPs) for integrable PDEs, and this method consists of two steps: (a) Construct an integral representation of the solution characterized via a matrix Riemann–Hilbert (RH) problem formulated in the complex k-plane, where k denotes the spectral parameter of the associated Lax pair. This representation involves, in general, some unknown boundary values, thus the solution formula is not yet effective. (b) Characterize the unknown boundary values by analysing a certain equation called the global relation. This analysis involves, in general, the solution of a nonlinear problem; however, for certain BVPs called linearizable, it is possible to determine the unknown boundary values using only linear operations. Here, we employ the above methodology for the investigation of certain BVPs for the elliptic version of the Ernst equation. For this problem, the main novelty is the occurrence of the spectral parameter in the form of a square root and this necessitates the introduction of a two-sheeted Riemann surface for the formulation of the relevant RH problem. As a concrete application of the general formalism, it is shown that the particular BVP corresponding to the physically significant case of a rotating disc is a linearizable BVP. In this way the remarkable results of Neugebauer and Meinel are recovered.

Well-posedness, linear perturbations, and mass conservation for the axisymmetric Einstein equations

For axially symmetric solutions of Einstein equations there exists a gauge which has the remarkable property that the total mass can be written as a conserved, positive definite, integral on the spacelike slices. The mass integral provides a nonlinear control of the variables along the whole evolution. In this gauge, Einstein equations reduce to a coupled hyperbolic-elliptic system which is formally singular at the axis. As a first step in analyzing this system of equations we study linear perturbations on a flat background. We prove that the linear equations reduce to a very simple system of equations which provide, though the mass formula, useful insight into the structure of the full system. However, the singular behavior of the coefficients at the axis makes the study of this linear system difficult from the analytical point of view. In order to understand the behavior of the solutions, we study the numerical evolution of them. We provide strong numerical evidence that the system is well-posed and that its solutions have the expected behavior. Finally, this linear system allows us to formulate a model problem which is physically interesting in itself, since it is connected with the linear stability of black hole solutions in axial symmetry. This model can contribute significantly to solve the nonlinear problem and at the same time it appears to be tractable.

Harrison transformation of hyperelliptic solutions and charged dust disks

We use a Harrison transformation on solutions to the stationary axisymmetric Einstein equations to generate solutions of the Einstein-Maxwell equations. The case of hyperelliptic solutions to the Ernst equation is studied in detail. Analytic expressions for the metric and the multipole moments are obtained. As an example we consider the transformation of a family of counter-rotating dust disks. The resulting solutions can be interpreted as disks with currents and matter with a purely azimuthal pressure or as two streams of freely moving charged particles. We discuss interesting limiting cases as the extreme limit where the charge becomes identical to the mass, and the ultrarelativistic limit where the central redshift diverges.

Ernst equation, Fay identities and variational formulas on hyperelliptic curves

We present a unified approach to theta-functional solutions of the stationary axisymmetric Einstein equations in vacuum. Using Fay's trisecant identity and variational formulas on hyperelliptic Riemann surfaces, we establish formulas for the metric functions, the Ernst potential and their derivatives.

Exact relativistic treatment of stationary counterrotating dust disks: Physical properties

This is the third in a series of papers on the construction of explicit solutions to the stationary axisymmetric Einstein equations which can be interpreted as counterrotating disks of dust. We discuss the physical properties of a class of solutions to the Einstein equations for disks with constant angular velocity and constant relative density which was constructed in the first part. The metric for these spacetimes is given in terms of theta functions on a Riemann surface of genus 2. It is parametrized by two physical parameters: the central redshift and the relative density of the two counterrotating streams in the disk. We discuss the dependence of the metric on these parameters using a combination of analytical and numerical methods. Interesting limiting cases are the Maclaurin disk in the Newtonian limit, the static limit which gives a solution of the Morgan and Morgan class and the limit of a disk without counterrotation. We study the mass and the angular momentum of the spacetime. At the disk we discuss the energy-momentum tensor, i.e., the angular velocities of the dust streams and the energy density of the disk. The solutions have ergospheres in strongly relativistic situations. The ultrarelativistic limit of the solution in which the central redshift diverges is discussed in detail: In the case of two counterrotating dust components in the disk, the solutions describe a disk with diverging central density but finite mass. In the case of a disk made up of one component, the exterior of the disks can be interpreted as the extreme Kerr solution.

Exact relativistic treatment of stationary counterrotating dust disks: Boundary value problems and solutions

This is the first in a series of papers on the construction of explicit solutions to the stationary axisymmetric Einstein equations which describe counter-rotating disks of dust. These disks can serve as models for certain galaxies and accretion disks in astrophysics. We review the Newtonian theory for disks using Riemann-Hilbert methods which can be extended to some extent to the relativistic case where they lead to modular functions on Riemann surfaces. In the case of compact surfaces these are Korotkin's finite gap solutions which we will discuss in this paper. On the axis we establish for general genus relations between the metric functions and hence the multipoles which are enforced by the underlying hyperelliptic Riemann surface. Generalizing these results to the whole spacetime we are able in principle to study the classes of boundary value problems which can be solved on a given Riemann surface. We investigate the cases of genus 1 and 2 of the Riemann surface in detail and construct the explicit solution for a family of disks with constant angular velocity and constant relative energy density which was announced in a previous Physical Review Letter.

Differentially Rotating Disks of Dust

We present a three-parameter family of solutions to the stationary axisymmetric Einstein equations that describe differentially rotating disks of dust. They have been constructed by generalizing the Neugebauer—Meinel solution of the problem of a rigidly rotating disk of dust. The solutions correspond to disks with angular velocities depending monotonically on the radial coordinate; both decreasing and increasing behaviour is exhibited. In general, the solutions are related mathematically to Jacobi's inversion problem and can be expressed in terms of Riemann theta functions. A particularly interesting two-parameter subfamily represents Bäcklund transformations to appropriate seed solutions of the Weyl class.

Exact Relativistic Gravitational Field of a Stationary Counterrotating Dust Disk

Disks of collisionless particles are important models for certain galaxies and accretion disks in astrophysics. We present here a solution to the stationary axisymmetric Einstein equations 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 previously given by 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.

Riemann-Hilbert problems for the Ernst equation and fibre bundles

Riemann-Hilbert techniques are used in the theory of completely integrable differential equations to generate solutions that contain a free function which can be used at least in principle to solve initial or boundary-value problems. The solution of a boundary-value problem is thus reduced to the identification of the jump data of the Riemann-Hilbert problem from the boundary data. But even if this can be achieved, it is very difficult to get explicit solutions since the matrix Riemann-Hilbert problem is equivalent to an integral equation. In the case of the Ernst equation (the stationary axisymmetric Einstein equations in vacuum), it was shown in a previous work that the matrix problem is gauge equivalent to a scalar problem on a Riemann surface. If the jump data of the original problem are rational functions, this surface will be compact which makes it possible to give explicit solutions in terms of hyperelliptic theta functions. In the present work, we discuss Riemann-Hilbert problems on Riemann surfaces in the framework of fibre bundles. This makes it possible to treat the compact and the non-compact case in the same setting and to apply general existence theorems.

Physically realistic solutions to the Ernst equation on hyperelliptic Riemann surfaces

We show that the class of hyperelliptic solutions to the Ernst equation (the stationary axisymmetric Einstein equations in vacuum) previously discovered by Korotkin and Meinel and Neugebauer can be derived via Riemann-Hilbert techniques. The present paper extends the discussion of the physical properties of these solutions that was begun in a previous Letter and supplies complete proofs. We identify a physically interesting subclass where the Ernst potential is everywhere regular except at a closed surface which might be identified with the surface of a body of revolution. The corresponding spacetimes are asymptotically flat and equatorially symmetric. This suggests that they could describe the exterior of an isolated body, for instance, a relativistic star or a galaxy. Within this class, one has the freedom to specify a real function and a set of complex parameters which can possibly be used to solve certain boundary value problems for the Ernst equation. The solutions can have ergoregions, a Minkowskian limit, and an ultrarelativistic limit where the metric approaches the extreme Kerr solution. We give explicit formulas for the potential on the axis and in the equatorial plane where the expressions simplify. Special attention is paid to the simplest nonstatic solutions (which are of genus 2) to which the rigidly rotating dust disk belongs.

On the Dirichlet Problem for Stationary and Axisymmetric Einstein Equations

In suitable coordinates Einstein's field equations for a rigidly rotating perfect fluid in equilibrium can be written as a semilinear system of purely elliptic partial differential equations of second order. Therefore, the formulation of a boundary value problem is appropriate in this situation. It is shown that the Dirichlet problem for the vacuum region outside a ball, and for a ball inside the matter region, has a unique regular solution if the boundary data are in a characteristic way limited by the “diameter” of the ball. This restriction seems to be closely connected with stability limits for rotating stars. Furthermore, the used mathematical methods are directly related to a numerical solution technique for such physical systems.

The Ernst equation and the Riemann-Hilbert problem on hyperelliptic Riemann surfaces

We discuss a class of solutions to the Ernst equation (the stationary axisymmetric Einstein equations) obtained as solutions of a generalized scalar Riemann-Hilbert problem on a hyperelliptic Riemann surface. The singular structure of these solutions is studied for arbitrary genus of the Riemann surface. A subclass is given for which the Ernst potential is everywhere regular besides at a contour that can be identified with the surface of a body of revolution. It turns out that the recently discussed rigidly rotating dust disk belongs to this class.

The Boundary Value Problem for the Stationary and Axisymmetric Einstein Equations is Generically Solvable.

We prove existence, uniqueness, and regularity for Dirichlet solutions of the stationary and axisymmetric Einstein equations in vacuum and in rigidly rotating ideal fluid matter for data (on a ball) whose absolute values are in a characteristic way limited by the ``diameter'' of the ball. These results have important consequences for the existence of exterior vacuum and interior matter solutions for rotating stars. The mathematical procedures and results have remarkable connections with a numerical solution technique for rotating stars.

Ehlers transformations and string effective action

We explicitly obtain the generalization of the Ehlers transformation for stationary axisymmetric Einstein equations to string theory. This is accomplished by finding the twist potential corresponding to the moduli fields in the effective two dimensional theory. Twist potential and symmetric moduli are shown to transform under an O( d, d) which is a manifest symmetry of the equations of motion. The non-trivial action of this O( d, d) is given by the Ehlers transformation and belongs to the set O(d)×O(d) O(d) .

Elliptic solutions of stationary axisymmetric Einstein equations

In this paper we present a more transparent version of an earlier construction of genus g algebraic-geometric solutions of the Einstein equation. For one of the metric coefficients we obtain a new expression that allows us to construct the coefficient in terms of derivatives of the function Psi ( lambda ) (solution of associative linear system). Finally, we proceed with an analysis of the two simplest genus 1 (elliptic) solutions.