Hartle's slow rotation formalism is developed in the presence of a
cosmological constant. We find the generalisation of the Hartle-Thorne vacuum
metric, the Hartle-Thorne-(anti)-de Sitter metric, and find that it is always
asymptotically (anti)-de Sitter. Next we consider Wahlquist's rotating perfect
fluid interior solution in Hartle's formalism and discuss its matching to the
Hartle-Thorne-(anti)-de Sitter metric. It is known that the Wahlquist solution
cannot be matched to an asymptotically flat region and therefore does not
provide a model of an isolated rotating body in this context. However, in the
presence of a cosmological term, we find that it can be matched to an
asymptotic (anti)-de Sitter space and we are able to interpret the Wahlquist
solution as a model of an isolated rotating body, to second order in the
angular velocity.
The linear solution for quadrupolar perturbations around de Sitter spacetime was recently constructed. In this paper, we provide the flux-balance laws for each background symmetry (dilatations, rotations, spatial translations and cosmological boosts) in terms of source moments at quadrupolar order. We write the dilatation flux-balance law in two distinct ways, which allows to contrast two distinct proposals for the negative definite energy flux. The standard Poincar\'e flux balance laws at future null infinity are recovered in the flat limit of the SO(1,4) flux-balance laws.
The asymptotic structure of the gravitational field of isolated systems has been analyzed in great detail in the case when the cosmological constant Λ is zero. The resulting framework lies at the foundation of research in diverse areas in gravitational science. Examples include: i) positive energy theorems in geometric analysis; ii) the coordinate invariant characterization of gravitational waves in full, non-linear gen-eral relativity; iii) computations of the energy-momentum emission in gravitational collapse and binary mergers in numerical relativity and relativistic astrophysics; and iv) constructions of asymptotic Hilbert spaces to calculate S-matrices and analyze the issue of information loss in the quantum evaporation of black holes. However, by now observations have established that Λ is positive in our universe. In this pa-per we show that, unfortunately, the standard framework does not extend from the Λ = 0 case to the Λ > 0 case in a physically useful manner. In particular, we do not have positive energy theorems, nor an invariant notion of gravitational waves in the non-linear regime, nor asymptotic Hilbert spaces in dynamical situations of semi-classical gravity. A suitable framework to address these conceptual issues of direct physical importance is developed in subsequent papers.
The second order perturbative field equations for slowly and rigidly rotating perfect fluid balls of Petrov type D are solved numerically. It is found that all the slowly and rigidly rotating perfect fluid balls up to second order, irrespective of Petrov type, may be matched to a possibly nonasymptotically flat stationary axisymmetric vacuum exterior. The Petrov type D interior solutions are characterized by five integration constants, corresponding to density and pressure of the zeroth order configuration, the magnitude of the vorticity, one more second order constant, and an independent spherically symmetric second order small perturbation of the central pressure. A four-dimensional subspace of this five-dimensional parameter space is identified for which the solutions can be matched to an asymptotically flat exterior vacuum region. Hence these solutions are completely determined by the spherical configuration and the magnitude of the vorticity. The physical properties, like equation of state, shape, and speed of sound, are determined for a number of solutions.
We use analytic perturbation theory to present a new approximate metric for a
rigidly rotating perfect fluid source with equation of state (EOS)
. This EOS includes the interesting cases of
strange matter and Wahlquist's metric. It is fully matched to its approximate
asymptotically flat exterior using Lichnerowicz junction conditions and it is
shown to be a totally general matching using Darmois-Israel conditions and
properties of the harmonic coordinates. Then we analyse the Petrov type of the
interior metric and show, first, that in the case corresponding to Wahlquist's
metric it can not be matched to the asymptotically flat exterior. Next, that
this kind of interior can only be of Petrov types I, D or (in the static case)
O and also that the non-static constant density case can only be of type I.
Finally, we check that it can not be a source of Kerr's metric.
The aim of this book is to introduce the reader to research work on a particular aspect of rotating fields in general relativity. The account begins with a short introduction to the relevant aspects of general relativity, written at a level accessible to a beginning graduate student in theoretical physics. There follows a detailed derivation of the Wehl-Lewis-Papapetrou form of the stationary axially symmetric metric. The Kerr and Tomimatsu-Sato forms of the rotating interior and exterior solutions of the Einstein equations are then discussed. The last three chapters of the book illustrate the applications of the theory to rotating neutral dust, rotating Einstein-Maxwell fields, and rotating charged dust. The author has drawn on his own research work to produce a timely discussion of this important area of research.
This is the first in a series of papers on the construction of explicit solutions to the stationary axisymmetric Einstein equations which describe counterrotating 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 an explicit solution for a family of disks with constant angular velocity and constant relative energy density which was announced in a previous Letter.
Assuming that the Kerr-Newman metric is the field of a layer of mass and charge distributed over the equatorial disk spanning the ring singularity, the source distribution on the disk is computed explicitly. In the uncharged case, this interpretation automatically excises the noncausal parts of the manifold, so that one obtains the unique source of the causally maximal extension of the vacuum metric. A Newtonian field which gives the same source distribution is exhibited, and shown to be closely analogous to the relativistic case. In particular, the Newtonian particle orbits show the same avoidance of the ring singularity that is such a remarkable feature of geodesics in the Kerr geometry. In the charged case, we examine how the gyromagnetic moment (which is equal to that of the Dirac electron) is reflected in the character of the source distribution.
We present an exhaustive integration of the type D vacuum and electrovac field equations with cosmological constant admitting a nonsingular aligned Maxwell field satisfying the generalized Goldberg–Sachs theorem. We derive a single expression for the general solution from which one may obtain all particular cases known until now in partial versions. We also investigate in detail the separability properties of the Hamilton–Jacobi equation for the charged particle orbits and of the Klein–Gordon equation for a massive charged spin‐zero test particle and their corresponding Killing tensors.
![Parameter values are λ=-0.008\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda = -0.008$$\end{document} which gives xbh≈1.9793\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_\mathrm{bh } \approx 1.9793$$\end{document}; initial conditions are xi=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_i = 2$$\end{document}, K2(xi)={10-2,10-3,10-4}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_2(x_i)=\{10^{-2},10^{-3},10^{-4}\}$$\end{document}, H2(xi)=10-3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_2(x_i)=10^{-3}$$\end{document}. The dashed lines on the right panel show the asymptotes of -H2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-H_2$$\end{document}](https://www.researchgate.net/profile/Christian-Boehmer-2/publication/268525720/figure/fig1/AS:961488344862728@1606248093974/Parameter-values-are-l-0008documentclass12ptminimal-usepackageamsmath_Q320.jpg)
![Parameter values are λ=1/18\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda = 1/18$$\end{document} which gives xbh≈2.1962\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_\mathrm{bh}\approx 2.1962$$\end{document} and xc=6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_\mathrm{c} = 6$$\end{document} with the following initial conditions: left panelsi=-10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_i = -10$$\end{document}, K2(si)={1,e2,e4}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_2(s_i)=\{1,e^2,e^4\}$$\end{document}, K2′(si)=-1/50\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_2'(s_i)=-1/50$$\end{document}; right panelsi=-10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_i=-10$$\end{document}, H2(si)={1,e4,e8}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_2(s_i)=\{1,e^4,e^8\}$$\end{document}, H2′(si)=-1/50\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_2'(s_i)=-1/50$$\end{document}. The dashed vertical line indicates the location of the cosmological horizon, the dotted lines on the right panel show the asymptotes of H2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_2$$\end{document}. We note that K2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_2$$\end{document} is regular everywhere, also at the horizon, while H2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_2$$\end{document} diverges at the horizon. K2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_2$$\end{document} changes its sign which explains the kink in the log–log plot](https://www.researchgate.net/profile/Christian-Boehmer-2/publication/268525720/figure/fig2/AS:961488349036546@1606248094098/Parameter-values-are-l1-18documentclass12ptminimal-usepackageamsmath_Q320.jpg)
![The result of numerically matching the functions h2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_2$$\end{document} and k2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_2$$\end{document} with parameter choices μ0=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _0=1$$\end{document}, κ=0.5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa =0.5$$\end{document}, r0=0.01\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_0=0.01$$\end{document} and Λ=-0.1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda =-0.1$$\end{document}. The dashed line indicates the radius of the zero pressure surface at r=2.65\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r=2.65$$\end{document}. As can be seen from the left figure, the matching conditions only require these functions to be continuous on the matching surface](https://www.researchgate.net/profile/Christian-Boehmer-2/publication/268525720/figure/fig3/AS:961488349057024@1606248094266/The-result-of-numerically-matching-the-functions-h2documentclass12ptminimal_Q320.jpg)
