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We derive a family of Post-Newtonian (PN) Dedekind ellipsoids to first order. They describe non-axially symmetric, homogeneous, and rotating figures of equilibrium. The sequence of the Newtonian Dedekind ellipsoids allows for an axially symmetric limit in which a uniformly rotating Maclaurin spheroid is recovered. However, the approach taken by Chandrasekhar & Elbert (1974) to find the PN Dedekind ellipsoids excludes such a limit. In G\"urlebeck & Petroff (2010), we considered an extension to their work that permits a limit of 1 PN Maclaurin ellipsoids. Here we further detail the sequence and demonstrate that a choice of parameters exists with which the singularity formerly found in Chandrasekhar & Elbert (1974) along the sequence of PN Dedekind ellipsoids is removed., Comment: 23 pages, 17 figures; changes in v2: first paragraph in introduction added, minor changes to agree with published version

We consider the post-Newtonian approximation for the Dedekind ellipsoids in the case of axisymmetry. The approach taken by Chandrasekhar & Elbert (1974, 1978) excludes the possibility of finding a uniformly rotating (deformed) spheroid in the axially symmetric limit, though the solution exists at the point of axisymmetry. We consider an extension to their work that permits the possibility of such a limit., Comment: 12 pages, v1: more discussion of comparison to Chandrasekhar's and Elbert's work, modified to agree with published version

In this paper uniformly rotating relativistic rings are investigated analytically utilizing two different approximations simultaneously: (1) an expansion about the thin ring limit (the cross-section is small compared with the size of the whole ring) (2) post-Newtonian expansions. The analytic results for rings are compared with numerical solutions., Comment: 12 pages, 7 figures, v1: 2 tables added, agrees with published version

We use surface adapted ellipsoidal coordinates to obtain a first order post-Newtonian (PN) approximation to Dedekind ellipsoids with the intention of proceeding to higher orders., Comment: 3 pages, no figures, contribution to the 12th Marcel Grossmann Meeting (Paris, July 12 - 18, 2009), submitted to the Conference Proceedings, v2: minor changes

A Roche model for describing uniformly rotating rings is presented and the results are compared with numerical solutions to the full problem for polytropic rings. In the thin ring limit, the surfaces of constant pressure including the surface of the ring itself are given in analytic terms, even in the mass-shedding case., Comment: 6 pages, 4 figures, v2: minor corrections

An analytical method is presented for treating the problem of a uniformly rotating, self-gravitating ring without a central body in Newtonian gravity. The method is based on an expansion about the thin ring limit, where the cross-section of the ring tends to a circle. The iterative scheme developed here is applied to homogeneous rings up to the 20th order and to polytropes with the index n=1 up to the third order. For other polytropic indices no analytic solutions are obtainable, but one can apply the method numerically. However, it is possible to derive a simple formula relating mass to the integrated pressure to leading order without specifying the equation of state. Our results are compared with those generated by highly accurate numerical methods to test their accuracy., Comment: 18 pages, 12 figures. This paper is essentially a combined version of arXiv:0802.0078 and arXiv:0802.0081 and was published in this form. (v2: minor corrections)

An iterative method is presented for solving the problem of a uniformly rotating, self-gravitating ring without a central body in Newtonian gravity by expanding about the thin ring limit. Using this method, a simple formula relating mass to the integrated pressure is derived to the leading order for a general equation of state. For polytropes with the index n=1, analytic coefficients of the iterative approach are determined up to the third order. Analogous coefficients are computed numerically for other polytropes. Our solutions are compared with those generated by highly accurate numerical methods to test their accuracy., Comment: 10 pages, 8 figures, minor corrections, references added. This paper has be combined with arXiv:0802.0078 and published in MNRAS, see arXiv:0808.2612

In this paper, we describe an analytical method for treating uniformly rotating homogeneous rings without a central body in Newtonian gravity. We employ series expansions about the thin ring limit and use the fact that in this limit the cross-section of the ring tends to a circle. The coefficients can in principle be determined up to an arbitrary order. Results are presented here to the 20th order and compared with numerical results., Comment: 10 pages, 5 figures, minor corrections. This paper has be combined with arXiv:0802.0081 and published in MNRAS, see arXiv:0808.2612

A highly accurate multi-domain spectral method is used to study axially symmetric and stationary spacetimes containing a black hole or disc of dust surrounded by a ring of matter. It is shown that the matter ring can affect the properties of the central object drastically. In particular, by virtue of the ring's frame dragging, the so-called Komar mass of the black hole or disc can become negative. A continuous transition from such discs to such black holes can be found., Comment: 3 pages, submitted to the Proceedings of the MGXI Meeting

The scheme developed by Hartle for describing slowly rotating bodies in 1967 was applied to the simple model of constant density by Chandrasekhar and Miller in 1974. The pivotal equation one has to solve turns out to be one of Heun's equations. After a brief discussion of this equation and the chances of finding a closed form solution, a quickly converging series solution of it is presented. A comparison with numerical solutions of the full Einstein equations allows one to truncate the series at an order appropriate to the slow rotation approximation. The truncated solution is then used to provide explicit expressions for the metric., Comment: 16 pages, uses document class iopart, v2: minor corrections

To investigate the imprint on the gravitational-wave emission from extreme mass-ratio inspirals in non-pure Kerr spacetimes, we have studied the ``kludge'' waveforms generated in highly-accurate, numerically-generated spacetimes containing a black hole and a self-gravitating, homogeneous torus with comparable mass and spin. In order to maximize their impact on the produced waveforms, we have considered tori that are compact, massive and close to the central black hole, investigating under what conditions the LISA experiment could detect their presence. Our results show that for a large portion of the space of parameters the waveforms produced by EMRIs in these black hole-torus systems are indistinguishable from pure-Kerr waveforms. Hence, a ``confusion problem'' will be present for observations carried out over a timescale below or comparable to the dephasing time., Comment: 21 pages, 12 figures. Short discussion on the accuracy of the spacetimes added, typos corrected. Accepted for publication in PRD

We study two types of axially symmetric, stationary and asymptotically flat spacetimes using highly accurate numerical methods. The one type contains a black hole surrounded by a perfect fluid ring and the other a rigidly rotating disc of dust surrounded by such a ring. Both types of spacetime are regular everywhere (outside of the horizon in the case of the black hole) and fulfil the requirements of the positive energy theorem. However, it is shown that both the black hole and the disc can have negative Komar mass. Furthermore, there exists a continuous transition from discs to black holes even when their Komar masses are negative., Comment: 7 pages, 2 figures, document class iopart. v2: changes made (including title) to coincide with published version

It is shown numerically that strange matter rings permit a continuous transition to the extreme Kerr black hole. The multipoles as defined by Geroch and Hansen are studied and suggest a universal behaviour for bodies approaching the extreme Kerr solution parametrically. The appearance of a `throat region', a distinctive feature of the extreme Kerr spacetime, is observed. With regard to stability, we verify for a large class of rings, that a particle sitting on the surface of the ring never has enough energy to escape to infinity along a geodesic., Comment: 16 pages, 11 figures, v3: minor changes so as to coincide with published version

A highly accurate computer program is used to study axially symmetric and stationary spacetimes containing a Black Hole surrounded by a ring of matter. It is shown that the matter ring affects the properties of the Black Hole drastically. In particular, the absolute value of the ratio of the Black Hole's angular momentum to the square of its mass not only exceeds one, but can be greater than ten thousand (|J|/M^2 > 10^4). Indeed, the numerical evidence suggests that this quantity is unbounded., Comment: 3 pages, 2 figures, v2: minor changes, one reference added

Highly accurate numerical solutions to the problem of Black Holes surrounded by uniformly rotating rings in axially symmetric, stationary spacetimes are presented. The numerical methods developed to handle the problem are discussed in some detail. Related Newtonian problems are described and numerical results provided, which show that configurations can reach an inner mass-shedding limit as the mass of the central object increases. Exemplary results for the full relativistic problem for rings of constant density are given and the deformation of the event horizon due to the presence of the ring is demonstrated. Finally, we provide an example of a system for which the angular momentum of the central Black Hole divided by the square of its mass exceeds one., Comment: 12 pages, 14 figures, revtex, v4: minor changes, Eq. (17) corrected, corresponds to version in PRD

In this paper, we develop an iterative scheme to enable the explicit calculation of an arbitrary post-Newtonian order for a relativistic body that reduces to the Maclaurin spheroid in the appropriate limit. This scheme allows for an analysis of the structure of the solution in the vicinity of bifurcation points along the Maclaurin sequence. The post-Newtonian expansion is solved explicitly to the fourth order and its accuracy and convergence are studied by comparing it to highly accurate numerical results., Comment: 31 pages, 3 figures