In this paper, we discuss general relativistic, self‐gravitating and uniformly rotating perfect fluid bodies with a toroidal topology (without central object). For the equations of state describing the fluid matter, we consider polytropic as well as completely degenerate, perfect Fermi gas models. We find that the corresponding configurations possess similar properties to the homogeneous relativistic Dyson rings. On one hand, there exists no limit to the mass for a given maximal mass–density inside the body. On the other hand, each model permits a quasi‐stationary transition to the extreme Kerr black hole.
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.
In this paper, uniformly rotating relativistic rings are investigated analytically utilizing two different approximations simultaneously: (1) an expansion about the thin-ring limit (the crosssection is small compared with the size of the whole ring) and (2) post-Newtonian expansions. The analytic results for rings are compared with numerical solutions.
A Roche model for describing uniformly rotating rings is presented, and the results are compared with the 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 analytical terms, even in the mass-shedding case.
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