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Register a new userElectrovacuum Static Counterrotating Relativistic Dust Disks
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Added by
Gonzalo Garc\\'ia-Reyes
Publication date
2008
fields
Physics
and research's language is
English
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A detailed study is presented of the counterrotating model (CRM) for generic electrovacuum static axially symmetric relativistic thin disks without radial pressure. We find a general constraint over the counterrotating tangential velocities needed to cast the surface energy-momentum tensor of the disk as the superposition of two counterrotating charged dust fluids. We also find explicit expressions for the energy densities, charge densities and velocities of the counterrotating fluids. We then show that this constraint can be satisfied if we take the two counterrotating streams as circulating along electro-geodesics. However, we show that, in general, it is not possible to take the two counterrotating fluids as circulating along electro-geodesics nor take the two counterrotating tangential velocities as equal and opposite. Four simple families of models of counterrotating charged disks based on Chazy-Curzon-like, Zipoy-Voorhees-like, Bonnor-Sackfield-like and Kerr-like electrovacuum solutions are considered where we obtain some disks with a CRM well behaved. The models are constructed using the well-known ``displace, cut and reflect method extended to solutions of vacuum Einstein-Maxwell equations.
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The interpretation of some electrovacuum spacetimes in terms of counterrotating perfect fluid discs is presented. The interpretation is mades by means of an inverse problem approach used to obtain disc sources of known static solutions of the Einstein-Maxwell equations. In order to do such interpretation, a detailed study is presented of the counterrotating model (CRM) for generic electrovacuum static axially symmetric relativistic thin discs with nonzero radial pressure. Four simple families of models of counterrotating charged discs based on Chazy-Curzon-type, Zipoy-Voorhees-type, Bonnor-Sackfield-type, and charged and magnetized Darmois electrovacuum metrics are considered where we obtain some discs with a CRM well behaved.
An infinite family of new exact solutions of the Einstein vacuum equations for static and axially symmetric spacetimes is presented. All the metric functions of the solutions are explicitly computed and the obtained expressions are simply written in terms of oblate spheroidal coordinates. Furthermore, the solutions are asymptotically flat and regular everywhere, as it is shown by computing all the curvature scalars. These solutions describe an infinite family of thin dust disks with a central inner edge, whose energy densities are everywhere positive and well behaved, in such a way that their energy-momentum tensor are in fully agreement with all the energy conditions. Now, although the disks are of infinite extension, all of them have finite mass. The superposition of the first member of this family with a Schwarzschild black hole was presented previously [G. A. Gonzalez and A. C. Gutierrez-Pi~neres, arXiv: 0811.3002v1 (2008)], whereas that in a subsequent paper a detailed analysis of the corresponding superposition for the full family will be presented.
The first fully integrated explicit exact solution of the Einstein field equations corresponding to the superposition of a counterrotating dust disk with a central black hole is presented. The obtained solution represents an infinite annular thin disk (a disk with an inner edge) around the Schwarszchild black hole. The mass of the disk is finite and the energy-momentum tensor agrees with all the energy conditions. Furthermore, the total mass of the disk when the black hole is present is less than the total mass of the disk alone. The solution can also be interpreted as describing a thin disk made of two counterrotanting dust fluids that are also in agreement with all the energy conditions. Additionally, as we will show shortly in a subsequent paper, the above solution is the first one of an infinite family of solutions.
In this paper is discussed a class of static spherically symmetric solutions of the general relativistic elasticity equations. The main point of discussion is the comparison of two matter models given in terms of their stored energy functionals, i.e., the rule which gives the amount of energy stored in the system when it is deformed. Both functionals mimic (and for small deformations approximate) the classical Kirchhoff-St.Venant materials but differ in the strain variable used. We discuss the behavior of the systems for large deformations.
An algorithm presented by K. Lake to obtain all static spherically symmetric perfect fluid solutions was recently extended by L. Herrera to the interesting case of locally anisotropic fluids (principal stresses unequal). In this work we develop an algorithm to construct all static spherically symmetric anisotropic solutions for general relativistic polytropes. Again the formalism requires the knowledge of only one function (instead of two) to generate all possible solutions. To illustrate the method some known cases are recovered.
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