Axially symmetric relativistic thin disks immersed in spheroidal matter haloes

Spanish Relativity Meeting: almost 100 years after Einstein's revolution

González, Guillermo A.^1 ; Pimentel, Oscar M.^1

Escuela de Física, Universidad Industrial de Santander, A. A. 678, Bucaramanga

68002, Colombia^1

关键词: Axially symmetric;    Cylindrical coordinates;    Einstein equations;    Energy-momentum tensor;    Finite discontinuity;    Geodesic equations;    Oblate spheroidal coordinates;    Particular solution;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/600/1/012015/pdf
DOI  :  10.1088/1742-6596/600/1/012015

来源: IOP

【 摘 要 】

Two infinite families of axially symmetric relativistic thin disks of dust immersed in spheroidal matter haloes are presented. The disks are obtained from solutions of the Einstein equations for an axially symmetric conformastaticspacetimein which the metric tensor is characterised only by one metric function. By introducing a finite discontinuity on the first derivatives of the metric tensor, solutions with a singularity of the delta function type are obtained, so describing thin disks. The nonzero components of the energy-momentum tensor, both for the disk and the halo, are obtained from the Einstein equations. In this way, the energy densities and pressures of the sources are determined. By imposing the fulfilment of all the energy conditions we obtain a constraint over the solutions, in such a way that the metric function can be properly expressed in terms of a solution of the Laplace equation. By using the solution to Laplace equation in cylindrical coordinates we find infinite disks and by using the solution to Laplace equation in oblate spheroidal coordinates we find finite disks. In both cases we obtain particular solutions with energy densities and pressures well behaved everywhere. We also show that the masses of de disks and the haloes are finite. Finally we solve the geodesic equation for circular orbits in the plane of the disk.