Radiating-collapsing models satisfying Karmarkar condition
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Regular Article - Theoretical Physics
Radiating-collapsing models satisfying Karmarkar condition Suresh C. Jaryala Department of Physics and Astronomical Science, Central University of Himachal Pradesh (CUHP), Dharamshala, Kangra, HP 176215, India
Received: 15 May 2020 / Accepted: 16 July 2020 © The Author(s) 2020
Abstract This paper presents a class of exact spherical symmetric solutions of the Einstein equations admitting heatconducting anisotropic fluid as a collapsing matter. The exterior spacetime is assumed to be the Vaidya metric. This class of solutions is shown to satisfy all the energy conditions throughout the interior of the star, and the luminosity is time independent, radiating uniformly throughout the collapse.
1 Introduction There has been extensive research in the field of gravitational collapse. Since the pioneering work on the gravitational collapse of homogeneous dust, [1,2], it is now accepted that for a gravitational collapse of homogeneous pressureless matter, the central singularity remains hidden behind the horizon, implying that the end state of the continual gravitational collapse of homogeneous dust cloud must be a black hole [3]. Further studies have examined various aspects of gravitationally collapsing stellar systems for different kinds of matter distributions, and details may be found in [4–7,11]. These studies have thrown light to many interesting facts which must hold for the collapse processes to be physically realistic. For example, for the continuous and smooth matching of the interior collapsed spacetime to the exterior Vaidya spacetime over the timelike hypersurface , the radial pressure must not vanish at the boundary of the collapsing radiant star, but instead be proportional to the heat flux [13,14]. In order to have a physically well behaved model of a gravitation collapse with generic energy momentum tensors, one not only needs to find physically consistent analytical solutions of the Einstein field equations, but also must ensure validity of the energy conditions as well. The practice usually followed are: to specify the spacetime symmetry, or the gravitational potentials, imposing an equation of state, or restricting the matter content to find the solutions of the gravitational a e-mail:
collapse. However, there exists a class I condition, which is useful to obtain classes of solutions. This condition arise from some well known geometric theorems as follows: First, an (n + 1)-dimensional space V n+1 can be embedded into a pseudo Euclidean space E n+2 of dimension (n + 2) [15], and that all the spherically symmetric spacetime are in general of class II. Next, the necessary and sufficient condition for any Riemannian space to be embedding class I is that it satisfies the Karmarkar condition [16,17]. Thus the Karmarkar condition is a useful condition which gives new solutions. Recently, there has been a renewed interest in obtaining solutions using these conditions, [18–20]. The study of the non-static radiating metric with timelike Karmarkar condition, when the tem
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