• Communications of the Korean Mathematical Society
• /
• v.39 no.1
• /
• pp.201-210
• /
• 2024

This paper is concerned with the study of spacetimes satisfying div 𝓜 = 0, where "div" denotes the divergence and 𝓜 is the m-projective curvature tensor. We establish that a perfect fluid spacetime with div 𝓜 = 0 is a generalized Robertson-Walker spacetime and vorticity free; whereas a four-dimensional perfect fluid spacetime becomes a Robertson-Walker spacetime. Moreover, we establish that a Ricci recurrent spacetime with div 𝓜 = 0 represents a generalized Robertson-Walker spacetime.

• Proceedings of the Korea Contents Association Conference
• /
• 2003.05a
• /
• pp.470-473
• /
• 2003

We have the index form of the multiply warped products manifold. From this result we will investigate the physical properties on the Robertson-Walker spacetime. The singularly thorem are usually interpreted as an indication that black hole exist and that there has been a big bang. This paper include necessary condition for the Robertson-Walker spacetime with singularly.

• Communications of the Korean Mathematical Society
• /
• v.37 no.3
• /
• pp.893-904
• /
• 2022

In this note, we apply a maximum principle related to volume growth of a complete noncompact Riemannian manifold, which was recently obtained by Alías, Caminha and do Nascimento in [4], to establish new uniqueness and nonexistence results concerning maximal spacelike hypersurfaces immersed in a generalized Robertson-Walker (GRW) spacetime obeying the timelike convergence condition. A study of entire solutions for the maximal hypersurface equation in GRW spacetimes is also made and, in particular, a new Calabi-Bernstein type result is presented.

• Communications of the Korean Mathematical Society
• /
• v.30 no.2
• /
• pp.109-121
• /
• 2015

In this paper, we investigate horizontal lightlike hypersurfaces of Robertson-Walker spacetimes. Some results involving the unique existence of the screen distribution and the symmetry of the induced Ricci curvature tensor of horizontal lightlike hypersurfaces are presented. We also obtain some properties concerning the symmetry and the parallelism of the second fundamental forms of such lightlike hypersurfaces.

• Kyungpook Mathematical Journal
• /
• v.60 no.3
• /
• pp.571-584
• /
• 2020

The object of the present paper is to characterize generalized Ricci recurrent (GR₄) spacetimes. Among others things, it is proved that a conformally flat GR₄ spacetime is a perfect fluid spacetime. We also prove that a GR₄ spacetime with a Codazzi type Ricci tensor is a generalized Robertson Walker spacetime with Einstein fiber. We further show that in a GR₄ spacetime with constant scalar curvature the energy momentum tensor is semisymmetric. Further, we obtain several corollaries. Finally, we cite some examples which are sufficient to demonstrate that the GR₄ spacetime is non-empty and a GR₄ spacetime is not a trivial case.

• Kyungpook Mathematical Journal
• /
• v.58 no.4
• /
• pp.761-779
• /
• 2018

The object of the present paper is to study weakly Z symmetric spacetimes $(WZS)_4$. At first we prove that a weakly Z symmetric spacetime is a quasi-Einstein spacetime and hence a perfect fluid spacetime. Next, we consider conformally flat $(WZS)_4$ spacetimes and prove that such a spacetime is infinitesimally spatially isotropic relative to the unit timelike vector field ${\rho}$. We also study $(WZS)_4$ spacetimes with divergence free conformal curvature tensor. Moreover, we characterize dust fluid and viscous fluid $(WZS)_4$ spacetimes. Finally, we construct an example of a $(WZS)_4$ spacetime.

• Publications of The Korean Astronomical Society
• /
• v.19 no.1
• /
• pp.1-10
• /
• 2004

On the framework of a linearly perturbed Friedmann-Robertson-Walker spacetime, we derive an expression for the cosmological angular diameter distance affected by scalar and tensor perturbations. Our expression is applicable in linear order to distances in general FRW models. We study the effect of a stocastic gravitaional wave background on the two-point correlation function of the angular diameter distance fluctuations and, on the basis of this we also derive an expression for the power spectrum of the angular diameter distance fluctuations.

• Journal of the Korean Mathematical Society
• /
• v.48 no.4
• /
• pp.669-689
• /
• 2011

The notion of quasi-Einstein manifolds arose during the study of exact solutions of the Einstein field equations as well as during considerations of quasi-umbilical hypersurfaces. For instance, the Robertson-Walker spacetimes are quasi-Einstein manifolds. The object of the present paper is to study Lorentzian quasi-Einstein manifolds. Some basic geometric properties of such a manifold are obtained. The applications of Lorentzian quasi-Einstein manifolds to the general relativity and cosmology are investigated. Theories of gravitational collapse and models of Supernova explosions [5] are based on a relativistic fluid model for the star. In the theories of galaxy formation, relativistic fluid models have been used in order to describe the evolution of perturbations of the baryon and radiation components of the cosmic medium [32]. Theories of the structure and stability of neutron stars assume that the medium can be treated as a relativistic perfectly conducting magneto fluid. Theories of relativistic stars (which would be models for supermassive stars) are also based on relativistic fluid models. The problem of accretion onto a neutron star or a black hole is usually set in the framework of relativistic fluid models. Among others it is shown that a quasi-Einstein spacetime represents perfect fluid spacetime model in cosmology and consequently such a spacetime determines the final phase in the evolution of the universe. Finally the existence of such manifolds is ensured by several examples constructed from various well known geometric structures.