Conformal Solutions of Static Plane Symmetric Cosmological Models in Cases of a Perfect Fluid and a Cosmic String Cloud
Ragab M. Gad, Awatif Al-Jedani, Shahad T. Alsulami
TL;DR
This work investigates static plane-symmetric cosmological spacetimes that admit conformal motions, solving Einstein's equations for two matter configurations: a perfect fluid with a dark-energy-like equation of state and a cosmic string cloud. By enforcing conformal symmetry, the metric is constrained to satisfy $A=B$ and $\psi = A' e^{A}$, and the resulting spacetimes are shown to be nonsingular, conformally flat, and, in each case, reduce to anti-De Sitter geometry. For the perfect fluid, the solution yields $\rho=-p=3m^2$ (i.e., $\rho+p=0$); for the cosmic string cloud, the densities decay along the string direction with $\mu=0$ and $\lambda=3/((\tfrac{3}{2}x+c_4)^2)$, and the metric acquires a similar AdS-like form. The authors also analyze conformal vector fields, finding an orthogonal CKV that exists in these models while proving there is no CKV parallel to the four-velocity, and they discuss physical properties such as acceleration and rotation of the resulting spacetimes.
Abstract
In this work, we obtained exact solutions of Einstein's field equations for plane symmetric cosmological models by assuming that thy admit conformal motion. The space-time geometry of these solutions is found to be nonsingular, non-vacuum and conformally flat. We have shown that in the case of a perfect fluid, these solutions have an energy-momentum tensor possessing dark energy with negative pressure and the energy equation of state is $ρ+p=0$. We have shown that a fluid has acceleration, rotation, shear-free, vanishing expansion, and rotation. In the case of a cosmic string cloud, we found that the tension density and particle density decrease as the fluid moves along the direction of the strings, then vanish at infinity. We shown that the exact conformal solution for a static plane symmetric model reduces to the the well-known anti-De Sitter space time. We obtained that the space-time under consideration admits a conformal vector field orthogonal to the four-velocity vector and does not admits a vector parallel to the four-velocity vector. Some physical and kinematic properties of the resulting models are also discussed.