Relativistic Magneto-fluid Spacetime and Conformal $\eta$-Ricci-Yamabe solitons

Abstract

The object of the research article is to investigate the geometric properties of a relativistic magneto-fluid spacetime if its metrics admits conformal $\eta$-Ricci-Yamabe soliton and gradient conformal $\eta$-Ricci-Yamabe soliton of type $(\kappa,l)$. A $\mathcal{C}^{\dag}}$-flat relativistic magneto-fluid spacetime, $\mathcal{P}_1$-flat relativistic magneto-fluid spacetime and $\mathrm{T}^{c}$-flat relativistic magneto-fluid spacetime filled with a magneto-fluid density $\rho$, magnetic field strength  $\widetilde{\mathcal{H}}$, and magnetic permeability $\alpha$ obeys the Einstein field equation with a concircular vector field are also investigated. Moreover, we illustrates some physical significance of conformal pressure $\tilde{\Omega}$ in view of a conformal $\eta$-Ricci-Yamabe soliton of type $(\kappa,l)$ along with concircular vector field on a relativistic magneto-fluid spacetime. Within this ongoing work, using such soliton, we analyze the various energy conditions, some black holes criteria, and Penrose’s singularity theorem on a relativistic magneto-fluid spacetime. In addition of this, we further investigate the generalized Liouville and Poisson equations associated with such soliton on a relativistic magneto-fluid spacetime Moreover, in the context of a relativistic magneto-fluid spacetime, we explore the harmonic aspects of conformal $\eta$-Ricci-Yamabe soliton of type $(\kappa,l)$ on such a relativistic magneto-fluid spacetime. Finally, we discuss some results on different types of magneto-fluid spacetimes coupled with such type of a soliton.