Investigating string cloud spacetimes with energy-momentum tensor constraints in general relativity

The objective of this study was to investigate the behavior of relativistic G(QE) ${}_n$ string cloud spacetime endowed with various forms of the string cloud energy-momentum tensor $\mathcal{T}$, which includes the string cloud fluid density $\alpha$ and string tension $\beta$. A G(QE) ${}_n$ string cloud spacetime with a covariantly constant energy-momentum tensor satisfies the equation of state $\frac{\alpha }{\beta }=-1$, and represents either a massive string cloud spacetime or a bulk viscous fluid spacetime. Moreover, when a string cloud spacetime is coupled with a covariantly constant energy-momentum tensor, the equation of state $\frac{\alpha }{\beta }=-1$ coincides with the state equation for a cloud of geometric strings or represents the quintessence era. G(QE) ${}_n$ string cloud spacetimes with Codazzi-type and pseudo-symmetric energy-momentum tensors were investigated. Furthermore, we characterized $\mathcal{T}$-recurrent, weakly $\mathcal{T}$-symmetric, specially weakly $\mathcal{T}$-symmetric, generalized $\mathcal{T}$-recurrent, semi-generalized $\mathcal{T}$-recurrent, and quadratic Killing-type energy-momentum tensors on a string cloud spacetime with a Killing velocity vector field, and concluded that the string cloud spacetime either represents a massive string cloud spacetime or corresponds to the quintessence era.