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.