Abstract
A novel analytical approach is used to achieve an exact solution of the Bianchi type- III spacetime in the context of \(f(R,T)\) gravity. Two functional forms are considered: \(f(R,T)=R+2f_1(T)=R+2\mu T\) and \(f(R,T)=f_1(R)+f_2(T)=R+\alpha R^2+\mu T\) . The solution is achieved assuming a barotropic fluid, which actually produces a vacuum energy. In mathematical terms, it is equivalent to the cosmological constant \(\Lambda\) . The resulting spacetime is found to be anisotropic and homogeneous. In addition, the stability of the model is examined, and numerous physical properties are discussed in detail for the case of \(f(R,T)=R+2\mu T\) . In contrast, the nonlinear case, \(f(R,T)=R+\alpha R^2+\mu T\) , reduces to a self-consistent vacuum solution characterized by \(A_1=\pm t\) , \(A_2=t\) , and \(A_3=1\) . In this configuration, the modified field equations reduce to those of general relativity in vacuum, demonstrating that the higher-order curvature term \(\alpha R^2\) and the matter coupling term \(\mu T\) do not influence the cosmic dynamics.
