Exact Solutions of Einstein Field Equation in Locally Conformally Flat Manifolds

We consider the pseudo-Euclidean space (R-n, g), n >= 3, with coordinates x = (x(1), . . ., x(n)) and metric components g(ij) = delta(ij)epsilon(i), 1 <= i, j <= n, where epsilon(i) = +/- 1, with at least one epsilon(i) = 1 and one diagonal (0,2)-tensors of the form T = Sigma(i) epsilon(i)h(i)(x)dx(i)(2). We obtain necessary and sufficient conditions for the existence of a metric (g) over bar, conformal to g, such that Ric((g) over bar) - (K) over bar /2 (g) over bar = T, where Ric((g) over bar) and (K) over bar are the Ricci tensor and scalar curvature of the metric (g) over bar, respectively. Using the results obtained, we construct an example of a static perfect fluid spacetime. Similar problems are considered for locally conformally flat manifolds.

Automated summary

This paper examines the existence conditions of metrics in pseudo-Euclidean spaces with specific metric components and tensor forms, and constructs an example of a static perfect fluid spacetime using the obtained results. Similar problems are also considered for locally conformally flat manifolds.