Abstract
We consider the pseudo-Euclidean space \((\mathbb {R}^n,g)\), \(n \ge 3\), with coordinates \(x=\left( x_1,...,x_n\right) \) and metric components \(g_{ij} = \delta _{ij}\epsilon _i\), \(1\le i, j\le n\), where \(\varepsilon _i=\pm 1\), with at least one \(\varepsilon _i=1\) and one diagonal (0,2)-tensors of the form \(T=\sum _i\epsilon _i{h_i(x)dx_i^2}\). We obtain necessary and sufficient conditions for the existence of a metric \(\bar{g}\), conformal to g, such that \(\hbox {Ric}_{\bar{g}}-\displaystyle \frac{\bar{K}}{2} \bar{g} =T\), where \(\hbox {Ric}_{\bar{g}}\) and \(\bar{K}\) are the Ricci tensor and scalar curvature of the metric \(\bar{g}\), respectively. Using the results obtained, we construct an example of a static perfect fluid spacetime. Similar problems are considered for locally conformally flat manifolds.
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The authors would like to thank the referee for his careful reading, relevant remarks, and valuable suggestion.
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Adriano, L.R., Menezes, I.F.d., Pieterzack, M.D. et al. Exact Solutions of Einstein Field Equation in Locally Conformally Flat Manifolds. Results Math 76, 175 (2021). https://doi.org/10.1007/s00025-021-01476-5
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DOI: https://doi.org/10.1007/s00025-021-01476-5