On a spacetime $(M,g)$ endowed with a density function h, we consider the vacuum weighted Einstein field equations: \[hρ-Hes_h+Δ h g=0.\] First, it is shown that the equation characterizes critical metrics for an appropriate action. Then, after describing locally conformally flat solutions in arbitrary dimension, four-dimensional solutions with harmonic curvature are classified.
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Brozos‐Vázquez et al. (2024) studied this question.
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