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<p>On a spacetime (<i>M</i>,&#xa0;<i>g</i>) endowed with a density function <i>h</i>, we consider the vacuum weighted Einstein field equations: <Equation ID="Equ40"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3747_Article_Equ40.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="166" /> </MediaObject> <EquationSource Format="TEX">\(\begin{aligned} h\rho -{{\,\mathrm{\mathrm Hes}\,}}_h+\Delta h g=0. \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>h</mi> <mi>ρ</mi> <mo>-</mo> <msub> <mrow> <mspace width="0.166667em" /> <mrow> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">es</mi> </mrow> <mspace width="0.166667em" /> </mrow> <mi>h</mi> </msub> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>h</mi> <mi>g</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>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.</p>

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The vacuum weighted Einstein field equations

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年5月3日

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  • M. Brozos-Vázquez,
  • D. Mojón-Álvarez

摘要

On a spacetime (M, g) endowed with a density function h, we consider the vacuum weighted Einstein field equations: \(\begin{aligned} h\rho -{{\,\mathrm{\mathrm Hes}\,}}_h+\Delta h g=0. \end{aligned}\) h ρ - H es 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.