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<p>We study the spherically symmetric gravitational collapse of the Einstein–scalar field system with a positive cosmological constant under double-null coordinates (<i>u</i>,&#xa0;<i>v</i>). The spacelike singularities arise when the area radius <i>r</i> vanishes. We derive new quantitative estimates, and obtain polynomial blow-up rates <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_2937_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="63" /> </InlineMediaObject> <EquationSource Format="TEX">\(O(1/r^N)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">/</mo> <msup> <mi>r</mi> <mi>N</mi> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for various quantities. These estimates also apply to the hard-phase model of the spherically symmetric Einstein-Euler system and give quantitative blow-up upper bounds of fluid velocity and density. Near timelike infinity, we link the precise blow-up rates of the Kretschmann scalar to the exponential Price’s law along the event horizon. In cosmological settings, this further reveals the mass-inflation phenomena along spacelike singularities.</p>

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Quantitative blow-up estimates for spacelike singularities in gravitational-collapse cosmological spacetimes

  • 期刊论文

  • 发表日期：

    2025年2月6日

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  • Xinliang An,
  • Haoyang Chen,
  • Taoran He

摘要

We study the spherically symmetric gravitational collapse of the Einstein–scalar field system with a positive cosmological constant under double-null coordinates (u, v). The spacelike singularities arise when the area radius r vanishes. We derive new quantitative estimates, and obtain polynomial blow-up rates \(O(1/r^N)\) O ( 1 / r N ) for various quantities. These estimates also apply to the hard-phase model of the spherically symmetric Einstein-Euler system and give quantitative blow-up upper bounds of fluid velocity and density. Near timelike infinity, we link the precise blow-up rates of the Kretschmann scalar to the exponential Price’s law along the event horizon. In cosmological settings, this further reveals the mass-inflation phenomena along spacelike singularities.