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<p>We construct geometric microstates for a class of two-dimensional flow geometries — spacetimes that interpolate from an asymptotic AdS<sub>2</sub> boundary to a dS<sub>2</sub> static patch in the interior — by inserting particles behind the horizon. We show that this mechanism produces dS microstates with an Einstein–Rosen bridge of infinite length behind the horizon. The state-counting of these microstates, including wormhole contributions, reproduces the Gibbons-Hawking entropy, <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math display="inline"> <msub> <mi>S</mi> <mi>dS</mi> </msub> <mo>=</mo> <msubsup> <mi>A</mi> <mtext>horizon</mtext> <mi>dS</mi> </msubsup> <mo>/</mo> <mn>4</mn> <mi>G</mi> </math></EquationSource> <EquationSource Format="TEX">\( {S}_{\mathrm{dS}}={A}_{\mathrm{horizon}}^{\mathrm{dS}}/4G \)</EquationSource> </InlineEquation>. Furthermore, we extend the microstate-counting method to the case of a finite-length Einstein–Rosen bridge. As a result, the Hilbert space of the dS horizon in the flow geometry can be spanned by states with a purely dS Einstein–Rosen bridge, containing no AdS portion on the time-symmetric slice. This provides a concrete realization of dS microstates within a controlled holographic framework.</p>

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Flow-geometry microstates

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年6月3日

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  • Ricardo Espíndola,
  • Shoichiro Miyashita

摘要

We construct geometric microstates for a class of two-dimensional flow geometries — spacetimes that interpolate from an asymptotic AdS2 boundary to a dS2 static patch in the interior — by inserting particles behind the horizon. We show that this mechanism produces dS microstates with an Einstein–Rosen bridge of infinite length behind the horizon. The state-counting of these microstates, including wormhole contributions, reproduces the Gibbons-Hawking entropy, S dS = A horizon dS / 4 G \( {S}_{\mathrm{dS}}={A}_{\mathrm{horizon}}^{\mathrm{dS}}/4G \) . Furthermore, we extend the microstate-counting method to the case of a finite-length Einstein–Rosen bridge. As a result, the Hilbert space of the dS horizon in the flow geometry can be spanned by states with a purely dS Einstein–Rosen bridge, containing no AdS portion on the time-symmetric slice. This provides a concrete realization of dS microstates within a controlled holographic framework.