<p>We investigate the quasi-local thermodynamics of rotating Kerr-AdS black holes enclosed by a finite timelike boundary. By employing the Brown-York stress tensor, we define a holographic pressure <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {P}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">P</mi> </math></EquationSource> </InlineEquation> and its conjugate area <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation> at a finite cutoff. We demonstrate that the inclusion of angular momentum introduces a momentum flux at the boundary, requiring a generalized first law of the form <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$dE = T_{\textrm{loc}}dS + \Omega _{\textrm{loc}}dJ - \mathcal {P}d\mathcal {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mi>E</mi> <mo>=</mo> <msub> <mi>T</mi> <mtext>loc</mtext> </msub> <mi>d</mi> <mi>S</mi> <mo>+</mo> <msub> <mi mathvariant="normal">Ω</mi> <mtext>loc</mtext> </msub> <mi>d</mi> <mi>J</mi> <mo>-</mo> <mi mathvariant="script">P</mi> <mi>d</mi> <mi mathvariant="script">A</mi> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Omega _{\textrm{loc}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Ω</mi> <mtext>loc</mtext> </msub> </math></EquationSource> </InlineEquation> accounts for frame-dragging effects. A central result of our study is the analysis of the extensivity parameter <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\eta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>η</mi> </math></EquationSource> </InlineEquation>. We show that while small rotating black holes exhibit non-extensive behavior due to gravitational self-interactions, extensivity is recovered in the large-size limit (<InlineEquation ID="IEq6"> <EquationSource Format="TEX">$r_{+} \gg \ell $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>r</mi> <mo>+</mo> </msub> <mo>≫</mo> <mi>ℓ</mi> </mrow> </math></EquationSource> </InlineEquation>), where the system satisfies the Euler relation. These findings provide robust evidence for the fluid-gravity correspondence at finite cutoff and offer a new perspective on the thermodynamic structure of rotating holographic duals.</p>

Holographic Pressure and Extensivity of Rotating Black Holes at Finite Cutoff

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发表日期：

2026年5月21日

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Hoang Van Quyet

摘要

We investigate the quasi-local thermodynamics of rotating Kerr-AdS black holes enclosed by a finite timelike boundary. By employing the Brown-York stress tensor, we define a holographic pressure $\mathcal {P}$ $P$ and its conjugate area $\mathcal {A}$ $A$ at a finite cutoff. We demonstrate that the inclusion of angular momentum introduces a momentum flux at the boundary, requiring a generalized first law of the form $dE = T_{\textrm{loc}}dS + \Omega _{\textrm{loc}}dJ - \mathcal {P}d\mathcal {A}$ $d E = T_{loc} d S + Ω_{loc} d J - P d A$ , where $\Omega _{\textrm{loc}}$ $Ω_{loc}$ accounts for frame-dragging effects. A central result of our study is the analysis of the extensivity parameter $\eta $ $η$ . We show that while small rotating black holes exhibit non-extensive behavior due to gravitational self-interactions, extensivity is recovered in the large-size limit ( $r_{+} \gg \ell $ $r_{+} ≫ ℓ$ ), where the system satisfies the Euler relation. These findings provide robust evidence for the fluid-gravity correspondence at finite cutoff and offer a new perspective on the thermodynamic structure of rotating holographic duals.