Black Hole from Entropy Maximization
Abstract: One quantum characterization of a black hole motivated by (local) holography and thermodynamics is that it maximizes thermodynamic entropy for a given surface area. In the context of quantum gravity, this could be more fundamental than the classical characterization by a horizon. As a step, we explore this possibility by solving the 4D semi-classical Einstein equation with many matter fields. For highly-excited spherically-symmetric static configurations, we apply local typicality and estimate the entropy including self-gravity to derive its upper bound. The saturation condition uniquely determines the entropy-maximized configuration: self-gravitating quanta condensate into a radially-uniform dense configuration with no horizon, where the self-gravity and a large quantum pressure induced by the curvatures are balanced and no singularity appears. The interior metric is a self-consistent and non-perturbative solution in Planck's constant. The maximum entropy, given by the volume integral of the entropy density, agrees with the Bekenstein-Hawking formula through self-gravity, deriving the Bousso bound for thermodynamic entropy. Finally, 10 future prospects are discussed, leading to a speculative view that the configuration represents a quantum-gravitational condensate in a semi-classical manner.
- See (45) for the relation of s𝑠sitalic_s and sμsuperscript𝑠𝜇s^{\mu}italic_s start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT.
- Y. Yokokura, [arXiv:2207.14274 [hep-th]].
- This also means that we don’t only consider static configurations where Tolman’s law holds Francesco . See also Sec.V.1.
- The proportionality to the area in (24) comes from ⟨−Tt⟩t=∂ra*(r)8πGr2∝1Gr2\langle-T^{t}{}_{t}\rangle=\frac{\partial_{r}a_{*}(r)}{8\pi Gr^{2}}\propto% \frac{1}{Gr^{2}}⟨ - italic_T start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_FLOATSUBSCRIPT italic_t end_FLOATSUBSCRIPT ⟩ = divide start_ARG ∂ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT * end_POSTSUBSCRIPT ( italic_r ) end_ARG start_ARG 8 italic_π italic_G italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ∝ divide start_ARG 1 end_ARG start_ARG italic_G italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG. Such an energy density also appears in Roberto .
- More precisely, we have RRicciScalar=−2L2+2r2subscript𝑅𝑅𝑖𝑐𝑐𝑖𝑆𝑐𝑎𝑙𝑎𝑟2superscript𝐿22superscript𝑟2R_{RicciScalar}=-\frac{2}{L^{2}}+\frac{2}{r^{2}}italic_R start_POSTSUBSCRIPT italic_R italic_i italic_c italic_c italic_i italic_S italic_c italic_a italic_l italic_a italic_r end_POSTSUBSCRIPT = - divide start_ARG 2 end_ARG start_ARG italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG + divide start_ARG 2 end_ARG start_ARG italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG. See KY4 ; KY5 for an explicit proof of AdS2×S2𝐴𝑑subscript𝑆2superscript𝑆2AdS_{2}\times S^{2}italic_A italic_d italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT × italic_S start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT.
- This non-global equilibrium makes the difference from Oppenheim .
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.