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The Horizon Quantum of Area

  Several independent semiclassical arguments converge on $$\boxed{\Delta A=8\pi l_{\mathrm{Pl}}^2}, \qquad l_{\mathrm{Pl}}^2=\frac{G\hbar}{c^3}$$ I got interested in this reading the 2018 paper  Bekenstein, I, and the quantum of black-hole surface area . Even though I disagree with Hod, its a great paper.  Honourable mentions also go to:   The Physical Interpretation of the Spectrum of Black Hole Quasinormal Modes , 2008  The Off-Shell Black Hole , 1994     On Black Hole Spectroscopy via Adiabatic Invariance , 2012 An Analytical Computation of Asymptotic Schwarzschild Quasinormal Frequencies , 2003  Asymptotic Black Hole Quasinormal Frequencies , 2003  1. Horizon Action Near any nonextremal horizon, imaginary time turns the normal geometry into a plane: $$ds_E^2\simeq d\rho^2+\rho^2d\Theta^2+\cdots.$$ Smoothness requires one horizon cycle: $$\oint d\Theta=2\pi.$$ The gravitational action identifies the momentum conjugate to $\The...
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What If Dark Gravity Is Just Entropy Winning?

 

A Minimal Entropic Re-Derivation of MOND-like Gravity

Motivation

Verlinde (2016) proposed that de Sitter space contains a volume-scaling entropy associated with dark energy. Matter removes part of this entropy. Beyond a critical scale, the remaining volume entropy dominates and produces an apparent dark gravitational component. Peach (2019) reformulated the idea using an entropic-screen construction. 

The argument below combines these ideas and fixes the force normalisation using Verlinde's spherical response coefficient.

Competing Entropies

For a sphere of radius $r$, the de Sitter entropy is

$$S_{\rm DE}(r) = \frac{r}{L}\frac{A(r)c^3}{4G\hbar}, \qquad A(r)=4\pi r^2,$$

while a central mass $M$ removes

$$S_M(r)=\frac{2\pi Mc}{\hbar}r.$$

Thus $S_{\rm DE}\propto r^3$ and $S_M\propto r$. Define $a_\Lambda=c^2/L$. Their ratio is

$$\frac{S_{\rm DE}}{S_M} = \frac{a_\Lambda r^2}{2GM} = \left(\frac{r}{r_c}\right)^2,$$

where

$$\boxed{r_c=\sqrt{\frac{2GM}{a_\Lambda}}}.$$

At this radius, $g_B(r_c)=a_\Lambda/2$. For $r>r_c$, matter no longer removes all the available volume entropy.

Information Response

Peach's screen construction gives a bulk response growing as $N_{\rm bulk}/N_{\rm screen} = r/r_c$. The entropy equations show that this scaling can be written as

$$\boxed{ \frac{N_{\rm bulk}}{N_{\rm screen}} = \sqrt{\frac{S_{\rm DE}}{S_M}} }.$$

This is the response amplitude associated with the entropy competition, rather than the direct ratio of volume entropy to surface information.

The entropy comparison fixes the scaling but not the complete force normalization. For the isotropic spherical response used by Verlinde in three spatial dimensions,

$$\boxed{ \left(\frac{g_D}{g_B}\right)^2 = \frac{1}{3}\frac{S_{\rm DE}}{S_M} }.$$

The factor $1/3$ is the geometric coefficient supplied by the tensorial spherical-response calculation. It is the factor absent from Peach's scalar force argument.

Dark Acceleration

Using $g_B=GM/r^2$ and the entropy ratio,

$$\frac{g_D^2}{g_B^2} = \frac{1}{3}\frac{a_\Lambda r^2}{2GM} = \frac{a_\Lambda}{6g_B}.$$

Therefore,

$$\boxed{g_D^2=\frac{a_\Lambda}{6}g_B}.$$

Defining the MOND acceleration scale $a_M=a_\Lambda/6$, we obtain

$$\boxed{g_D^2=a_Mg_B}, \qquad \boxed{g_D=\frac{\sqrt{GMa_M}}{r}}.$$

The coefficient separates naturally as

$$\frac{a_M}{a_\Lambda} = \underbrace{\frac{1}{2}}{\text{entropy comparison}} \underbrace{\frac{1}{3}}{\text{spherical response}}$$

This is Verlinde's point-mass result.

Immediate Consequences

For a circular orbit in the regime $g_D\gg g_B$, $v_f^2/r\simeq g_D$, so

$$\boxed{v_f^4=GMa_M}.$$

This is the baryonic Tully–Fisher relation.

Writing the same acceleration as $g_D=GM_D(r)/r^2$ gives

$$\boxed{ M_D(r)=r\sqrt{\frac{a_MM}{G}} }.$$

Outside a compact baryonic source, where $M$ is constant,

$$\rho_D(r) = \frac{1}{4\pi r^2}\frac{dM_D}{dr} = \boxed{ \frac{1}{4\pi r^2}\sqrt{\frac{a_MM}{G}} } \propto\frac{1}{r^2}.$$

This produces approximately flat rotation curves.

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