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What If Dark Gravity Is Just Entropy Winning?

  A Minimal Entropic 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_\...
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What If Dark Gravity Is Just Entropy Winning?

 

A Minimal Entropic 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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