Verlinde ( https://arxiv.org/abs/1611.02269 ) tries to deduce MOND from emergent gravity. Can you help?
Emergent or entropic gravity goes back to Jacobson. He starts with
- the entropy-area connection $S= A\,{kc^3}/{4G\hbar }$ (Bekenstein-Hawking)
- the temperature-acceleration connection $T= a \, {\hbar}/{2\pi kc}$ (Unruh)
- the heat and entropy connection $\delta Q =T \delta S$.
Then he uses the thermodynamic horizon relation ("energy change is heat change")
\begin{equation}
\delta E = \delta Q
\end{equation}
to get
\begin{equation}
\delta E = \frac{c^2}{8\pi G} a \delta A
\end{equation}
He uses the energy-momentum tensor $T_{ab}$ to write
\begin{equation}
\int T_{ab} k^a {\rm d} \Sigma^b = \frac{c^2}{8\pi G} a \delta A
\end{equation}
where ${\rm d} \Sigma^b$ is the general surface element and $k^a$ is the Killing vector that generates the horizon. Then he uses the Raychaudhuri equation to rewrite this as
\begin{equation}
\int T_{ab} k^a {\rm d} \Sigma^b = \frac{c^4}{8\pi G} \int R_{ab} k^a {\rm d} \Sigma^b
\end{equation}
where $R_{ab}$ is the Ricci tensor describing space-time curvature. This equality implies
\begin{equation}
T_{ab} = \frac{c^4}{8\pi G} ( R_{ab} - (R/2 + \Lambda) g_{ab} )
\end{equation}
where $R$ is the Ricci scalar and $\Lambda$ is a famous constant of integration: the cosmological constant. These are Einsteins field equations.
Ok, this was the summary of Jacobson's derivation, which is now often called emergent gravity or entropic gravity.
Now, Verlinde changes this derivation by stating that the Bekenstein-Hawking entropy-area connection $$S= A\,\frac{kc^3}{4G\hbar }$$ is not correct in de Sitter space, but has to be changed (here I might be wrong) to something involving the de Sitter radius $L$. It seems that he implies, for a black hole of radius $r$, something like
$$S= A(r)\,\frac{kc^3}{4G\hbar } - \frac{r}{L}A(r)\,\frac{kc^3}{4G\hbar }$$
where $L$ is the de Sitter radius. The black hole entropy is changed by a small effect of the de Sitter space (horizon). As I said, I might be reading him incorrectly, and maybe his expression differs.
Verlinde claims that Milgrom's MOND arises from the effect of $\Lambda$, or equivalently of the size of the horizon $L$.
Verlinde appears to say that the horizon $L$ changes gravity. Why does he not write (using $r=\sqrt{A/4\pi}$) for the entropy
$$S= A\left (1-\frac{\sqrt{A/4\pi}}{L} \right)\,\frac{kc^3}{4G\hbar }$$
and just continues with Jacobson's derivation? He could use this expression for entropy $S$ as a modified Bekenstein-Hawking relation and repeat the derivation by Jacobson. He would directly get new/improved/corrected field equations for gravitation ("Verlinde's field equations") that take into account $\Lambda$. They could be tested against experiment. And they should contain MOND, even a relativistic version of it. What is wrong with this approach?
This post imported from StackExchange Physics at 2020-06-25 21:55 (UTC), posted by SE-user Christian
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