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Blurring the horizon - the quantum width of the cosmic event horizon

A 2021 paper by Zurek applied a random walk argument to a black hole horizon.

  Credit, Zurek, 2021

Zurek called this a blurring of the horizon (a fuzzy, or uncertain horizon), and also went through some equivalent derivations, which basically supported the idea this length scale is the quantum uncertainty in the position of the BH horizon, aka a dynamic quantum width of an event horizon (a concept which would therefore also apply to the universe's own CEH). The Bekenstein-Hawking entropy gives the number of quantum degrees of freedom that can fluctuate. Below, we step out our own cosmic de-Sitter derivation of the random walk argument, obtaining the same result as Zurek did, so its certainly correct!

\begin{equation}
\Delta x^2 =2 DT
\end{equation} In this equation, $\Delta x$ is the position uncertainty, $D$ is the Einstein diffusion coefficient and $T$ is the time between measurements (aka the relaxation time).  

 So, lets look at the Einstein diffusion coefficient. $\mu=v_d/F$ is the mobility and $v_d$ is the drift velocity. We can write this as  $\mu=v_d\kappa$, as the inverse of the Einstein gravitational constant $\kappa$ is a local force, i.e. $F = c^4/8\pi G=1/\kappa$. Drift velocity is acceleration $\times$ relaxation time, the time between two successive collisions. With $l_{\Lambda}=2L_p$ (minimum de sitter length scale),  if we take the de Sitter acceleration $a=\frac{c^2}{l_{\Lambda}}$ and relaxation time as proportional to the inverse of the Hubble constant, i.e. $T=l_{\Lambda}/2c$, we get $v_d=c/2$.  Then:

\begin{equation}
D = \mu k_B T_{dS} = \frac{l_{\Lambda}}{2}c
\end{equation}   We can then work out that $\Delta x=\sqrt{2} \ L_p$ where $L_p$ is the Planck Length! Or we can write the nice equation (a geometric mean) or seesaw-like relation (in 4 dimensions):

\begin{equation}
\Delta x =\sqrt{L_p{l_\Lambda}}
\end{equation}

Not so coincidentally, this is the same relation that Bousso and Penington (B&P) finds that the protrusion distance outside the horizon of an entanglement island from a 4D Schwarzschild black hole! In fact, this is also what you get from from the position uncertainty of a QH oscillator. We have talked before about that - just use the zero-point energy $m_s=\hbar/cl_{\Lambda}$ and $w_B=c/l_{\Lambda}$ and presto!

\begin{equation}
    \Delta x = \sqrt{\frac{\hbar}{2 m_s \omega_B}} = \sqrt{L_pl_{\Lambda}}
\end{equation}

Also, consider the Bekenstein black hole quantum of area:

\begin{equation}
    \Delta A =  8 \pi L_p^2 = 2 \pi l_{\Lambda}^2
\end{equation} Surface area of a sphere, where radius $r$ so $r^2=\Delta x^2=2L_p^2$ 

\begin{equation}
    \Delta A =  4 \pi r^2 = 8 \pi L_p^2
\end{equation}   Zurek also  pointed out that the equality $S_{BH}=S_{ent}$ is known to be true only in certain systems, however, there is evidence this equality holds more generally.

This derivation invites a comparison to fluid dynamics. We defined a mobility $\mu$. If we accept the result for $\Delta x$ then we are effectively assigning a minimum kinematic viscosity to vacuum spacetime:\begin{equation}
    \nu_{vac} = D \ = \frac{l_{\Lambda}}{2}c 
\end{equation}

This matches the KSS conjecture from AdS/CFT correspondence (dynamic viscosity to bulk entropy density ratio  $\eta / s \geqslant \hbar / 4 \pi k_{B}$). If the horizon is a fluid, this diffusion calculation shows that it flows with maximum quantum efficiency. 

We can also recover the quantum limit of diffusion as $D\ge\hbar/m^*$.  In our view where classical GR is simply an effective hydrodynamical description, this connects the Misner–Sharp effective gravitational mass to the CEH mass, i.e. $m^*=m_{CEH}$.  Of course, you have to use the minimum de Sitter length scale in the effective mass relation for the quantum limit.    

This is also equivalent to Nelson’s stochastic formulation of QM, or alternatively Schrödinger's equation considered as a diffusion equation, where  \begin{equation}  D =\hbar/2m \end{equation} Here we substitute $m\to m_s$ (the fundamental quantum) in the kinetic term, remembering from this post that at minimum dS scales, where $m_{CEH}=m_{Planck}$ we get $2m_s=2\frac{c^2l_{\Lambda}}{4G}=m_{CEH}$. 

 





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