Skip to main content

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. 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} = L_pc
\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_pl_{\Lambda}}
\end{equation}

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 $m_0$ and $w_B$ and presto!

Not so coincidently, 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! 

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.

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 observable (effective) 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 $m\to m_0$ in the kinetic term, remembering zero-point energy $2m_0=2\frac{c^2l_{\Lambda}}{4G}=m_{CEH}$, where we are considering the minimum effective mass CEH.  

 





Comments

Post a Comment

Popular posts from this blog

Remarks on the cosmological constant and minimal acceleration

As Lineweaver explained, because our Universe is expanding at an accelerating rate, our Universe has a cosmic event horizon (CEH).    Its an event horizon Jim, but not as we know it...Image credit: DALL.E2 by SR Anderson Events beyond the CEH will never be observed. The CEH is also the source of de Sitter radiation, which has a specific temperature $T_{dS}$. It is the minimum possible temperature of the Universe, and, it is not absolute zero (zero Kelvins). Numerically $T_{dS} \approx 2.4 \times 10^{-30}K$, the universal minimum (black body) `absolute cold' local temperature of the future dS state. As a comparison, in 2020 the NASA Cold Atom lab was able to cool an atom to a record low $\sim 2 \times 10^{-7}K$.  Now, in any theory one may think of temperature as an energy, and from the semi-classical Unruh relationship, temperature $\sim$ acceleration. Therefore:   \begin{equation} \notag E_{dS} = T_{ds} \ k_B= \frac {\hslash a}{2\pi c} \end{equation} What we g...
Post a Comment
Read more

Dark photons and the Universal ground-state energy

 In a previous post we introduced the idea that our current Universe has boundary conditions.    Classical 1+1 dS space-time visualized as basketball hoop, with the up/down slam-dunk direction being the time dimension, and the hoop circumference the space dimension.    We also showed a diagram similar to Figure 1 below. Except here, we are once again thinking about the future dS state . Figure 1 . For an observer at O inside the cosmic event horizon (CEH) with radius $l_{\Lambda}$, the universe can be divided into two sub-vacuums, $(A)$ inside the CEH, and $(B)$, outside. The horizon surface $\Sigma$ has entanglement entropy $S_{dS}$ and rest energy $E_H$   Figure 2. The maximum entropy of the Universe (credit: Lineweaver ).   Now, a comoving volume partition of the Universe can be treated as a closed system for which $dS \geq0$. The maximum entropy of a closed system, in this case  ( Figure 2 ) with $L=2 \pi l_{\Lambda}$, the circum...
Post a Comment
Read more