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

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

Credit, Zurek, 2021

Zurek (Snowmass White Paper: Observational Signatures of Quantum Gravity) called this a blurring of the horizon: a fuzzy, or uncertain horizon, and went through derivations supporting the idea that this length scale is the quantum uncertainty in the position of the black hole horizon: a dynamic quantum width of an event horizon. This is a concept which fundamentally applies to the Universe's own Cosmic Event Horizon (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, giving us the exact phenomenological signature Zurek found:

$$\Delta x^2 = 2DT$$

In this equation, $\Delta x$ is the position uncertainty, $D$ is the Einstein diffusion coefficient, and $T$ is the time between measurements (the relaxation time).

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Einstein Diffusion Coefficient

So, let's 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.

$$F = \frac{c^4}{8\pi G} = \frac{1}{\kappa}$$

Drift velocity is acceleration $\times$ relaxation time, i.e. the time between two successive collisions. If the UV cutoff is $l_{\Lambda} = 2L_p$, and we take the maximal de Sitter acceleration

$$a = \frac{c^2}{l_{\Lambda}}$$

and minimal relaxation time i.e.

$$T = \frac{l_{\Lambda}}{2c}$$

we get

$$v_d = aT = \frac{c^2}{l_{\Lambda}} \cdot \frac{l_{\Lambda}}{2c} = \frac{c}{2}$$

The effective drift velocity $v_d$ arises because microscopic spacetime fluctuations are locally isotropic, but a causal horizon admits only outward‑directed null steps; the mean outward projection of a fixed‑magnitude null velocity $c$ over the accessible hemisphere is therefore $\langle v_z\rangle = c/2$, making the factor $1/2$ a universal consequence of isotropy combined with one‑sided causal projection, independent of curvature or microscopic details.

Then:

$$D(l_{\Lambda}) = \frac{2cL_{p}^{2}}{l_{\Lambda}}$$ So, with $$   k_B T(l) = \frac{\hbar c}{2\pi l}$$ we get

$$D_{UV} = L_p c$$ The UV diffusion constant of the spacetime vacuum.

 

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 Fluid Dynamics

This derivation invites a direct translation into quantum-critical fluid dynamics. We defined a mobility $\mu$. If we accept the result for $D_{UV}$, we are effectively assigning a fundamental minimum kinematic viscosity to the vacuum spacetime:

$$\nu_{UV} = L_p c$$

This means the Schmidt number of the dS vacuum is one. If the horizon is a fluid, this diffusion calculation shows that it flows with maximum quantum efficiency. Also, the vacuum has no scale because it is at the RG fixed point.  

Because we are dealing with a topologically protected quantum-critical fluid, its macroscopic behaviour is governed by the scaled macroscopic kinematic viscosity:

$$\nu_{IR} = \frac{l_{IR}}{2}c = \frac{c^2}{2H}$$

We can evaluate its macroscopic shear viscosity using the macroscopic density $\rho_{\Lambda} = \frac{3H^2}{8\pi G}$:

$$\eta_{vac} = \rho_{\Lambda}\nu_{IR} = \frac{3H^2}{8\pi G}\cdot\frac{c^2}{2H} = \frac{3c^2 H}{16\pi G} = \frac{3\sigma(0)H^3}{16\pi G}$$

where $\sigma(0) = l_{IR}^2$ is a geometric identity, showing a membrane-paradigm / AdS-CFT dictionary. Note $\eta_{vac}$ is not the volume viscosity $\zeta$. Then, as the bulk entropy density $s$ (dimension $k_B L^{-3}$) is:

$$s = \frac{S_{dS}}{V} = \frac{3}{4}\frac{k_B c^3}{\hbar G l_{IR}}$$

we recover the KSS conjecture from the AdS/CFT correspondence (shear viscosity to bulk entropy density ratio $\eta/s \ge \hbar/4\pi k_B$). The reason this works seamlessly across 60 orders of magnitude is UV/IR conformality: the ratio of degrees of freedom to dissipation is topologically protected.

Interestingly, \begin{equation}
    \lambda_L \cdot \frac{\eta}{s} = \frac{T dS}{2}
\end{equation} where $\lambda_L=H$ is the MSS bound on chaos and $T_{BH}=T_{dS}/2$. In Planckian fluids, the butterfly velocity $v_B$ is determined by the relationship between the diffusion of energy/momentum and the chaos parameters, with $C$ being a numerical constant of order unity. \begin{equation}
    D = C \frac{v_B^2}{\lambda_L}
\end{equation}Here of course, we can easily see that $$ v_B = \frac{c}{\sqrt{2}}$$

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Quantum of Circulation

We can also see at the fundamental limit:

$$\nu_{UV} m_s = \frac{\hbar}{2}$$

Onsager noted the ratio of Planck's constant to mass has the same dimension as kinematic viscosity. This implies that the effective kinematic viscosity — aka the quantum of circulation — associated with coarse-grained dynamics can be written as the universal critical velocity $v_c$ associated with oscillatory flow generated by an oscillating body: $$v_c^2 = \nu_{IR} H = \frac{c^2}{2}$$ $$ i.e. $v_B=v_c$. The butterfly is the vortex! In a relativistic conformal fluid the sound speed is $c_s = c/\sqrt{3}$, however, in a holographic universe, information must scramble faster than sound can propagate.

The associated Reynolds Number is $R_e = v_c L / \nu$, where $v_c$ is the critical velocity, $L$ is the characteristic length scale, and $\nu$ is the kinematic viscosity. Contrary to naive expectations, a Reynolds number holds even for superfluids, where for sufficiently rapid flows, perfect inviscid flow breaks down and an effective viscosity emerges dynamically via the nucleation of quantised vortices.

Indeed, because this is a quantum-critical fluid, we can evaluate this at any generalised scale $l_{\Lambda}$. Associating the characteristic length scale $L = l_{\Lambda}/2$ and viscosity $\nu = l_{\Lambda}c/2$, the length scales completely vanish:

$$R_e = \frac{v_c L}{\nu} = \frac{(c/\sqrt{2})(l_{\Lambda}/2)}{l_{\Lambda}c/2} = \frac{1}{\sqrt{2}} \approx 0.7$$

The resulting vacuum $R_e$ is a scale-invariant topological constant, identical to that experimentally (Reeves et al, 2014, Identifying a superfluid Reynolds number via dynamical similarity), found as a universal superfluid Reynolds number. Also, turbulence here is not just chaotic geometry, it would be quantised vortex nucleation. Of course, the presence of massive particles breaks conformal invariance (creates drag on the CEH), so this turbulence threshold only applies to pure de Sitter.   

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Quantum Limit of Diffusion

We can also recover the quantum limit of diffusion as:

$$D \ge \frac{\hbar}{m^*}$$

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. $m^* = m_{CEH}$.  

If we consider Schrödinger's equation considered as a diffusion equation, where

$$D = \frac{\hbar}{2m}$$

Now, there is a single scale-dependent mass quantum, $m_s(\ell)$:

\begin{equation}
    m_s(\ell) = \frac{\hbar}{c\,\ell},
\end{equation} whose value at the UV (Planck scale) and IR (dS horizon scale) is:

\begin{equation}
    m_{s}^{\mathrm{UV}} = \hbar / \left( 2 c L_{p} \right) 
\end{equation} \begin{equation}
    m_{s}^{\mathrm{IR}} = \hbar H_{\Lambda} / c^{2} = \hbar / (c l_{\mathrm{IR}})  
\end{equation} If the macroscopic CEH behaves as a single quantum object, the effective stochastic mass in diffusion is $m_{s}^{\mathrm{IR}}$ rather than $m_{CEH}$. Then:\begin{equation}D_{IR} = \frac{\hbar}{2 m^{IR}_s} = \frac{c^2}{2 H_\Lambda}\end{equation}

Thus, a causal horizon is a quantum-critical stretched fluid whose primitive density is the universal area entropy density $$\sigma_S = \frac{k_B}{4L_p^2}$$ Bulk spacetime thermodynamics is obtained by coarse-graining this surface density over the causal diamond $$s_{\rm bulk} = \frac{3\sigma_S}{L}$$ The de Sitter vacuum energy is then not independent matter energy but thermalized horizon information$$\rho_\Lambda c^2 = T_{dS}s_{\rm bulk}$$ The same horizon fluid saturates the KSS viscosity bound and the MSS chaos bound$$
\frac{\eta}{s} = \frac{\hbar}{4\pi k_B},
\qquad
\lambda_L = H,
\qquad
v_B = \frac{c}{\sqrt{2}}.
$$ Its quantum width obeys a UV/IR random-walk law$$w^2 \sim L_p L$$ Thus, the bulk is not filled with entropy; the bulk is the hydrodynamic shadow of horizon entropy.

 

Classical GR is the hydrodynamic (the low energy, long wavelength) regime of a more fundamental microscopic theory of spacetime.