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The Horizon's Poisson Ratio: Extremal by Necessity


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Yes, you can define an effective Poisson ratio for the spacetime horizon, and the 3D bulk.  

What a Poisson ratio really measures

The Poisson ratio (ν) is defined (in ordinary elasticity) as

$$\nu = - \frac{\text{transverse strain}}{\text{longitudinal strain}}.$$

Equivalently, in terms of elastic moduli:

$$\nu = \frac{3K - 2\mu}{2(3K + \mu)} \quad \text{(3D bulk)},$$

or

$$\nu_{2D} = \frac{K_{2D} - \mu_{2D}}{K_{2D} + \mu_{2D}} \quad \text{(membrane)}.$$

So to even ask for a Poisson ratio, we need:

  • a bulk (or area) modulus (K),
  • a shear modulus (μ).

This is where horizons are special.

Horizon fluids: what moduli exist?

From the membrane paradigm, AdS/CFT, and our own derivations:

  • Horizons have:
    • finite shear viscosity (η),
    • no elastic shear modulus ((μ = 0)),
    • negative bulk viscosity in GR, but effectively incompressible area when quantum fluctuations are included.

This is crucial:
A horizon is not an elastic solid. It is a maximally dissipative, incompressible quantum fluid.
That already tells us what (ν) must be.

 2D membrane viewpoint  

The cosmological event horizon is a 2D membrane embedded in spacetime.

For a 2D incompressible membrane:

$$K_{2D} \to \infty \quad \Rightarrow \quad \nu_{2D} \to 1.$$

$$\boxed{\nu_{2D} = 1}$$

It cannot be "thinned" or "thickened" locally; it can only redistribute its area. 

The 3D Bulk: The Auxetic Limit

However, the 3D bulk is governed by holographic scaling, which we wrote about previously. This relates the radial (longitudinal) scale $l_{IR}$ to the transverse quantum uncertainty (the “blurring” width) $\Delta x $:

$
\Delta x = \sqrt{2L_p l_{IR}}
$

As the universe expands longitudinally $l_{IR}$ increases, the transverse width of the vacuum fluctuations $\Delta x $ also increases. We define the 3D strains as:

  • Longitudinal Strain $\epsilon_L$ : $ d(\ln l_{IR})$  
  • Transverse Strain $\epsilon_T$: $d(\ln \Delta x)$

Taking the derivative of the scaling relation:
\[
\ln(\Delta x) = \frac{1}{2} \ln(l_{IR}) + \text{constant}
\]
\[
\frac{d(\ln \Delta x)}{d(\ln l_{IR})} = \frac{1}{2}
\]
The 3D Poisson ratio is the negative ratio of these strains:
\[
\nu_{3D} = -\frac{\epsilon_T}{\epsilon_L} = -0.5
\] Conclusion: The 3D vacuum bulk is a dilational auxetic medium with \( \nu_{3D} = -0.5 \). Unlike standard matter, stretching the vacuum causes it to “fatten” transversely through increased quantum  uncertainty. 

The Moduli of the Vacuum

Using the Young’s Modulus we previously derived from the Einstein constant and the cosmological constant \( \Lambda \):
\[
Y_\Lambda = \frac{\Lambda c^4}{8\pi G}
\]
We calculate the Bulk Modulus (K) and Shear Modulus (G\_s) for the 3D bulk (\( \nu = -0.5 \)):

- Bulk Modulus:
  \[
  K = \frac{Y}{3(1-2\nu)} = \frac{Y}{3(1 - (-1))} = \frac{1}{6} Y_\Lambda
  \]
- Shear Modulus:
  \[
  G_s = \frac{Y}{2(1+\nu)} = \frac{Y}{2(1 - 0.5)} = Y_\Lambda
  \]
This demonstrates a stiffness hierarchy:  
the vacuum is six times more resistant to shear (metric distortion) than to volume expansion.

The Vacuum Speed of Sound

The non-relativistic elastic formula is not relevant here. So, the (longitudinal) speed of sound \( c_s \) in a relativistic holographic fluid is determined by the Conformal Limit. In a medium where the trace of the stress-energy tensor vanishes (consistent with the UV/IR conformality of the derivation), the pressure \( P \) and energy density \( \rho \) are related by:
\[
P = \frac{1}{3} \rho c^2
\]
The sound speed is the square root of the derivative of pressure with respect to density:
\[
c_s = \sqrt{\frac{\partial P}{\partial \rho}} = \frac{c}{\sqrt{3}} \approx 0.577c
\]

Reconciling \( c_s \) with Elasticity

In our auxetic bulk, the ratio of the “Holographic Sound Speed” (\( c_s \)) to the “Shear Speed” (which, for massless gravitons, is \( c \)) is:
\[
\frac{c_s}{c} = \frac{1}{\sqrt{3}}
\]
This matches the hydrodynamic limit of the AdS/CFT correspondence. It implies that while gravitational information (shear, i.e. transverse) propagates at \( c \), density fluctuations (sound) are constrained by the 3D degrees of freedom available in the holographic bulk.

Interestingly, this connection between auxeticity and holographic blurring identifies Dark Energy not as a substance, but as the elastic response of a manifold stretched into its non-linear regime.  

 

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