Skip to main content

The Horizon's Poisson Ratio: Extremal by Necessity


Generated image 

Yes, you can define an effective Poisson ratio for the horizon—but only in a very specific, emergent, hydrodynamic sense—and it takes its extremal value.

The value you get is not arbitrary; it is forced on you by horizon incompressibility, maximal entropy, and diffusion physics.

1. 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.

2. Horizon fluids: what moduli exist?

From the membrane paradigm, AdS/CFT, and your 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.

3. Incompressibility fixes the Poisson ratio

(a) 3D effective description (bulk viewpoint)

For an incompressible medium:

$$K \to \infty.$$

Taking the limit K ≫ μ in 3D elasticity:

$$\nu \longrightarrow \frac{1}{2}.$$

$$\boxed{\nu_{\text{eff}} = \tfrac{1}{2}}$$

This is the maximum physically allowed Poisson ratio in 3D. It corresponds to:

  • constant volume,
  • all longitudinal deformation being converted into transverse deformation.

This is exactly what the cosmic event horizon does: it cannot change its entropy density per Planck area.

(b) 2D membrane viewpoint (more appropriate)

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

For a 2D incompressible membrane:

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

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

This is even stronger:

  • any stretch in one direction produces an equal contraction in the orthogonal direction,
  • the area remains fixed.

This matches perfectly with:

$$\Delta A = 8\pi L_p^2$$

being the minimum and quantized area change.

4. Why this is consistent with a diffusion result

Previously our random-walk / diffusion derivation gave:

$$\Delta x^2 = L_p l_\Lambda.$$

Key point:

  • fluctuations are radial,
  • but the entropy lives in area.

Thus:

  • radial uncertainty produces tangential strain,
  • but the area law prevents net areal compression.

That is exactly the mechanical meaning of:

  • (ν = 1/2) (3D incompressible),
  • or (ν₂D = 1) (membrane).

So the diffusion constant

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

is not just a transport coefficient—it is telling you the horizon sits at the incompressible quantum limit.

5. Connection to KSS, Nelson, and quantum limits

All of the following now line up:

  • KSS bound $$\frac{\eta}{s} = \frac{\hbar}{4\pi k_B}$$ → maximally efficient momentum transport.

  • Quantum diffusion bound $$D \geq \frac{\hbar}{m^*}, \quad m^* = m_{\mathrm{CEH}}$$

  • Nelson / Schrödinger diffusion $$D = \frac{\hbar}{2m_s}$$

These are not independent facts. They are all consequences of the same statement:

The horizon is a maximally incompressible, maximally dissipative quantum fluid.

An incompressible fluid has:

  • infinite bulk modulus,
  • extremal Poisson ratio,
  • no independent elastic shear response.

6. Conclusion

It makes sense to define a Poisson ratio for the horizon as an emergent hydrodynamic quantity, not as a microscopic elastic constant.

We find:

  • 3D effective description: $$\boxed{\nu = \tfrac{1}{2}}$$

  • 2D membrane (horizon) description: $$\boxed{\nu_{2D} = 1}$$

These are extremal values, signalling:

  • incompressibility,
  • saturation of quantum bounds,
  • and consistency with Zurek's horizon blurring, Bousso–Penington islands, and your diffusion-based derivation.

In short: the Poisson ratio is not arbitrary—it is fixed by horizon thermodynamics and quantum uncertainty. The Poisson ratio of spacetime is undefined in flat SR, but emerges as an extremal hydrodynamic diagnostic when spacetime becomes an entropy-constrained system (e.g. horizons). 

Comments

Post a Comment

Popular posts from this blog

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 co...
Post a Comment
Read more

Our cosmic event horizon on a string

In this post, we introduced the idea that in the presence of a positive cosmological constant, there is a minimum (local) mass $m_{GR}$   McDormand swinging a cosmic light-like mass-energy $m_{GR}$, with a "cosmic string" of radius $l_{\Lambda}$, giving a centripetal force $F_{local}=m_{GR} \ c^2/l_{\Lambda}$.    Let's think about that string for a bit. In fact, a great number of physicists have spent their entire careers tied up unraveling string theory . For a classical string, which lives in D = 10 dimensions, associated with Nambu-Goto action, the the string tension $T_G$ is a local force, or energy per unit length (dimensions $MLT^{-2}$): \begin{equation} \notag T_G = \frac{1}{2\pi \alpha \prime} \end{equation}$\alpha \prime$ is the Regge slope parameter, set here with dimensions of inverse force.  The wavelength of the stringy mass-energy standing wave (such that it does not interfere with itself), is the circumference of the circle, and we know it moves...
Post a Comment
Read more

The Virial Horizon

  In general relativity, defining "energy" is notoriously difficult. A specific point of confusion often arises in de Sitter space: Why is the quasi-local energy of the horizon exactly double the effective gravitational mass? By imposing a fundamental consistency condition— Horizon Uncertainty—we can re-derive the thermodynamics of the de Sitter horizon. We find that the horizon behaves exactly like a quantum-stretched membrane, where the "factor of two" is simply the result of the Virial Theorem applied to spacetime itself. Horizon Position Uncertainty Whether through holographic arguments, diffusion models, or entanglement entropy, a causal horizon is never sharp. Its position fluctuates. The variance of the horizon position $\Delta x^2$  scales with the geometric mean of the Planck length $L_p$ ​ and the horizon radius $l_{\Lambda}$ ​ :  \begin{equation}     \Delta x^2 = L_P \, l_\Lambda, \qquad L_P = \sqrt{\frac{G\hbar}{c^3}} \end{equation}    W...
Post a Comment
Read more