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The Horizon Quantum of Area

  Several independent semiclassical arguments converge on $$\boxed{\Delta A=8\pi l_{\mathrm{Pl}}^2}, \qquad l_{\mathrm{Pl}}^2=\frac{G\hbar}{c^3}$$ I got interested in this reading the 2018 paper  Bekenstein, I, and the quantum of black-hole surface area . Even though I disagree with Hod, its a great paper.  Honourable mentions also go to:   The Physical Interpretation of the Spectrum of Black Hole Quasinormal Modes , 2008  The Off-Shell Black Hole , 1994     On Black Hole Spectroscopy via Adiabatic Invariance , 2012 An Analytical Computation of Asymptotic Schwarzschild Quasinormal Frequencies , 2003  Asymptotic Black Hole Quasinormal Frequencies , 2003  1. Horizon Action Near any nonextremal horizon, imaginary time turns the normal geometry into a plane: $$ds_E^2\simeq d\rho^2+\rho^2d\Theta^2+\cdots.$$ Smoothness requires one horizon cycle: $$\oint d\Theta=2\pi.$$ The gravitational action identifies the momentum conjugate to $\The...
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The fabric of space-time: stiffer than steel or weaker than jello?

 

The notorious rubber sheet analogy of spacetime teaches one concept and once concept only: Mass-energy causes curvature of space-time.

 When a gravitating mass recedes from a region of space-time the curvature diminishes. The field equations of General Relativity don’t have an explicit term for this elastic property, but the framework as a whole does have that property. As very large mass-energies are required to generate gravitational waves (ripples in space-time), the elastic property of space-time is generally regarded as very stiff

One other interesting consideration here is that elasticity is an emergent phenomenon. There is a great deal of interest in the idea that gravity is similarly emergent.  

In 2018, McDonald quantified the classical stiffness of space-time via an effective Youngs Modulus.

 Classical answer: Youngs Modulus of space-time $\sim$ 20 orders of magnitude greater than steel.  

DALL.E2 depiction of classical space-time. (Credit SR Anderson)  

In a previous post we discussed de Sitter space-time, which is space-time with no matter, just vacuum, and its associated energy-density, the cosmological constant. 

In field theory, spacetime is the geometric manifold (the stage), while the vacuum is the state of lowest energy of the quantum fields residing on it (the actors)   

 In a quantum Universe, the Youngs Modulus of space-time is frequency independent, limited by the energy density of the medium through which the gravity wave propagates. 

In dS space, the medium is vacuum, and the energy density is the cosmological constant. In our current Universe, which includes matter, the energy density is the critical density.  

Either way, what this means, as Melissinos pointed out in a rejoinder to McDonald:

Quantum answer: Youngs Modulus of space-time $\sim$ 14 orders of magnitude less than jello! 

 

Why the massive discrepancy between these two answers? It is because the results are just another way of stating the cosmological constant problem.     

  Generated image
The smart money is on jello 

Youngs Modulus can also be related to the spring constant $k$ (the measure of a spring's stiffness), of a quantum harmonic oscillator via Hooke's Law, which holds up to the so-called proportionality limit i.e. when extension is linearly proportional to the load applied. The relation is:   \begin{equation}
\notag
Y_\Lambda = \frac{k \ L}{A}\
\end{equation} So, with a "cosmic string" of the de Sitter characteristic length $l_{\Lambda}$ and a cross-sectional vector area $A=1/ \Lambda$, with a quantum Youngs Modulus of vacuum: $Y_\Lambda = \Lambda c^4 / 8 \pi G$, the spring constant of vacuum $k$ is:

\begin{equation}
\notag
k = \frac{c^4}{8 \pi G \ l_{\Lambda}}
\end{equation} The term $c^4 / 8 \pi G$ is the inverse of the Einstein gravitational constant.  In classical elasticity, Hooke's Law relates stress (force per unit area) to strain (geometric deformation) via an elastic modulus: Stress = Modulus $\times$ Strain. If we look at the Einstein Field Equations in 4D $$T_{\mu\nu} = \frac{c^4}{8\pi G} G_{\mu\nu}$$

1. The Stress: The stress-energy tensor $T_{\mu\nu}$ represents energy density, which dimensionally reduces to Force/Area.
2. The Strain: The Einstein tensor $G_{\mu\nu}$ represents curvature, with units of $1/L^2$.
3. The Modulus: The proportionality constant linking them is $\kappa = \frac{c^4}{8\pi G}$.
 

Notice that unlike a standard material where the modulus has units of Force/Area, the spacetime modulus $\kappa$ has units of Force. Spacetime has a scale-dependent effective elastic response. 

Another way to look at this:  

  • Classical Stiffness (Steel): This measures the resistance of the spacetime manifold to dynamic shear strain (gravitational waves). This is governed by the inverse of the Einstein gravitational constant, which is an enormously massive number. It dictates that it takes catastrophic amounts of energy, e.g colliding black holes, to cause even microscopic ripples in spacetime. 
  • Quantum Softness (Jello): This measures the static background bulk pressure of the vacuum state (the cosmological constant). Because the observed expansion of the universe is slow and cold, this energy density is remarkably low. 

Spacetime (the geometry) is extraordinarily stiff; the vacuum (the field state) is extraordinarily dilute. 

 

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