Skip to main content

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 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. 

 

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. 

 

Comments

Post a Comment

Popular posts from this blog

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 2021 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. To do this, let $l_{\Lambda}$ represent the generalised de Sitter horizon scale. Due to the holographic UV/IR correspondence, this scale manifests dually: at the fundamental microscopic limit as $l_{UV} = 2L_p$ (the gravitational/casual limit, aka the Schwarzschild radi...
Post a Comment
Read more

The Cosmic Strange Metal

       In condensed matter physics, a class of strongly correlated systems share three remarkable transport properties simultaneously. Strange metals, quantum spin liquids, and Sachdev-Ye-Kitaev (SYK) models all exhibit (i) a shear-viscosity-to-entropy ratio at or near the conjectured KSS minimum $\eta/s = \hbar/(4\pi k_B)$, (ii) a Lyapunov exponent saturating the Maldacena-Shenker-Stanford (MSS) chaos bound $\lambda_L = 2\pi k_B T/\hbar$, and (iii) a Planckian dissipation time $\tau = \hbar/(k_BT)$ governing all transport. These systems have no quasiparticle description. Their low-energy physics is instead captured, in known cases, by a gravitational dual via holography. This post argues that the cosmological horizon of a de Sitter (dS) universe belongs to this same universality class, with every quantitative bound matched, provided one applies the correspondence to the horizon membrane rather than the bulk spacetime. The membrane paradigm as the physical basis Th...
Post a Comment
Read more

Our cosmic event horizon on a string

A Cosmic Stringy Adventure ! As we previously discussed, our spacetime characterised by a positive cosmological constant $\Lambda$. The natural bounds are then a minimal  ultraviolet (UV) length $l_{UV} = 2L_P$ and an infrared (IR) cosmological horizon $l_{\Lambda}$.  This dual-boundary spacetime enforces a fundamental Compton–gravitational duality . Every geometric scale $r$ carries two natural mass definitions: $$m_C(r) = \frac{\hbar}{rc}, \qquad m_G(r) = \frac{c^2}{4G}\ r$$ The product of these masses, $m_C \ m_G = M_P^2/4$, is scale-independent. They intersect exclusively at the UV boundary $r = l_{UV}$, defining a maximal local force in GR: $F_{max} = c^4 / 4G$. At the opposite extreme, the Compton mass evaluated at the IR horizon yields the fundamental  spectral gap  (not a particle) of the universe: $m_s = \hbar / (l_{\Lambda} c)$.  In this post, to explore how energy propagates through this dual-scale geometry, we model the mass gap $m_s$ as a null-ener...
Post a Comment
Read more