If space-time is considered as a kind of material, then what properties make sense? Properties that can be observed, measured.
In the context of General relativity, (aggregate) properties related to curvature and rippling. Like elasticity, stiffness.
In the context of quantum physics, (discrete) properties related to quanta, energy density, charge, spin, entanglement. Like gradients, flux – “jitter and jive.” So, …
• How stiff is space-time?
(quotes) … space-time is extremely stiff … even rapid motions involving large amounts of mass [e.g., mergers of two black holes] produce only tiny wiggles in space-time.
Space-time can be distorted, but it’s very hard work.
Upon combining general relativity with quantum mechanics, we calculate that space is a kind of quivering Jell-O, in constant motion.
In gravitational waves, bending of space-time in some directions causes bending in others. – Wilczek, Frank. Fundamentals – Ten Keys to Reality. (Chapters 8, 2, 1, 8)
• Well, what bends space-time?
(quote) Bending space-time requires energy, and energy causes space-time to bend. – ibid. Chapter 8.
The cosmological constant merely recognizes the possibility that space-time itself, a material that general relativity allows to bend, push, and shake, might also have inertia. – ibid. Chapter 9.
How does the interaction of quanta at a discrete level result in bending of space-time at an aggregate level? (Like pressure on a flexible membrane.)
• How can we tell if space-time is bent?
[List of experiments]
Time gradients [a better term?] [notable experiments]
Models and metaphors, levels of description
• “Times are legion” … “a different rhythm in every different place” – dynamic “spiderweb of times”
As Carlo Rovelli remarks in his book The Order of Time, “Times are legion: a different one for every point in space.” 
* Quanta Magazine > “An Ultra-Precise Clock Shows How to Link the Quantum World with Gravity” by Katie McCormick (October 25, 2021)
(quote) In a paper posted earlier this month to the scientific preprint server arxiv.org, researchers from the lab of Jun Ye, a physicist at JILA in Boulder, Colorado, measured the difference in the flow of time between the top and the bottom of a millimeter-tall cloud of [100,000 ultracold strontium] atoms.
* Science News > “An atomic clock measured how general relativity warps time across a millimeter” by Emily Conover (October 18, 2021)
Does the variation in time flux (the different rate at which time passes at locations in space-time) bend space-time, or does bending in space-time cause variation in time flux? [A better question is …]
• “The incessant flux of quantum Grid” – dynamic equilibrium
Wilczek’s characterization of space-time is predicated on a model which he calls the Grid, a multilayer structure, a dynamic medium (“aboil with virtual particles”), whose properties are quite different from the classical ether.
(quote) The entity we perceive as empty space is a multilayered, multicolored superconductor. What an amazing, astonishing, beautiful, breathtaking concept. Extraordinary, too. – Wilczek, Frank. The Lightness of Being (p. 97). Basic Books. Kindle Edition.
And his metaphor of the Grid as an exotic (cosmic) superconductor. With condensates. 
Like the gauge bosons, all these particles would be massless but for Grid superconductivity. But Grid superconductivity gives them mass [neutrinos are a special case] and also allows the heavier ones to mix with, and thereby decay into, lighter ones in complicated ways. – ibid. (p. 169)
Emergence in space-time
From the incessant flux of the Grid and the legion of times, meaningful patterns emerge. An order via confinement. A dynamic stability via chaos.
(quote) To get the most out of the big bang theory, we need to refine our assumption that the distribution of matter early on was completely uniform. Small deviations from uniformity will do, because they get amplified by gravitational instability. – Wilczek. ibid. Chapter 7.
 How might such stiffness be quantified? By sampling reality.
We can make a pretty reliable estimate of what it takes to knock loose a piece of the condensate responsible for the Grid’s (electroweak) superconductivity. The weak force is short-ranged, but not infinitely so. The W and Z bosons are heavy, but not infinitely so. The observed range of the force, and mass of the force carriers, give us good handles on the stiffness of the condensate responsible for those effects. Knowing the stiffness, we can estimate how much energy we need to concentrate in order to break off individual pieces (quanta) of the condensate … or whatever kind of new stuff … that makes the Grid a cosmic superconductor. – Wilczek, Frank. The Lightness of Being (p. 194). Basic Books. Kindle Edition.
Also, as Wilczek remarks, the Grid’s total density is likely small; otherwise, there would be no sentient beings to observe the universe. The fact that, sort of like fish, we “swim” in the Grid easily.
And, as noted in another comment, here’s a link to a paper (PDF) by Wilczek for the MIT Physics Annual 2009, “What is Space?” The paper is a useful recap of the history of space — so-called empty space, the void, the plenum, ether; and ends with a brief overview of a toy model of quantum reality.
 See Rovelli, Carlo. The Order of Time (pp. 14-18). Penguin Publishing Group. Kindle Edition.
But if different clocks mark different times, as we have seen above, what does ‘t’ indicate? When the two friends meet up again after one has lived in the mountains and the other at sea level, the watches on their wrists will show different times. Which of the two is ‘t’?
In a physics laboratory, a clock on a table and another on the ground run at different speeds. Which of the two tells the time? How do we describe the difference between them? Should we say that the clock on the ground has slowed relative to the real time recorded on the table? Or that the clock on the table runs faster than the real time measured on the ground?
The question is meaningless. … There is no “truer” time; there are two times and they change relative to each other. Neither is truer than the other.
But there are not just two times. Times are legion: a different one for every point in space. There is not one single time; there is a vast multitude of them.
Time has lost its first aspect or layer: its unity. It has a different rhythm in every different place and passes here differently from there. The things of this world interweave dances made to different rhythms.
 In particular, Chapter 8 in Wilczek’s book The Lightness of Being is all about the Grid. His Recapitulation at the end of the chapter lists key Grid properties. For example, a metric field for space-time rigidity; a universal (nonzero) density.
And the dynamics of interactions across Grid layers (quantum dimensions).
You can visualize the quantum dimensions as new layers of the Grid. When a particle hops into these layers its spin changes, and so does its mass. Its charges—electric, color, and weak—stay the same. – Wilczek, Frank. The Lightness of Being (p. 188). Basic Books. Kindle Edition.
8 thoughts on “How stiff is space-time?”
Here’s a sketch, my attempt at a simple visualization (cartoon) of Wilczek’s Grid. Grid dynamics interplay.
Perhaps a visualization, a conceptualized framework, similar to a geographic information system (GIS)? A Grid “map” with overlays for spatio-temporal location and corresponding attribute information. Overlays (which may be toggled) for continuous fields and discrete objects. Thematic layers. Topological relationships.
Cf. Hydrological and Cartographic modeling with the dimension of time.
I like this visualization (below), although the article uses the diagram in a different context.
• Phys.org > “Quantum physics in proteins: AI affords unprecedented insights into how biomolecules work” by Deutsches Elektronen-Synchrotron (11-3-2021)
Here’s my attempt to “package” the structure of Wilczek’s Grid, using the metaphor of the Grid as a layered superconductor.
0. space-time substrate
0.1 virtual energy stratum
0.2 field-condensate strata 
0.3 quasi-particle stratum
[Wilczek, Frank. The Lightness of Being. Basic Books. Kindle Edition.]
 As noted in Wilczek’s The Lightness of Being, “We discussed how empty space is presently filled with various material condensates” (p. 249).
And the various (relative) densities of condensates (pp. 109-110):
And stiffness (p. 194):
And strata (p. 214):
Regarding visualization of the Grid’s multilayer structure – subspace.
• Scientific American > “Could Gravity’s Quantum Origins Explain Dark Energy?” by Conor Purcell (October 28, 2021)
Earlier this year, Ethan Siegel wrote an excellent recap on the reality of virtual particles: “… does the mere fact that we have quantum fields in our universe mean that empty space is actually filled with something physical?”
The distinction between observable effects of the quantum vacuum and scattering off the quantum vacuum. The concept of vacuum polarization. Vacuum birefringence. The Casimir effect. Proton soup.
• Forbes > “Ask Ethan: Do Virtual Particles Really Exist?” by Ethan Siegel, Senior Contributor (May 7, 2021) – The effects of virtual particles are real, but the particles themselves are not!
Interacting Grid layers?
• As noted in my June 12, 2019 comment for “Acceleration causes gravity, gravity causes acceleration” post:
So, applying Rovelli’s notion that “The gravitational field doesn’t live in space, but is space itself” makes this NASA APOD caption more about spacetime energy-density (energy-momentum) and geometric stress – stretching and bending (as in fluid dynamics) – rather than attraction and repulsion.
• NASA > APOD > “Dark Matter in a Simulated Universe” (October 31, 2021)
Here’s some further speculation about the Grid, particularly interactive processes between layers and the emergence of localized properties.
• Inertia (the origin of inertia) as “knots” in spacetime flux-topology?
Metaphorically as in fluid dynamics, visualize laminar (smooth) vs. turbulent (rough) flow within and between Grid layers.
Asymmetries in “laminar” flow produce imperfections (defects) in energy flux of the quantum vacuum. Or, a disruption in flow between Grid layers (which must be treated probabilistically), with non-zero vorticity.
And as to why neutrinos have such negligible inertia – the simplest of twists or knots.
• Mach’s principle as a framework for spacetime geometry.
• Relational theory à la Smolin, and relational quantum mechanics à la Rovelli.
Video #13 in Sean Carroll’s “Biggest Ideas” chat series includes an introduction to the Riemann Curvature Tensor.
Also, in topology, winding numbers, topological defects (in cosmology) in homotopy groups:
π 0 -> Domain Walls (field theory)
π 1 -> Cosmic String
π 2 -> Monopoles
π 3 -> Textures
• YouTube > Sean Carroll > The Biggest Ideas in the Universe > #13 “Geometry and Topology” (June 20, 2020)
Riemann Curvature Tensor
Topological defect (topological soliton)
In my original post “How stiff is space-time?” (October 25, 2021), I noted two articles on preprint research which measured “the difference in the flow of time between the top and the bottom of a millimeter-tall cloud of [100,000 ultracold strontium] atoms.”
Here’s another article on that research published in the February 22, 2022, issue of Nature.
• Science Mews for Students > “A new clock shows how gravity warps time — even over tiny distances” by Kendra Redmond (June 2, 2022) – Using an elaborate optical lattice single-clock design (which spans two rooms), JILA researchers measured time dilation within a tiny cloud of strontium atoms over a distance of one millimeter.
Credit: Pixabay/CC0 Public Domain
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