What Kind of Medium

The substrate’s superfluid vacuum forms enough structure for static shear and to hold the metric for gravity and light: one medium with order and coherence, like a supersolid.

The Superfluid as a Supersolid

This section asks the question whether a superfluid vacuum is too soft, as compared with the elastic-solid vacuum like Kleinert’s world crystal.

The term “Elastic solid” is a claim about structure and “Superfluid” is a claim about phase coherence. As independent properties, a medium can have neither, either, or both. An ordinary liquid has neither. A crystal has order without coherence. Bulk helium-II resting in a beaker has coherence without order as expected in laboratory superfluids. Labs have measured both today in covered in the supersolids chapter.

Under this lens, the substrate is a superfluid that produces an elastic solid as an emergent structure — a rotating condensate whose circulation has quantized into lines and whose lines have locked into a triangular lattice with genuine elasticity, shear waves, defects, and grain boundaries. It thereby gets the elastic column, and keeps the quantum column that the elastic solid cannot reach. This addresses the objection that “superfluids do not resist static shear”. That would be true of a featureless superfluid in a lab, but is not true of the vortex crystal the substrate forms.

The Superfluid in the Lab

Liquid helium-II in a stationary vessel is isotropic, featureless, and genuinely unable to support a static shear stress. Set it spinning, and it becomes a different thing. It cannot rotate as a body, because its flow is irrotational; it accommodates the rotation by minting quantized vortex lines, and above a few lines per square centimetre those lines do what lines of like circulation do — they repel, they equilibrate, and they crystallize into Abrikosov’s triangle. That lattice has a shear modulus. Tkachenko computed it in 1966; Baym put it in the modern stiff-limit form; rotating condensates have been photographed self-assembling into it; and neutron stars announce it in public every time a pulsar glitches, because glitch recovery is the pinned vortex lattice transferring angular momentum through its own elastic response.

So superfluids with vortices do support static shear with a measurable result.

What Rigidity Costs

The vacuum carries transverse waves with two polarizations, it deflects light around masses, and it holds a metric that requires a stiff medium.

The elastic solid assumes rigidity at the constituent level. Its constituents have equilibrium positions; disturbances are displacements about those positions; shear is resisted because the positions resist rearrangement. This coherent construction fromes from:

  1. A preferred frame. A medium whose constituents have fixed positions has lattice vectors, and lattice vectors define a rest frame. Michelson and Morley looked for and could not find this reference frame. The classical aether failed because empty space would expose this frame and does not.
  2. A Bragg comb. A medium scatters a probe with momentum transfers to its density profile, so scattering must track the structure factor S(\mathbf q). A crystal’s S(\mathbf q) is a comb of sharp spikes at reciprocal-lattice vectors — which is to say a diffraction grating. Starlight crossing gigaparsecs shows no orders, no dimming, no angle-dependent splitting. A vacuum crystal has to explain the silence.
  3. A second wave speed. A Cauchy solid at Poisson ratio 0.25 carries a longitudinal compression branch at \sqrt3\,c alongside its transverse branch at c. Nothing has ever been seen moving at \sqrt3\,c. The solid must explain why its own second branch is invisible.

In contrast, the substrate gets rigidity from quantized circulatione:

  1. No frame, because the excitations ride an acoustic metric rather than the constituents. Any barotropic, inviscid, irrotational flow produces an effective metric formally identical to a curved Lorentzian spacetime (Barceló, Liberati & Visser 2005), and for low-energy quasiparticles in ^3He the emergent Lorentz group is exact to all orders below the gap (Volovik, Ch. 7). A measurement of the acoustic metric by acoustic instruments cannot find that metric’s own rest frame. Michelson–Morley is not survived; it is predicted.
  2. No comb, because the lattice is a domain glass of triangular crystallites — locally ordered, globally disordered, and disordered hyperuniform by the same \eta=1 requirement that gives the bridge equation its isotropy. Such a texture has S(\mathbf q)\to 0 at small \mathbf q — no long-range scattering, no fog — and no reciprocal lattice to orient against the Earth’s motion. Stealthy hyperuniform materials are, for exactly this reason, transparent at high density (Leseur, Pierrat & Carminati 2016).
  3. No unexplained second speed, because the extra branch is slow and decoupled rather than fast and missing. The substrate’s elastic branch is the Tkachenko wave at c_T \approx 9 km/s \approx 3\times10^{-5}c — five orders of magnitude below light, coupling to neither the photon field nor the gravitational-wave sector at leading order, which is why nothing has seen it. That is a zero-parameter prediction with a stated frequency (f_T \approx 3{,}700 Hz), not a branch that has to be explained away.

The superfluid gets the metric using these theorems about flow that keeps the frame hidden.

The Shear Modulus

The vortex lattice has three elastic moduli, and the Higgs chapter uses all three. The tilt modulus c_{44} is the cost of bending vortex lines out of the lattice plane — very stiff, the spring axis of the mattress. The compressional modulus c_{11} is the cost of squeezing the lattice in-plane. And the shear modulus is

c_{66} = \frac{\rho\,\kappa\,\Omega}{8\pi}

This is the cost of sliding adjacent rows of the triangular lattice past each other, the softest of the three, with no logarithmic enhancement. It is a static shear modulus in the ordinary sense: displace the rows and a restoring stress appears, proportional to the displacement, with a well-defined spring constant set by the circulation quantum and the rotation rate. Its Goldstone mode is the Tkachenko wave, and its speed in the stiff limit is c_T = \sqrt{\hbar\Omega/(4 m_1)}, giving the 9 km/s quoted above. The hierarchy c_{44} \gg c_{11} \sim c_{66} is itself load-bearing in the framework: the lattice rearranges in-plane rather than tilting, which is why the polar-jet coupling provides a disproportionately strong righting moment.

Three independent columns have measured this modulus in media that are unambiguously superfluids:

  • Theory, since Tkachenko (1966) and in the modern form Baym (2003) uses.
  • The bench. Rotating condensates assemble the triangle on camera, and the 2019 dipolar supersolids resolved a compressional oscillation splitting into two branches across the transition — the superfluid branch and the crystal branch shearing apart, watched directly (Tanzi et al., Nature 574, 382, 2019; Guo et al., Nature 574, 386; Natale et al., PRL 123, 193002). Two restoring forces, one medium.
  • The sky. Pulsar glitch recovery is the standard observational handle on a pinned vortex lattice’s elastic and frictional response, and the neutron-star chapter reads it as boundary-layer-limited mutual friction.

Where the Energy Went

Ultimately, a medium that hides that much energy in shear boundaries is not the soft candidate — it is stiffer than an elastic solid and it is silent, which is the combination the sky demands and the one an assumed rigidity cannot produce.