Hydrodynamics of dilation and spin currents

This paper formulates a relativistic hydrodynamic theory for fluids with spin and intrinsic dilation charges, deriving constitutive relations that reveal gapped dilation excitations and scale-anomaly-induced transport effects, which reduce to microstretch fluid dynamics in the nonrelativistic limit and apply to rapidly expanding or contracting nearly conformal systems.

The Gist

Imagine the universe as a giant, chaotic dance floor. Usually, when physicists study how fluids (like water, air, or even the super-hot soup of particles created in particle colliders) move, they treat them like a simple crowd of people shuffling around. They track where the crowd is going (flow), how fast they are moving (velocity), and how hot the room is (temperature).

But this new paper suggests we've been missing two very important details about the dancers: how they are spinning and how they are stretching.

Here is the breakdown of the paper's big ideas, translated into everyday language:

1. The Two New Dancers: Spin and Stretch

In standard physics, a fluid element is like a solid ball rolling down a hill. It has a position and a speed.

• Spin Hydrodynamics (The Spinning Top): Recent experiments showed that particles in heavy-ion collisions (smashing atoms together) actually spin like tiny tops. This paper adds "spin" to the fluid equations. Imagine the fluid isn't just a blob of water, but a swarm of tiny, spinning tops. Their rotation affects how the whole fluid flows.
• Dilation Currents (The Stretchy Band): This is the paper's big new idea. Imagine the fluid elements aren't just spinning tops, but also stretchy rubber bands. They can expand and contract internally, even if the whole fluid isn't moving.
  • The Analogy: Think of a crowd of people. In normal fluid theory, they just walk forward. In this new theory, the people are also constantly inflating and deflating like balloons while they walk. The paper calls this "intrinsic dilation."

2. The "Rubber Band" Effect (Viscosity)

Usually, when a fluid expands, it resists it (like blowing up a balloon). This resistance is called "bulk viscosity."

• The Twist: In a "scale-invariant" fluid (one that looks the same at any size, like a fractal), the fluid shouldn't resist expanding at all. It should be perfectly happy to grow.
• The Discovery: The authors found that even though the fluid doesn't resist the overall expansion, it does resist the internal stretching of its own "rubber bands."
• The Metaphor: Imagine a group of dancers holding hands. If the whole group spreads out, that's fine. But if individual dancers try to stretch their own arms out wider than their neighbors, there's a friction. The paper identifies a new "friction" (conductivity) that governs how fast these internal stretchy arms snap back to their normal size.

3. The "Freeze-Out" (The Cosmic Horizon)

One of the coolest findings is about how sound waves travel in this fluid.

• The Scenario: Imagine the fluid is expanding very fast, like the universe did right after the Big Bang.
• The Phenomenon: The paper predicts that if the fluid expands fast enough, sound waves (ripples in the fluid) will get "frozen." They won't be able to travel anymore.
• The Analogy: Think of a runner trying to run on a treadmill that is speeding up. If the treadmill speeds up faster than the runner can run, the runner stays in the same spot relative to the room, even though they are running hard.
  • In this fluid, the "treadmill" is the expansion of space. If the expansion is too fast, sound waves get stuck. They can't cross the "horizon" (the edge of the expanding region). This is very similar to what happens in cosmology with the "superhorizon" modes in the early universe.

4. The Electric Spark (Scale Anomaly)

The paper also asks: "What happens if we add electricity to this spinning, stretching fluid?"

• The Problem: In the quantum world, perfect symmetry (like perfect scale invariance) is often broken by tiny quantum effects. This is called an "anomaly."
• The Result: This breaking of symmetry creates a new kind of electric current. It's like a hidden battery inside the fluid that generates electricity just because the fluid is expanding or contracting in a specific way. The paper maps out exactly how this new current, the energy, and the stretching interact.

5. Why Should We Care?

This theory isn't just math for math's sake. It applies to two extreme places in our universe:

1. The Quark-Gluon Plasma (QGP): The state of matter created in particle accelerators (like the Large Hadron Collider) where protons and neutrons melt into a hot soup. This soup spins and expands incredibly fast. This theory helps us understand how that soup behaves.
2. The Early Universe: Right after the Big Bang, the universe was a hot, expanding fluid dominated by radiation. This theory helps us model how that early universe evolved, especially regarding how sound waves behaved in those first moments.

Summary

In short, this paper upgrades our "fluid map."

• Old Map: Fluids flow and spin.
• New Map: Fluids flow, spin, and stretch.
• The Consequence: When these fluids expand rapidly, they develop new types of friction, new ways to conduct electricity, and sound waves that can get "frozen" in place, unable to travel.

It's like realizing that the ocean isn't just waves moving across the surface; the water molecules themselves are also stretching and shrinking, and that tiny detail changes how the entire ocean behaves during a storm.

Technical Summary

1. Problem Statement

Relativistic hydrodynamics is the standard macroscopic framework for describing systems near local thermodynamic equilibrium, such as the quark-gluon plasma (QGP) and cosmological fluids. Recent observations of spin polarization in heavy-ion collisions have necessitated the extension of hydrodynamics to include spin (intrinsic angular momentum), leading to the framework of spin hydrodynamics.

However, many physical systems of interest (e.g., early universe radiation, high-energy QGP) are nearly scale-invariant (conformal). While standard conformal hydrodynamics assumes the energy-momentum tensor is traceless and symmetric, this often requires fixing a specific "pseudo-gauge" that eliminates intrinsic spin and dilation currents as independent dynamical variables.

The authors address the gap in formulating a relativistic hydrodynamic theory that:

1. Preserves spin and intrinsic dilation as independent dynamical degrees of freedom.
2. Extends the symmetry group from Poincaré to the Weyl-Poincaré group (including dilations).
3. Describes fluids undergoing rapid expansion or contraction where scale invariance is approximately preserved but intrinsic dilation plays a crucial role.

2. Methodology

The authors employ a phenomenological approach based on the entropy-current analysis to derive constitutive relations.

• Symmetry Framework: They consider a quantum field theory in 4D Minkowski spacetime invariant under the Weyl-Poincaré group. The associated Noether currents are:

  • Energy-momentum tensor: $\Theta^{\mu\nu}$
  • Spin tensor: $\Sigma^{\mu\rho\sigma}$
  • Dilation (scale/virial) current: $S^\mu = \Theta^{\mu\rho}x_\rho + \Upsilon^\mu$
  • Note: They explicitly avoid choosing a pseudo-gauge that sets the intrinsic dilation current $\Upsilon^\mu$ and spin tensor $\Sigma^{\mu\rho\sigma}$ to zero, treating them as dynamical variables.

• Thermodynamics & Variables:

  • The system is characterized by energy density ($\varepsilon$), fluid velocity ($u^\mu$), spin density ($\sigma^\mu$), and dilation density ($\upsilon$).
  • They adopt the Landau-Lifshitz frame.
  • The first law of thermodynamics is generalized: $Tds = d\varepsilon - \omega_\mu d\sigma^\mu - \lambda d\upsilon$, where $\lambda$ is the dilation potential.

• Power Counting Scheme:

  • To handle rapid expansion/contraction, they treat the scalar expansion $\theta = \partial_\mu u^\mu$ as a zeroth-order quantity (unlike standard hydrodynamics where it is first-order).
  • This allows the theory to describe global equilibrium states with uniform expansion (solutions to the conformal Killing equation).
  • Spin potential $\omega_\mu$ and spin density $\sigma^\mu$ are treated as higher-order (quantum) corrections.

• Entropy Production:

  • They construct the entropy current $s^\mu$ and calculate the divergence $\partial_\mu s^\mu$.
  • By enforcing the second law of thermodynamics ($\partial_\mu s^\mu \geq 0$), they derive the dissipative constitutive relations.

3. Key Contributions and Results

A. Constitutive Relations and New Transport Coefficients

The analysis yields a set of constitutive relations containing new transport coefficients:

1. Shear Viscosity ($\eta$): Standard dissipation.
2. Rotational Viscosity ($\eta_s$): Governs spin relaxation.
3. Bulk Viscosity ($\zeta$): In a scale-invariant fluid, the conventional bulk viscosity vanishes. However, here $\zeta$ is finite and governs the relaxation of the intrinsic dilation of a fluid cell toward the equilibrium value set by the macroscopic expansion ($\theta/3$).
4. Dilation Conductivity ($\kappa_d$): A new coefficient governing the diffusion of dilation density driven by the gradient of the dilation potential ($\nabla(\lambda/T)$).

The dissipative corrections are:
$$\Pi = -\zeta(\theta - 3\lambda)$$
$$\delta\Upsilon^\mu = -\kappa_d T \nabla^\mu \Lambda$$
(where $\Lambda = \beta\lambda$).

B. Linear Mode Analysis

The authors linearized the hydrodynamic equations around a global equilibrium with uniform expansion/contraction. The analysis reveals:

1. Gapped Dilation Mode: A new collective excitation associated with intrinsic dilation. It is gapped (has a non-zero imaginary part at zero momentum) and damps with a lifetime $\approx 1/(3D_d)$.
2. Freeze-out of Sound Modes: Long-wavelength sound modes exhibit a "freeze-out" phenomenon. There exists a critical momentum $k_c$ below which perturbations do not propagate.
   • This is analogous to the superhorizon freeze-out in cosmological inflation.
   • In an expanding background, $\lambda_0$ acts like a Hubble constant; perturbations with wavelengths larger than the horizon radius $R \sim c_s/|\lambda_0|$ cannot propagate across the horizon.

C. Nonrelativistic Limit

In the nonrelativistic limit ($u^\mu \to (1, \mathbf{v})$), the theory reduces exactly to the theory of microstretch fluids (a generalization of micropolar fluids).

• Spin corresponds to microrotation.
• Intrinsic Dilation corresponds to microstretch (the ability of fluid elements to expand/contract locally).
  This establishes a rigorous relativistic foundation for microstretch fluid theory.

D. Scale Anomaly and Electromagnetic Coupling

When coupling to an electromagnetic field, the scale invariance is broken at the quantum level (Scale Anomaly).

• The anomaly introduces a term proportional to $F_{\mu\nu}F^{\mu\nu}$ in the dilation balance equation.
• To preserve the second law of thermodynamics, the authors introduce non-dissipative, anomaly-induced corrections to the electric current, energy-momentum tensor, and dilation density.
• Specifically, the anomaly induces a charge transport term $\delta j^\mu_A = 4C_T \partial_\nu(F^{\mu\nu}\Lambda)$, which must be accompanied by corrections to the energy-momentum tensor and dilation current to ensure entropy production remains positive semi-definite.

4. Significance

• Theoretical Unification: The paper successfully unifies spin hydrodynamics with scale-invariant hydrodynamics, treating intrinsic dilation as a fundamental dynamical variable rather than a gauge artifact.
• Cosmological Analogy: The discovery of the "freeze-out" of sound modes in expanding fluids provides a hydrodynamic analogue to cosmological horizon physics, potentially offering new insights into the early universe's radiation-dominated era.
• Heavy-Ion Physics: The framework is directly applicable to the QGP created in ultra-relativistic heavy-ion collisions, which undergoes rapid expansion and is nearly conformal. It suggests that intrinsic dilation effects could be observable in the relaxation dynamics of the plasma.
• Microstretch Fluids: It provides the relativistic parent theory for non-relativistic microstretch fluids, clarifying the physical origin of the "stretch" degree of freedom.
• Anomaly Physics: It extends the understanding of quantum anomalies (specifically the scale anomaly) to hydrodynamic transport, identifying new non-dissipative currents similar to those found in chiral anomaly physics.

In summary, this work establishes a comprehensive relativistic hydrodynamic framework for fluids with spin and intrinsic dilation, revealing novel collective modes and transport phenomena relevant to both high-energy nuclear physics and cosmology.