Causal UV completions of relativistic hydrodynamics

This paper demonstrates that standalone relativistic hydrodynamic effective field theories are inherently acausal and necessitates the inclusion of transient ultraviolet modes to restore causality while preserving the correct late-time hydrodynamic behavior.

The Gist

Imagine you are trying to predict how a crowd of people moves through a city square. If you only look at the crowd from a very high altitude, far away, and only care about the general flow (where the crowd is dense, where it's thin), you can use a simple set of rules called hydrodynamics. These rules work great for describing the "big picture" of the crowd moving slowly and smoothly.

However, this paper argues that if you try to use only these simple rules to describe the crowd in a world where nothing can move faster than the speed of light (a relativistic world), you run into a massive logical problem: the rules break causality.

Here is the breakdown of the paper's findings using simple analogies:

1. The "Instant Message" Problem (Acausality)

In our everyday world, if you drop a stone in a pond, the ripples spread out slowly. But the simple hydrodynamic rules used in physics (like Fick's diffusion equation) act like a magical stone. If you drop it, the ripples would instantly appear everywhere in the pond at the exact same moment, even on the other side of the universe.

In physics terms, this means the theory predicts that a signal can travel faster than light. The paper proves mathematically that any standalone theory of fluid flow that includes friction or heat (dissipation) will always have this "instant message" flaw. It's like trying to build a house on a foundation that is guaranteed to crumble; the theory is inherently broken if used alone.

2. The "Ghostly Tails" (Exponential Decay)

So, if the simple rules are broken, why do they work so well in real life? The paper explains that the "broken" part of the theory only happens in the "ghostly tails" of the signal—places very far away from the center of the action.

Imagine a lighthouse beam. The main beam is bright and clear. But if you look at the very edge of the light, it fades away exponentially (it gets dimmer and dimmer very fast). The paper shows that in a fluid, the part of the signal that violates the speed of light is like this fading edge. It exists, but it dies out incredibly fast as you move away from the center of the light cone (the area where light can reach).

Because this "bad" part fades away so quickly, it opens a door for a solution.

3. The "UV Completion" (Adding the Missing Pieces)

To fix the broken theory, the paper suggests we need to add "transient UV modes." Think of these as hidden gears inside a clock.

• The Hydrodynamic View: You only see the clock hands moving.
• The UV View: You see the tiny, fast-moving gears inside that make the hands move.

The paper proves that you can always add these hidden gears (transient modes) to the theory. These gears move so fast and die out so quickly that they cancel out the "instant message" error.

• The Result: You get a new, perfect theory that respects the speed of light.
• The Catch: If you zoom out and look at the clock from far away (low energy/late times), those hidden gears are invisible. You only see the smooth, slow-moving hands. The simple hydrodynamic rules re-emerge naturally, but now they are part of a larger, causal system.

4. The "Speed Limit" Requirement

There is one strict rule for this fix to work: The fluid must have a "sound speed" (how fast a wave travels through it) that is slower than light.

• If the fluid's internal waves try to travel at or faster than light, the math says you cannot fix the causality problem.
• If the fluid is slower than light, the paper proves you can always construct a "causal completion" (a fix) that keeps the simple rules working for the slow, big-picture stuff while adding the necessary fast-moving parts to keep physics honest.

5. What About the Hidden Gears? (Non-Hydrodynamic Modes)

The paper also asks: "What do these hidden gears actually look like?"
The answer is a bit surprising. The paper shows that the only thing the simple hydrodynamic rules tell us about these hidden gears is how long they must last before disappearing.

• They must disappear fast enough to cancel out the "instant message" error.
• Beyond that, the simple rules don't tell us exactly what the gears are. They could be anything, as long as they vanish quickly enough.

However, the paper uses a specific example (pure diffusion) to show that if you demand the theory be perfectly causal, the "hidden gears" end up looking like specific mathematical shapes (branch cuts) that are unique to that system. It's like saying: "If you want your house to be earthquake-proof, the foundation must have a specific shape, even if the rest of the house can be built in many ways."

Summary

The paper is a mathematical proof that:

1. Simple fluid rules are broken because they allow signals to travel faster than light.
2. They can be fixed by adding fast, short-lived "hidden modes" (UV modes).
3. These fixes always exist as long as the fluid moves slower than light.
4. The simple rules are still valid for slow, late-time observations because the "fix" parts fade away so quickly they become invisible.

Essentially, the paper provides a "patch" for the software of fluid dynamics, ensuring it doesn't crash when you try to run it in a relativistic universe.

Technical Summary

Technical Summary: Causal UV Completions of Relativistic Hydrodynamics

Problem Statement
Relativistic hydrodynamics serves as a successful effective field theory (EFT) for describing the low-energy, long-wavelength regime of out-of-equilibrium systems. However, the paper identifies a fundamental flaw: stand-alone dissipative hydrodynamic theories are inherently acausal. While hydrodynamic modes (associated with conserved quantities) vanish as $k \to 0$, the resulting correlation functions often exhibit support outside the forward lightcone ($|x| > t$). A canonical example is Fick's diffusion equation, where a local perturbation instantaneously spreads across all space. While previous approaches, such as the Israel-Stewart (MIS) theory or the BDNK framework, attempt to restore causality by introducing transient modes, these methods typically modify the hydrodynamic dispersion relation itself in the infrared (IR) regime via higher-gradient corrections. The paper seeks to determine if it is possible to construct a causal ultraviolet (UV) completion that preserves the exact hydrodynamic dispersion relation in the IR while introducing additional transient UV modes to enforce causality.

Methodology
The authors employ a "bootstrap" approach, treating causality as an overarching principle to constrain the structure of UV completions without assuming specific microscopic details. The analysis relies on the mathematical properties of retarded correlation functions $G(t, x)$ and their singularities (modes) in complex frequency space.

The core methodology involves three main steps:

1. Proof of Inherent Acausality: The authors mathematically demonstrate that any dissipative hydrodynamic theory, if treated as a stand-alone EFT with a single hydrodynamic mode, must violate causality. This is established by analyzing the analyticity of the dispersion relation $\omega(k)$ and the resulting support of the correlation function.
2. Analysis of Hydrodynamic Decay: Using saddle-point (steepest descent) methods, the paper investigates the late-time behavior of the hydrodynamic correlation function along radial lines $x = vt$. It is shown that outside the sound cone ($|v| \neq c_s$), the hydrodynamic contribution decays exponentially in time, with a rate $\Gamma(v)$ that increases with the distance from the sound speed.
3. Construction of Causal UV Completions: Leveraging the exponential decay of the hydrodynamic tails outside the lightcone, the authors construct a causal correlation function. This is achieved by "cutting off" the hydrodynamic mode in momentum space (removing the unstable or acausal tails outside a region $K$ containing the origin) and redistributing the removed contribution within the lightcone. This redistribution introduces non-hydrodynamic (UV) modes that cancel the acausal support.

Key Contributions and Results

• Theorem 1 (Acausality of Hydrodynamics): The paper proves that for any dissipative hydrodynamic dispersion relation $\omega(k)$ (where $\omega(0)=0$ and $\text{Im}\,\omega(k) < 0$ for $k \neq 0$), the correlation function defined solely by this mode has support outside the lightcone. Even if the dispersion relation satisfies the causality bound $\text{Im}\,\omega(k) \le |\text{Im}\,k|$ locally, the single-mode correlation function remains acausal unless supplemented by other modes to remove singularities (e.g., branch cuts).
• Theorem 2 (Decay Rate): The authors establish that for analytic dispersion relations with a finite imaginary gap away from $k=0$, the hydrodynamic correlation function decays exponentially in time for all velocities $v \neq c_s$. The decay rate $\Gamma(v)$ is strictly positive and monotonically increases with $|v - c_s|$. This exponential suppression is the mechanism that allows for a causal completion.
• Theorem 3 (Causal UV Completion): The paper proves that for any hydrodynamic dispersion relation with subluminal sound speed ($|c_s| < 1$) and a finite radius of convergence, there exists a causal UV completion. In this completion, the hydrodynamic mode remains an exact, single-pole solution for small $|k|$ (within an interval $K$), while the acausal tails are canceled by additional transient UV modes.
• Explicit Example (Pure Diffusion): The authors apply this framework to pure diffusion. They derive an explicit causal UV completion where the non-hydrodynamic sector consists of branch cuts starting at $\omega = \pm k - i/(4D)$. This structure ensures that the correlation function vanishes outside the lightcone while preserving the diffusive mode $\omega = -iDk^2$ in the IR. The result recovers known structures from kinetic theory (e.g., RTA models) but derives them strictly from causality and the IR dispersion relation.

Significance and Claims
The paper claims to provide a rigorous mathematical foundation for understanding the necessity and structure of UV completions in relativistic hydrodynamics. Its primary significance lies in shifting the perspective from "fixing" hydrodynamics by modifying its IR equations to "completing" it by adding UV modes that leave the IR physics untouched.

Key implications highlighted by the authors include:

• Necessity of UV Modes: Dissipative hydrodynamics cannot be causal on its own; transient UV modes are strictly required to restore causality.
• Constraints on the Non-Hydrodynamic Sector: While the specific microphysics of the UV sector is not fixed, causality imposes a lower bound on the thermalization timescale of the non-hydrodynamic modes. These modes must be sufficiently long-lived to cancel the hydrodynamic tails outside the lightcone.
• Regime of Applicability: The paper argues that hydrodynamics is not restricted by the dispersion relation itself, but rather by the regime of applicability (the interval $K$ in momentum space) where the hydrodynamic mode is the slowest decaying.
• Hydrohedron Bounds: The derived exponential decay rates offer a potential method to improve existing causal bounds on transport coefficients (the "hydrohedron"), suggesting that causality restricts the domain of hydrodynamics rather than the coefficients themselves.

The authors maintain a modest stance, noting that while the existence of such completions is proven, the specific distribution of UV modes (beyond the minimal set required for causality) depends on the specific microscopic system and is not uniquely determined by the hydrodynamic EFT alone.