Planar Black holes and Entanglement Entropy in Analog Gravity Models

This paper demonstrates that all planar black holes conformal to a Painlevé–Gullstrand line element can be realized as analogue metrics by constructing an explicit Lagrangian, while also introducing a concept of holographic entanglement entropy valid for arbitrary conformal and blackening factors.

The Gist

The Big Idea: Building a Black Hole in a Bathtub

Imagine you want to study a black hole. In real life, they are terrifyingly far away, incredibly massive, and impossible to touch. You can't put one in a lab to see how it behaves.

But what if you could build a miniature, fake black hole right here on your kitchen table?

This is the core concept of Analog Gravity. The authors of this paper, Neven and Tobias, are essentially saying: "We can't build a real black hole out of stars and gravity, but we can build one out of flowing water (or cold atoms) that 'pretends' to be a black hole."

In their experiment, sound waves traveling through a special fluid act exactly like light waves traveling near a real black hole. If the fluid flows fast enough, the sound waves get trapped, just like light gets trapped by a black hole's event horizon.

The Problem: Too Many Shapes, Too Few Fluids

For a long time, scientists could only simulate very specific, simple types of black holes with these fluid models. It was like having a 3D printer that could only print perfect spheres. But real black holes (and the math describing them) come in all shapes and sizes.

The authors asked: "Can we make our fluid mimic any flat, planar black hole, not just the simple ones?"

They found the answer is yes. They figured out how to design the "recipe" (the Lagrangian) for the fluid so that it can mimic a vast family of black holes, provided the fluid flows in a specific way.

The Metaphor: The River and the Waterfall

To understand how they did it, imagine a river flowing toward a waterfall.

1. The River (The Fluid): This is your lab setup.
2. The Waterfall (The Black Hole): This is the point of no return.
3. The Fish (The Sound Waves): Imagine a fish trying to swim upstream against the current.
   • If the river flows slowly, the fish can swim upstream easily.
   • If the river flows faster than the fish can swim, the fish gets swept over the edge.
   • The Event Horizon: The exact spot where the river's speed equals the fish's swimming speed. Beyond this point, the fish cannot escape.

In this paper, the authors figured out how to shape the riverbed and control the water speed so that the "fish" (sound waves) experience the exact same physics as light near a complex, mathematical black hole. They showed that by tweaking the fluid's density and pressure (the "recipe"), they can create a "river" that mimics almost any flat black hole geometry.

The Twist: The "Holographic" Entanglement

Once they built their fake black hole, they wanted to measure something very mysterious called Entanglement Entropy.

What is Entanglement Entropy?
Imagine you have a pair of magic dice. No matter how far apart you are, if you roll a 6, your friend's die instantly shows a 6. They are "entangled." If you only look at your own die, it looks random. But if you look at both, there is a hidden connection.

In the universe, space itself is "entangled." Parts of space are connected in ways we can't see. When a black hole forms, it cuts space in half. The part inside the hole is cut off from the part outside. This "cut" creates a specific amount of "missing information" or entropy.

The Holographic Trick:
The paper introduces a concept called Holographic Entanglement Entropy. Think of it like a hologram on a credit card. The 3D image is stored on a flat, 2D surface. Similarly, the authors suggest that the complex 3D physics happening inside their fluid "black hole" can be understood by looking at a 2D surface on the edge of the fluid.

They calculated this entropy for their fluid model. They found that the amount of "entanglement" (the hidden connection) follows a specific rule: The more area the "cut" has, the more entanglement there is. This matches a famous rule in physics called the "Area Law," which says that the entropy of a black hole is proportional to its surface area, not its volume.

Why Does This Matter?

1. Lab Experiments: It means we can test theories about black holes and the early universe in a lab using cold atoms or water, rather than waiting for a telescope to see a real one.
2. New Math: They provided a "universal recipe" (a specific Lagrangian) that works for almost any flat black hole, vastly expanding the list of things we can simulate.
3. Connecting Worlds: This work bridges the gap between Condensed Matter Physics (stuff like fluids and cold atoms) and Gravity (black holes and spacetime). It suggests that the deep laws of the universe might be the same whether you are looking at a galaxy or a drop of water.

Summary in One Sentence

The authors figured out how to design a fluid flow in a lab that acts like a universal "black hole simulator," allowing scientists to study the mysterious "entanglement" of space-time using simple sound waves and water, proving that the physics of the cosmos can be mimicked in a tabletop experiment.

Technical Summary

1. Problem Statement

The field of Analog Gravity seeks to simulate gravitational phenomena (such as black holes and Hawking radiation) using condensed matter systems, where perturbations in a fluid obey equations analogous to fields in curved spacetime. However, a fundamental limitation exists:

• Degrees of Freedom Mismatch: General Relativity (GR) in $3+1$ dimensions possesses 4 degrees of freedom per spacetime point (10 metric components minus 6 Einstein constraints). In contrast, standard analog models based on a single scalar field (potential flow) typically depend on only two independent functions (the scalar potential $\theta$ and the speed of sound $c$).
• Scope Limitation: Consequently, not all interesting geometries can be mimicked by analog systems. Previous work (e.g., Ref. [10]) successfully modeled a specific Planar AdS$_5$ black hole with a fixed "blackening factor" ($\gamma(z) = 1 - z^4/\ell^4$), but a general framework for arbitrary stationary planar black holes was lacking.
• Entanglement Entropy: While holographic entanglement entropy is a key concept in AdS/CFT duality, its calculation and interpretation within general analog gravity models for planar black holes had not been systematically established.

The authors aim to:

1. Generalize the construction of analog metrics to cover any stationary planar black hole metric conformal to a Painlevé–Gullstrand type line element, regardless of the specific blackening factor.
2. Define and compute the holographic entanglement entropy for these general analog spacetimes.

2. Methodology

A. Theoretical Framework: Conformal Rescaling

The authors start with a generic stationary planar black hole metric in $n+1$ dimensions, conformally rescaled by a factor $\Omega(t, x, z)$:
$$ds^2 = \Omega^2 \sqrt{1-\gamma(z)} \left[ -\gamma(z)dt^2 + \frac{dz^2}{\gamma(z)} + dx^2 \right]$$
where $\gamma(z)$ is the "blackening factor" (vanishing at the horizon $z=\ell$). They utilize the property that a scalar field $\phi$ propagating in this metric can be conformally rescaled to a field $\tilde{\phi}$ propagating in a metric $\tilde{G}_{\mu\nu} = \Omega^{-2}G_{\mu\nu}$ with a modified effective mass. This allows the separation of the conformal factor from the core acoustic geometry.

B. Fluid Dynamics and Lagrangian Construction

The authors employ a Lagrangian formalism for non-isentropic fluids:
$$\mathcal{L} = F(\chi) - V(\theta, t, x, y, z)$$
where $\chi = -g^{\mu\nu}\partial_\mu\theta\partial_\nu\theta$ is the kinetic term, $\theta$ is the velocity potential, and $V$ is an external potential.

• Acoustic Metric: Linear perturbations ($\delta\theta$) around a background solution satisfy a Klein-Gordon equation in an effective acoustic metric $\tilde{G}_{\mu\nu}$.
• Mapping: The goal is to match this acoustic metric to the rescaled black hole metric (4).

C. Coordinate Transformation and Parameter Matching

To map the fluid variables to the black hole geometry, the authors perform a coordinate transformation $(t, z) \to (\tilde{t}, \tilde{z})$ to bring the black hole metric into a form resembling the Painlevé–Gullstrand structure.

• They derive expressions for the fluid velocity $u^\mu$, specific enthalpy $w$, and particle number density $n$ in terms of the blackening factor $\gamma(z)$ and the sound speed $c$.
• Key Constraint: They derive a differential equation for the sound speed $c^2$ dependent on $\gamma(z)$:
  $$c^2 = c_1(1-\gamma) + \frac{1}{2}$$
  where $c_1$ is an integration constant. This relationship dictates the equation of state $p(\rho)$ required to simulate the specific black hole geometry.

D. Potential and Effective Mass

The external potential $V(\theta)$ is constructed to satisfy the background equations of motion. Crucially, the authors show that by adding a deformation term proportional to $(\theta - \theta_0)^2$ to the Lagrangian, one can tune the effective mass ($m_{eff}$) of the perturbation to any desired value without altering the acoustic metric itself.

E. Holographic Entanglement Entropy

The authors define the entanglement entropy for the analog system by dividing the spacetime into two regions:

• Region A: From the cutoff $z_{min}$ (where the analog model is valid) to the horizon $z=\ell$.
• Region B: From $z=0$ to $z_{min}$.
  Using the Ryu-Takayanagi prescription (area law), they calculate the entropy $S$ as the area of a minimal surface $\Sigma$ extending into the bulk, bounded by a strip of width $d$ on the boundary.

3. Key Contributions

1. Generalization of Analog Metrics: The paper proves that any planar black hole metric conformal to a Painlevé–Gullstrand type (with an arbitrary blackening factor $\gamma(z)$) can be realized as an analog metric. This vastly extends the class of simulatable spacetimes beyond the specific AdS$_5$ case.
2. Explicit Lagrangian Derivation: They provide the explicit functional form of the fluid Lagrangian $F(\chi)$ and the necessary conditions for the potential $V(\theta)$ to reproduce these geometries.
   • The derived equation of state is $p \propto (w^2 + \text{const})^{3/2}$.
   • They identify the constraints on the integration constant $c_1$, showing that a single choice of $c_1$ may not cover the entire physical range $z>0$, necessitating a cutoff $z_{min}$ (typically where $\gamma(z) = 2/3$ or similar, depending on $c_1$).
3. Holographic Entanglement Entropy in Analog Gravity: They introduce the concept of holographic entanglement entropy for analog planar black holes. They derive the parametric equations relating the entropy $S$ to the strip width $d$ for a general $\gamma(z)$.
4. Numerical Validation: They numerically compute the entropy $S(d)$ for a planar AdS$_5$ black hole, demonstrating that in the large width limit, the entropy obeys the area law with a specific subleading term, consistent with AdS/CFT expectations.

4. Results

• Metric Realization: The acoustic perturbation in a fluid with the derived Lagrangian (Eq. 51) propagates exactly as a scalar field in the target curved spacetime (Eq. 4).
• Sound Speed Profile: The sound speed must vary spatially according to $c^2 = c_1(1-\gamma) + 1/2$. This implies that to simulate a black hole, the fluid's equation of state must be carefully engineered to vary with position.
• Validity Range: The analog model is valid only for $z \geq z_{min}$, where $z_{min}$ is determined by the condition $\gamma(z_{min}) = 2c_1 + 1$ (derived from $c^2 \leq 1$). For AdS$_5$ with $c_1=1/2$, the model breaks down at $z_{min} = \ell/3^{1/4}$.
• Entropy Behavior:
  • The entanglement entropy $S$ diverges logarithmically as the strip width $d$ approaches the horizon limit.
  • For large $d$, $S$ asymptotically approaches a linear function (Area Law):
    $$S_{lim} = \frac{L}{2\ell} \left( 3^{1/4} - 1 + \frac{d}{2\ell} \right)$$
  • This confirms that the analog system reproduces the expected area-law behavior of black hole entropy, with the "healing length" of the fluid playing the role of the Planck length.

5. Significance

• Bridging Gravity and Condensed Matter: This work significantly broadens the scope of analog gravity, moving from specific, isolated examples to a general framework for planar black holes. This allows experimentalists to simulate a wider variety of gravitational phenomena using tunable fluid systems (e.g., Bose-Einstein Condensates).
• Testing Quantum Gravity Phenomena: By enabling the simulation of arbitrary blackening factors and effective masses, the model provides a robust platform for testing semiclassical effects like Hawking radiation and entanglement entropy in a laboratory setting.
• AdS/CFT and Holography: The successful calculation of holographic entanglement entropy in an analog setting strengthens the link between analog gravity and the AdS/CFT correspondence. It suggests that the "bulk" geometry of holographic theories can be effectively probed using condensed matter analogs.
• Phenomenological Applications: The results are relevant for high-energy physics scenarios involving relativistic fluids, such as ultrarelativistic heavy-ion collisions, where the effective spacetime is often 1+1 dimensional (planar geometry).

In conclusion, Bilić and Zingg provide a rigorous theoretical foundation for realizing a vast class of planar black hole spacetimes in the lab, complete with a method to calculate their holographic entanglement entropy, thereby deepening the connection between fluid dynamics, general relativity, and quantum information theory.