Traversable wormhole with double trace deformations via gravitational shear and sound channels

This paper demonstrates that double trace deformations between the boundaries of an AdS$_5$ black brane induce gravitational shear and sound channel perturbations that violate the Averaged Null Energy Condition, thereby creating a traversable wormhole whose duration and decay behavior depend on the sound speed and coupling configurations.

The Gist

The Big Picture: Opening a Cosmic Shortcut

Imagine the universe as a giant, folded piece of paper. Usually, to get from one side to the other, you have to walk all the way around the edge. A wormhole is like a shortcut tunnel that connects the two sides directly.

However, in standard physics, these tunnels are usually unstable. They pinch off instantly, or they require "exotic matter" (a magical, impossible substance) to keep them open. This paper explores a way to open a wormhole using gravity itself, without needing magic ingredients.

The researchers are working within a framework called AdS/CFT, which is like a hologram. Imagine a 3D object (the wormhole in space) being projected from a 2D surface (the boundary of the universe). By manipulating the 2D surface, you can change the 3D object.

The Mechanism: The "Double-Trace" Deformation

To open the wormhole, the scientists use a specific trick called a double trace deformation.

• The Analogy: Imagine two people standing on opposite sides of a canyon (the two boundaries of the universe). They are holding walkie-talkies. Usually, they can't talk because the canyon is too deep.
• The Trick: The scientists program the walkie-talkies to send a specific, synchronized signal between the two people at the exact same time. This signal creates a "negative energy" shockwave.
• The Result: In the world of gravity, negative energy acts like a repulsive force. It pushes the walls of the canyon apart just enough to create a temporary bridge. This bridge is the traversable wormhole.

The Two Channels: Shear vs. Sound

The paper investigates how this works when the "signal" is made of gravitational waves (ripples in spacetime) rather than simple particles. They look at two different ways these ripples behave, which they call "channels":

1. The Shear Channel (The "Slippery" Mode):

   • Analogy: Imagine pushing a deck of cards sideways. The cards slide over each other but don't compress. This is "shear."
   • What happens: The researchers tested three different ways to send the signal (different coupling configurations). They found that the "slippery" mode works well to open the wormhole. It behaves somewhat like a slow, spreading drop of ink in water (diffusion).

2. The Sound Channel (The "Wave" Mode):

   • Analogy: Imagine clapping your hands. A sound wave travels through the air, compressing and expanding. This is the "sound" mode.
   • What happens: This mode is more complex. It travels like a wave rather than just spreading out.
   • The Speed Limit: The researchers found that if the "sound" travels too fast (faster than light), the wormhole opens for such a tiny fraction of a second that nothing can actually get through. It's like a door that slams shut before you can step through.
   • The Attenuation: They also looked at what happens if the sound gets "muffled" (attenuated) as it travels. They found that muffled sounds change when the wormhole is most open, shifting the timing of the best moment to cross.

The "Power-Law" Remnant

One of the most interesting findings is about how long the wormhole stays open.

• The Analogy: Think of a bell. When you hit it, it rings loudly at first, then fades away.
• The Finding: The researchers found that the "fading" of the wormhole's opening doesn't just happen at a steady rate. Instead, it leaves behind a "tail" that fades in a specific mathematical pattern (a power-law).
• Why it matters: This tail is different depending on whether you are using the "Shear" (slippery) or "Sound" (wave) mode.
  • At low speeds, the sound mode acts like the shear mode (spreading out slowly).
  • At high speeds, the sound mode acts more like a pure wave, and the "tail" fades away much faster, making the wormhole harder to use for information transfer.

The Bottom Line

This paper proves that you can open a wormhole using pure gravity (specifically, gravitational waves) by connecting two sides of the universe with a synchronized signal.

• It works: The wormhole opens, and signals can pass through.
• It depends on the type of wave: "Slippery" waves (shear) and "Sound" waves behave differently.
• Timing is everything: If the sound waves move too fast, the wormhole closes too quickly to be useful.
• No magic needed: The "negative energy" required to hold the door open comes naturally from the quantum interactions of the gravitational waves, meaning no exotic, impossible matter is required.

The study suggests that the way information travels through these wormholes leaves a specific "fingerprint" (the power-law tail) that tells us whether the universe is behaving like a slow, spreading fluid or a fast, traveling wave.

Technical Summary

Technical Summary: Traversable Wormhole with Double Trace Deformations via Gravitational Shear and Sound Channels

Problem Statement
This work investigates the mechanism by which traversable wormholes can be constructed in the context of the AdS/CFT correspondence, specifically extending the Gao-Jafferis-Wall (GJW) protocol. While previous studies have demonstrated traversability using scalar and vector (U(1) conserved current) operators via double trace deformations, this paper addresses the gap in understanding gravitational (tensor) perturbations. The central problem is to determine if non-local couplings between the two asymptotic boundaries of an AdS$_5$ black brane, mediated by the stress-energy tensor (dual to metric perturbations), can violate the Averaged Null Energy Condition (ANEC) and render the wormhole traversable. The authors specifically analyze the distinct hydrodynamic behaviors of gravitational perturbations, which include diffusive shear modes and propagating sound modes, contrasting them with the previously studied scalar and vector cases.

Methodology
The authors employ a semiclassical approximation within the AdS$_5$/CFT$_4$ framework. The methodology proceeds through the following steps:

1. Gravitational Setup: The background is an AdS$_5$ black brane. The metric is perturbed by tensor fields $h_{MN}$, which are treated as quantum operators in the interaction picture. The perturbations are classified into three channels based on spatial rotations: scalar ($h_{xy}$), shear ($h_{tx}, h_{zx}$), and sound ($h_{tt}, h_{tz}, h_{xx}, h_{zz}$). The study focuses on the shear and sound channels, as they exhibit hydrodynamic poles in their retarded Green's functions.
2. Double Trace Deformation: A non-local double trace deformation is introduced between the left (L) and right (R) CFT boundaries:
   $$\delta H(t) = \int d^3x \, Q_{\mu\nu\alpha\beta}(t, \vec{x}) T^{(L)}_{h,\mu\nu}(-t, \vec{x}) T^{(R)}_{h,\alpha\beta}(t, \vec{x})$$
   where $T_{h,\mu\nu}$ is the boundary stress-energy tensor. The coupling function $Q$ is chosen to be negative-valued to induce negative energy shockwaves.
3. Backreaction and ANEC Calculation: The linearized gravitational perturbations $h_{MN}$ source a second-order metric perturbation $\gamma_{MN} \sim O(h^2)$. The authors solve the linearized Einstein equations in the hydrodynamic limit (low energy, low temperature) to find the bulk-to-boundary propagators (retarded and Wightman functions).
4. Quantum Expectation Value: The ANEC violation is calculated by evaluating the quantum expectation value of the bulk stress-energy tensor component $\langle \hat{T}^{(2)}_{VV} \rangle$ in the Kruskal coordinate $V$. This involves computing the first-order correction to the Green's function due to the double trace deformation and integrating over the null geodesic.
5. Channel-Specific Analysis:
   • Shear Channel: Three distinct coupling configurations (Case I, II, III) are analyzed. The authors derive a generalized expression for the tensor diffusion constant $D_T$ for an arbitrary blackening function $f(u)$.
   • Sound Channel: The analysis varies the speed of sound ($v_s$) and the attenuation constant ($\Gamma_s$). The authors also investigate the superluminal regime ($v_s > 1$).
6. Data Fitting: A fitting model is developed to describe the late-time behavior of the ANEC integral, comparing the results against known power-law and exponential decay behaviors.

Key Contributions and Results

• Traversability via Gravitational Perturbations: The paper demonstrates that double trace deformations involving the stress-energy tensor (gravitational perturbations) can successfully violate the ANEC, rendering the wormhole traversable in both shear and sound channels. The violation is driven by the quantum expectation value of the second-order stress tensor, $\langle \hat{T}^{(2)}_{VV} \rangle$.
• Shear Channel Dynamics:
  • Three coupling configurations were tested. Case I (coupling $t-x$ components) yields the strongest traversability, followed by Case II ($t-x$ and $z-x$) and Case III ($z-x$).
  • The authors generalize the tensor diffusion constant $D_T$ for arbitrary black hole backgrounds, recovering the standard result $D_T = \beta/4\pi$ for Schwarzschild-AdS$_5$.
  • The wormhole opening $\Delta U$ exhibits a negative value (indicating traversability) that depends on the insertion time of the deformation, with maximal traversability occurring at $t_0 = 0$.
• Sound Channel Dynamics:
  • The sound channel also renders the wormhole traversable, but the degree of traversability is generally lower than in the shear channels.
  • Speed of Sound: As the speed of sound $v_s$ increases, the late-time behavior shifts from diffusive (power-law remnants) to propagating (exponential decay). Higher $v_s$ requires earlier insertion times ($t_0 < 0$) for maximal traversability.
  • Attenuation: The attenuation constant $\Gamma_s$ plays a complex role; increasing it suppresses traversability at early times but enhances it at later times.
  • Superluminality: For superluminal sound speeds ($v_s > 1$), the wormhole opens only for an extremely brief duration at late insertion times, effectively becoming non-traversable for signals.
• Late-Time Power-Law Remnants: The authors propose a generalized fitting function for the ANEC integral $A(V_0)$ that applies to both shear and sound channels:
  $$A_{new}(V_0) = -\frac{V_0}{V_0^{c+1} + 1} a [\ln(V_0 + 1/V_0)]^b$$
  • In the shear channel, the parameter $c \approx 1$, consistent with previous vector field results, indicating a dominant exponential decay with a power-law prefactor.
  • In the sound channel, as $v_s$ increases, the exponent $b$ decreases and $c$ deviates from unity. This indicates a transition where the power-law prefactor is suppressed, and the late-time behavior becomes increasingly dominated by pure exponential decay, reflecting the shift from diffusive to propagating dynamics.

Significance and Claims
The paper claims to extend the GJW protocol to the gravitational sector, demonstrating that dynamical metric perturbations alone can facilitate information transfer without exotic matter, provided the double trace deformation induces the necessary ANEC violation. The emergence of the gravitational constant $G_N$ in the calculation highlights the purely gravitational origin of the effect.

The work provides a unified framework for understanding traversability across different hydrodynamic modes (diffusive vs. propagating). It suggests that the nature of the wormhole opening is sensitive to the transport properties of the dual field theory (diffusion constants vs. sound speeds). The identification of power-law remnants in the shear channel and their suppression in the sound channel as $v_s$ increases offers a quantitative link between the hydrodynamic regime of the boundary theory and the geometric traversability of the bulk wormhole.

The authors modestly note that while the deformation is technically "irrelevant" in the renormalization group sense, it remains effective in the hydrodynamic regime. They also identify the study of fermionic operators and rotating black hole backgrounds as natural future directions, noting that fermions might offer advantages in avoiding superradiance issues in rotating geometries. The paper does not propose immediate experimental realizations but frames the results as a step toward understanding holographic quantum gravity and potential connections to inflationary cosmology via primordial gravitational perturbations.