⚛️ high-energy theory

Analytic Holographic Pseudo-Entropy in a Boosted BTZ Geometry

This paper derives an exact closed-form expression for holographic entanglement entropy in a boosted BTZ geometry and demonstrates that specific analytic continuations of the boost parameter and conserved charges establish a direct connection between this entropy and holographic pseudo-entropy for moving strongly coupled plasmas.

Original authors: M. M. Daryaei Goki, M. Ali-Akbari

Published 2026-09-25

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CC0 1.0

Original authors: M. M. Daryaei Goki, M. Ali-Akbari

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of modern physics, there exists a profound idea known as the holographic principle. It suggests that the complex, three-dimensional universe we experience might be a projection of information stored on a distant, two-dimensional surface, much like a hologram on a credit card contains a three-dimensional image within a flat film. This concept has become a powerful tool for physicists trying to understand the most extreme environments in nature, such as the swirling, super-hot fluids found inside particle colliders or the crushing gravity of black holes. By translating these difficult problems into the language of geometry, scientists can visualize how information is organized and how quantum particles become entangled, or deeply linked, across space. One of the most important measures of this connection is entanglement entropy, a number that tells us how much information is lost when we look at only a part of a system rather than the whole. When two parts of a system are highly entangled, this number is large; when they are separate, it is zero.

Recently, researchers have begun to explore a more subtle version of this idea called pseudo-entropy. While standard entanglement entropy measures the connection within a single state of a system, pseudo-entropy measures the relationship between two different states. It asks a different question: how distinguishable are two different configurations of a system when viewed from a limited perspective? This concept is particularly useful for studying systems that are changing rapidly or are in motion, where the standard rules of static physics do not apply. However, calculating this quantity is notoriously difficult, especially when the system is moving at high speeds. A team of physicists at Shahid Beheshti University in Iran has now taken a significant step forward by solving this problem for a specific, moving system, revealing a surprising mathematical bridge between motion and the complex nature of quantum information.

The researchers focused their attention on a theoretical model of a black hole in a universe with two spatial dimensions and one time dimension, known as a BTZ black hole. In the world of holography, this black hole acts as a gravitational mirror for a hot, dense fluid of particles moving along a line. To simulate a fluid that is not just sitting still but is rushing forward, the scientists applied a mathematical transformation called a Lorentz boost. This is the same type of transformation that describes how time and space change for an observer moving at high speeds. By boosting the black hole geometry, they created a model of a thermal plasma that is moving uniformly. The challenge was to calculate the entanglement entropy for a specific segment of this moving fluid. In a stationary system, this calculation involves finding the shortest path, or geodesic, through the curved space of the black hole. But because the fluid is moving, the geometry becomes twisted, and the path is no longer simple. The researchers had to solve a complex set of equations to find the exact shape of this path, accounting for the fact that time and space are now mixed together.

After working through the mathematics, the team derived a precise, closed-form formula for the entanglement entropy of this moving fluid. This formula depends on several key factors: the length of the segment being measured, the speed of the fluid, the temperature of the system, and specific quantities that remain constant as the system evolves. The result was a major success because it passed rigorous tests. When the scientists set the speed of the fluid to zero, their new formula perfectly matched the known results for a stationary black hole. Similarly, when they removed the heat, the formula smoothly transitioned to the known result for empty space. These checks confirmed that their complex new equation was correct and consistent with established physics. The formula provided a complete, analytic description of how entanglement behaves in a moving, hot environment, filling a gap that previous numerical approximations had left open.

The most striking part of their work came when they asked what would happen if they treated the moving system not just as a physical object, but as a mathematical puzzle to be solved in a different way. They performed a procedure known as analytic continuation, which involves shifting certain numbers in their equations into the realm of complex numbers. This is a standard technique in physics that often reveals hidden structures. When they applied this to their moving black hole, they discovered two distinct possibilities. In the first scenario, they adjusted the speed and time parameters in a way that kept the physical boundaries of their system real and tangible. In this case, the resulting entropy remained a real number, just like the standard entanglement entropy we are used to.

However, in the second scenario, they allowed the boundary of the system to become imaginary while keeping the rest of the math consistent. This choice led to a result that was fundamentally different: the entropy became a complex number, possessing both a real part and an imaginary part. In the language of physics, a complex entropy is not a measurement of a single static state but a measure of the relationship between two different states. The researchers identified this complex value as the holographic version of pseudo-entropy. The real part of this number describes the usual correlations, while the imaginary part encodes the subtle interference and overlap between the two different quantum states. This finding is significant because it provides a concrete geometric picture of how pseudo-entropy arises. It shows that the complex nature of this information measure is not an abstract mathematical trick, but a direct consequence of the geometry of spacetime when viewed through the lens of a moving, thermal system.

By connecting the motion of a black hole to the complex structure of quantum information, this work extends the toolkit available to physicists studying the quantum nature of reality. It demonstrates that the holographic principle can handle not just static, frozen snapshots of the universe, but also dynamic, moving ones. The ability to calculate these quantities exactly, rather than relying on computer simulations, allows for a deeper understanding of how information is preserved and transformed in extreme conditions. The study confirms that the bridge between the geometry of gravity and the entanglement of quantum particles is robust enough to withstand the introduction of motion and the exploration of complex numbers. This clarity offers a new way to think about how the universe might encode the differences between various states of matter, suggesting that the "ghostly" imaginary parts of quantum information have a very real, geometric home in the fabric of spacetime.

Technical Summary: Analytic Holographic Pseudo-Entropy in a Boosted BTZ Geometry

Problem Statement
The paper investigates the calculation of Holographic Entanglement Entropy (HEE) and its generalization to pseudo-entropy within the framework of gauge-gravity duality. Specifically, the authors address the challenge of computing these quantities for a $(1+1)$ -dimensional strongly coupled plasma in a moving frame, which is dual to a boosted BTZ black hole geometry. While HEE for static thermal states and vacuum configurations is well-established, and analytic continuations for rotating BTZ geometries have been studied, a closed-form analytic expression for HEE in a Lorentz-boosted spacetime—and its subsequent connection to holographic pseudo-entropy—remains an open technical problem. The presence of off-diagonal metric components in the boosted geometry complicates the determination of extremal surfaces compared to static backgrounds.

Methodology
The authors employ the Ryu-Takayanagi (RT) prescription, generalized to time-dependent or boosted backgrounds via the Hubeny-Rangamani-Takayanagi (HRT) formalism. The methodology proceeds in three main stages:

Static Baseline: The authors first re-derive the HEE for static BTZ black holes at both zero and finite temperatures. This serves as a consistency check, recovering the standard vacuum logarithmic scaling and the finite-temperature thermal entropy formula.
Boosted Geometry Construction: They construct the gravitational dual of a moving thermal state by applying a Lorentz boost to the static BTZ metric. This introduces an off-diagonal metric component ( $g_{tx}$ ), coupling temporal and spatial directions.
Extremal Surface Analysis: Unlike the static case where the extremal surface is described by a single embedding function $x(z)$ , the boosted case requires two independent functions, $x(z)$ and $t(z)$ , due to the lack of a static Killing vector. The authors solve the variational problem by identifying two conserved quantities, $E$ (associated with time translation) and $P$ (associated with spatial translation), derived from the Lagrangian of the geodesic length. They analytically solve for the turning point of the geodesic and integrate the length functional to obtain a closed-form expression for the HEE.
Analytic Continuation: To access pseudo-entropy, the authors perform analytic continuations to Euclidean signature. They identify that a direct continuation ( $t \to it_E$ $t \to i t_{E}$ ) renders the metric complex due to the off-diagonal term. To maintain a real Euclidean metric, they simultaneously continue the boost velocity ( $v \to iv_E$ $v \to i v_{E}$ ). They explore two distinct continuation schemes:
- Scheme A: Continuing $t$ , $v$ , and the conserved charge $E$ . This keeps the boundary interval real and yields a real-valued entropy.
- Scheme B: Continuing only $t$ and $v$ . This results in an imaginary boundary interval and a complex-valued entropy, which the authors interpret as the holographic dual of pseudo-entropy.

Key Contributions and Results

Exact Analytic Formula for Boosted HEE: The primary result is the derivation of an exact, closed-form expression for the HEE of a boosted BTZ black hole (Equation 52). This formula expresses the entropy as a function of the boundary interval length ( $\Delta x$ ), boost velocity ( $v$ ), temperature ( $T$ ), and the conserved momenta ( $E, P$ ).
Consistency Checks: The derived formula is shown to correctly reproduce known limits:
- In the vanishing boost limit ( $v \to 0$ ), it reduces to the standard thermal BTZ entanglement entropy.
- In the zero-temperature limit ( $T \to 0$ ), it smoothly reduces to the vacuum conformal result.
Comparison with Literature: The authors compare their result with a previous numerical study [29] and an analytic derivation based on boosting the thermal solution [28]. They demonstrate that their formula agrees numerically with [28] in the limit $P \gg E$ , though they note that a direct analytical mapping between their conserved-charge formulation and the boundary-data formulation of [28] remains non-trivial.
Holographic Pseudo-Entropy Construction: By applying the specific analytic continuation where the boundary interval becomes imaginary (Scheme B), the authors derive a complex-valued entropy. They decompose this into real and imaginary parts (Equations 63 and 64), identifying the complex quantity as the holographic pseudo-entropy for the boosted thermal state. This provides a geometric realization of pseudo-entropy in Lorentz-boosted spacetimes, extending previous work on rotating black holes.

Significance and Claims
The paper claims to establish a direct connection between boosted HEE and holographic pseudo-entropy. Its significance lies in:

Analytic Transparency: Providing a step-by-step derivation of the HEE in a boosted background, expressed explicitly in terms of the underlying conserved charges ( $E, P$ ), which offers a clearer handle on the dependence of entropy on these quantities compared to purely numerical treatments.
Extension of Frameworks: Extending the analytic framework for pseudo-entropy, previously developed for rotating BTZ geometries, to Lorentz-boosted spacetimes.
Geometric Interpretation: Demonstrating that the non-Hermitian nature of the transition matrix in pseudo-entropy (leading to complex values) finds a natural geometric counterpart in the analytic continuation of the bulk extremal surface in a boosted background.

The authors maintain a modest tone, noting that while their result reproduces known physics in limiting cases and agrees numerically with existing literature, the precise analytical relationship between their conserved-charge formulation and other boundary-data formulations remains an area where the mapping is not fully explicit. They do not propose new experimental applications but focus on the theoretical consistency and geometric interpretation of entropic measures in non-static holographic duals.

Technical Summary: Analytic Holographic Pseudo-Entropy in a Boosted BTZ Geometry

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