⚛️ high-energy theory

Classical double copy of black strings in an Anti-de Sitter background

This paper demonstrates that the classical double copy framework consistently applies to static and charged black string solutions in Anti-de Sitter spacetime by constructing their corresponding gauge and scalar single and zeroth copies via a Kerr-Schild formulation.

Original authors: G. Alencar, C. R. Muniz, M. S. Oliveira

Published 2026-02-02

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CC BY 4.0

Original authors: G. Alencar, C. R. Muniz, M. S. Oliveira

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ 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

Imagine the universe as a giant, complex machine. For a long time, physicists have been trying to figure out how the two biggest gears in this machine—Gravity (which pulls planets and stars together) and Gauge Theory (which explains how light and electricity work)—are actually connected.

This paper is like a detective story where the authors try to prove that these two seemingly different forces are actually just two sides of the same coin. They use a clever mathematical trick called the "Double Copy."

Here is the simple breakdown of what they did and what they found:

The Big Idea: The "Double Copy" Recipe

Think of gravity as a complicated, heavy meal. The "Double Copy" theory suggests you can make this meal by taking a simpler recipe (gauge theory/electromagnetism) and "copying" it twice.

Zeroth Copy: The basic ingredients (a simple scalar field, like a temperature map).
Single Copy: The sauce (a gauge field, like an electric field).
Double Copy: The full meal (gravity).

The authors wanted to see if this recipe works for a very specific, weird object: a Black String.

The Object: What is a Black String?

Usually, when we think of a black hole, we imagine a sphere, like a ball. But in this paper, the authors look at a Black String.

Analogy: Imagine a black hole stretched out into an infinite, long noodle. Instead of being a ball in space, it's a long, cylindrical tube that goes on forever in one direction.
The Setting: This noodle isn't floating in empty space; it's sitting in a special kind of curved space called Anti-de Sitter (AdS) space. You can think of AdS space as a bowl with curved walls, rather than a flat table.

The Investigation: Breaking Down the Noodle

The authors took the mathematical description of this "Black String Noodle" and tried to fit it into their "Double Copy" recipe. They had to rewrite the math to show that the gravity of the noodle is just a "copy" of a simpler electric field.

1. The Neutral Noodle (No Charge)
First, they looked at a plain black string with no electric charge.

The Gravity Side: They found that the gravity of this noodle can be described as a simple "ripple" on top of the curved background space.
The Single Copy (The Electric Field): When they applied the "copy" trick, the gravity turned into an electric field. But instead of a point charge (like a single electron), this field looked like an infinite line of color charge running down the center of the noodle.
- Analogy: If a normal black hole is like a single lightbulb, this black string is like a giant, glowing neon tube. The electric field radiates out from the tube in all directions, just like the gravity does.
The Zeroth Copy (The Scalar Field): The simplest version of this is a scalar field (like a heat map). They found that this field behaves exactly as if it were "conformally coupled" to the curved walls of the AdS space.
- Key Finding: The curvature of the space itself acts like the "source" for this field. You don't need to add extra ingredients; the shape of the universe does the work.

2. The Charged Noodle (With Electric Charge)
Next, they added electric charge to the black string.

The Change: Adding charge changed the "recipe" slightly. The math showed that the electric field now has two parts: one part coming from the mass of the string (the "noodle" itself) and a new part coming from the electric charge.
The Result: In the "Single Copy" (the gauge theory side), this created a distributed current.
- Analogy: In the neutral case, the charge was just a line right in the middle. In the charged case, the electric field creates a "cloud" of current flowing through the space around the string, in addition to the line in the middle.
The Zeroth Copy: Even with this extra charge, the simplest scalar field still behaved perfectly according to the rules of the curved space. No new "magic ingredients" were needed to make the math work.

The Conclusion

The authors successfully proved that the "Double Copy" recipe works for these long, cylindrical black strings in curved space.

Gravity (the black string) is perfectly mirrored by Electromagnetism (an infinite line of charge).
The Curved Space (AdS) plays a crucial role, acting as a natural container that helps the math work out without needing extra sources.

In short: They took a weird, stretched-out black hole, broke it down into its simplest parts, and showed that it follows the same "copy-paste" rules as simpler black holes. This confirms that the Double Copy idea is robust enough to handle extended objects (like strings) and curved universes, not just simple spheres in flat space.

Technical Summary: Classical Double Copy of Black Strings in an Anti-de Sitter Background

Problem Statement
The classical double copy formalism, which maps gravitational solutions to gauge theory and scalar field configurations, has been extensively validated for asymptotically flat spacetimes and spherically symmetric objects (e.g., Schwarzschild and Kerr black holes). However, generalizing this correspondence to curved spacetimes, specifically Anti-de Sitter (AdS) backgrounds, and to extended gravitational objects with non-spherical horizon topologies remains a significant challenge. This paper addresses the consistency of the classical double copy for static black strings—topological generalizations of black holes with horizon topology $\mathbb{R} \times S^1$ —within a 3+1-dimensional AdS background. The primary objective is to verify if the generalized Kerr–Schild (KS) construction holds for these cylindrical geometries and to determine the nature of the effective sources in the corresponding gauge and scalar sectors.

Methodology
The authors employ the Kerr–Schild ansatz, expressing the spacetime metric as a linear deformation of a fixed background: $g_{\mu\nu} = \bar{g}_{\mu\nu} + 2H(r)k_\mu k_\nu$ , where $\bar{g}_{\mu\nu}$ is the background AdS metric and $k_\mu$ is a null, geodesic vector field.

Metric Reformulation: The static black string metric (both neutral and charged) is cast into the KS form over a cylindrical AdS geometry. This involves transforming the standard coordinates into ingoing Eddington–Finkelstein (EF) coordinates to isolate the perturbation function $H(r)$ .
Construction of Copies:
- Single Copy: The gravitational null vector $k_\mu$ is replaced by a constant color vector $c^a$ to construct the gauge potential $A^a_\mu = c^a H(r) k_\mu$ .
- Zeroth Copy: The scalar profile is identified directly as the KS scalar function $H(r)$ .
Field Equations Verification: The authors verify that the constructed single copy satisfies Maxwell's equations (with appropriate sources) on the AdS background and that the zeroth copy satisfies the equation for a scalar field conformally coupled to the AdS curvature.
Extension to Charged Solutions: The analysis is extended to charged black strings (Einstein–Maxwell solutions with $\Lambda < 0$ ) to identify modifications in the gauge sector induced by the electromagnetic energy-momentum tensor.

Key Contributions and Results

Neutral Black String:
- The KS scalar function is identified as $H(r) = \frac{\lambda}{2\alpha r}$ , where $\lambda$ is the mass density and $\alpha^2 = -\Lambda/3$ .
- Single Copy: The resulting gauge potential corresponds to a purely temporal component, $A^a_\tau \propto 1/r$ . The field strength tensor describes a cylindrical electrostatic field. The authors demonstrate that for $r \neq 0$ , the solution is source-free, but the singularity at the axis implies a localized source. In the distributional sense, this corresponds to an infinite line of color charge along the string's axis, analogous to the point mass in the Schwarzschild case.
- Zeroth Copy: The scalar profile $H(r)$ satisfies the equation of a conformally coupled scalar field, $(\bar{\nabla}^2 - \bar{R}/6)H = 0$ . Notably, no localized scalar source is required; the background curvature itself effectively acts as the source term, ensuring the equation is satisfied.
Charged Black String:
- The KS function is modified to include the charge density parameter $\beta$ : $H(r) = \frac{\lambda}{2\alpha r} - \frac{\beta^2}{2\alpha^2 r^2}$ .
- Single Copy: The gauge potential retains the $1/r$ term (mass) but acquires a $1/r^2$ term (charge). The resulting field strength includes a term proportional to $1/r^3$ .
- Source Structure: Unlike the neutral case, the charged solution is not vacuum in the bulk. The authors show that the gauge theory side requires a distributed current source in the bulk, $J^\tau_a \propto 1/r^4$ , arising from the electromagnetic energy-momentum tensor of the gravitational solution. Additionally, the $1/r$ term in the potential still implies a localized line of color charge at the axis.
- Zeroth Copy: The modified scalar profile continues to satisfy the conformally coupled scalar field equation without the need for additional localized scalar sources.

Significance and Claims
The paper claims that the classical double copy consistently applies to extended gravitational objects in curved spacetimes. Specifically, it establishes that:

The generalized Kerr–Schild construction is valid for cylindrical black strings in AdS backgrounds.
The correspondence correctly maps the gravitational mass and charge parameters to effective color charge distributions (both localized lines and distributed bulk currents).
The zeroth copy for these geometries is naturally described by a scalar field conformally coupled to the AdS background, where the spacetime curvature plays a crucial role in satisfying the field equations without artificial source terms.

The authors conclude that these results extend the scope of the classical double copy beyond point-like sources and flat backgrounds, providing a framework for understanding the gauge-gravity duality for topologically non-trivial, extended objects. They suggest future work could explore rotating black strings and regular black string solutions (e.g., black-bounce regularizations) to further test the formalism with smoothed sources.

The Big Idea: The "Double Copy" Recipe

The Object: What is a Black String?

The Investigation: Breaking Down the Noodle

The Conclusion

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