The classical double copy for Taub-NUT spacetime

Abstract: The double copy is a much-studied relationship between gauge theory and gravity amplitudes. Recently, this was generalised to an infinite family of classical solutions to Einstein's equations, namely stationary Kerr-Schild geometries. In this paper, we extend this to the Taub-NUT solution in gravity, which has a double Kerr-Schild form. The single copy of this solution is a dyon, whose electric and magnetic charges are related to the mass and NUT charge in the gravity theory. Finally, we find hints that the classical double copy extends to curved background geometries.

• The paper extends the classical double copy to map Taub-NUT spacetime to a gauge theory dyon using a double Kerr-Schild decomposition.
• It generalizes the single Kerr-Schild metric by employing two null vectors to linearize Einstein’s equations for complex gravitational solutions.
• The study demonstrates that the double copy framework remains valid in curved backgrounds like (Anti)-de Sitter spaces, reinforcing gauge-gravity duality.

The Classical Double Copy for Taub-NUT Spacetime

The exploration of relationships between gauge theory and gravity continues to intrigue researchers, particularly through the concept of the "double copy." This concept provides a framework where solutions in gauge theories can be linked to solutions in gravity, extending beyond the field of perturbative scattering amplitudes to include classical solutions as well. The paper at hand makes a significant contribution to this discussion by examining the possibility of extending the classical double copy to the Taub-NUT solution in general relativity, specifically utilizing a double Kerr-Schild form.

Main Contributions

1. Extension to Taub-NUT Solutions: The authors extend the classical double copy to the Taub-NUT spacetime, a well-known solution of Einstein's equations. The Taub-NUT solution is notable for its gravitational analog to magnetic monopoles due to its NUT charge. The paper demonstrates how this solution can be mapped to a gauge theory dyon, consisting of both electric and magnetic charges. This mapping is achieved by employing a double Kerr-Schild decomposition, which allows the general relativity solution to maintain its linear form, a property leveraged to achieve the double copy.
2. Double Kerr-Schild Metrics: The work generalizes the single Kerr-Schild form, which has been pivotal in prior studies, to accommodate double Kerr-Schild metrics. While the standard approach involves a single vector in the Kerr-Schild metric that linearizes Einstein’s equations, the double Kerr-Schild form utilizes two null vectors, facilitating more complex solutions like the Taub-NUT. The successful application of this form to the Taub-NUT solution demonstrates its potential utility in linking more intricate gravitational solutions to their gauge theory counterparts.
3. Applications to Curved Backgrounds: Another significant aspect of the paper is the extension of the classical double copy to curved backgrounds, specifically (Anti)-de Sitter spaces. The paper asserts that the single copy, when applied to the Taub-NUT solution with a (Anti)-de Sitter background, remains valid, thus highlighting the robustness of the double copy principles beyond flat space assumptions. The paper finds that the associated gauge field satisfies curved space Maxwell equations, suggesting the adaptability of double copy methods to non-Minkowski backgrounds.
4. Zeroth Copy Insights: With the help of biadjoint scalar theory, the authors show that the scalar field corresponds to solutions of the linearized biadjoint scalar equation, akin to the gauge theory. This zeroth copy process ensures continuity with the theoretical framework for scattering amplitudes and serves as a mechanism to determine the single copy's form. For de Sitter backgrounds, it necessitates a coupling term indicating conformal symmetry, pointing to intriguing connections between scalar propagators and gravitational backgrounds.

Implications and Future Development

The paper's findings have theoretical implications for understanding the interplay between gauge and gravity theories through the double copy concept. It establishes the viability of applying double copy techniques to non-trivial and curved spacetime solutions, potentially unlocking new pathways for exploring both perturbative and non-perturbative aspects of high energy physics.

In the broader context of theoretical developments, this work lays groundwork for further studies on complex gravitational solutions and their gauge theory counterparts. Specifically, exploring additional examples where double Kerr-Schild forms can be applied or extended in higher dimensions could lead to deeper insights. The implications of the double copy in non-flat backgrounds might also reveal new perspectives on gauge and gravity dualities, particularly in frameworks involving cosmological constants or other curved spacetimes.

Thus, this paper not only extends the current understanding of the double copy for classical solutions but also prompts future inquiries into both its theoretical underpinnings and practical applications in high-energy physics.