Symmetry Groups of Plane-Fronted Gravitational Waves in Teleparallel Gravity: A Comprehensive Analysis

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This research paper explores the symmetry groups of plane-fronted gravitational waves (pp-waves) within the framework of teleparallel gravity, an alternative theory of gravity. The authors utilize symmetry methods and the Cartan-Karlhede algorithm to identify all permissible symmetry groups for these spacetimes, uncovering previously overlooked solutions in both teleparallel gravity and general relativity.

Resumen

• Bibliographic Information: Landry, A., & McNutt, D. D. (2024). All symmetry groups of pp-waves in teleparallel gravity. arXiv preprint arXiv:2411.11420v1.
• Research Objective: This study aims to determine the symmetry groups admissible for plane-fronted gravitational wave (pp-wave) solutions within the framework of teleparallel gravity.
• Methodology: The authors employ a combination of symmetry methods, including Killing vector fields and the Cartan-Karlhede algorithm, to analyze the teleparallel analogues of pp-wave spacetimes. They leverage the fact that pp-wave solutions in general relativity belong to the class of Vanishing Scalar Invariants (VSI) spacetimes, and extend this concept to teleparallel gravity.
• Key Findings: The research identifies all permissible symmetry groups for teleparallel pp-wave spacetimes with dimensions less than three and trivial isotropy groups. Additionally, it explores subclasses with non-trivial linear isotropy, utilizing the Cartan-Karlhede algorithm to pinpoint specific cases with the largest possible symmetry groups. Notably, the study uncovers two previously overlooked solutions applicable to both teleparallel gravity and general relativity.
• Main Conclusions: The authors successfully determine the complete set of symmetry groups for pp-waves in teleparallel gravity, providing a comprehensive classification of these spacetimes. The discovery of new solutions highlights the efficacy of symmetry methods in exploring alternative gravity theories.
• Significance: This research significantly contributes to the understanding of gravitational waves in alternative theories of gravity. The comprehensive analysis of symmetry groups provides valuable insights into the mathematical structure and potential physical implications of teleparallel gravity.
• Limitations and Future Research: The study primarily focuses on pp-wave spacetimes with specific symmetry properties. Further research could explore more general classes of solutions in teleparallel gravity and investigate the physical characteristics of the newly identified solutions.

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All symmetry groups of pp-waves in teleparallel gravity

by A. Landry, D... a las arxiv.org 11-19-2024

https://arxiv.org/pdf/2411.11420.pdf

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How do the symmetry properties of pp-waves in teleparallel gravity differ from those in other modified theories of gravity?

While the provided text focuses on the symmetry properties of pp-waves specifically within the framework of Teleparallel Gravity (TPG), it doesn't delve into the comparative analysis with other modified theories of gravity.
However, we can draw some insights based on the unique characteristics of TPG and its implications for symmetry considerations:

Frame-based vs. Metric-based: Unlike General Relativity (GR) and some modified gravity theories that are metric-based, TPG is fundamentally frame-based. This means that the fundamental geometric objects are the coframe fields ($h^a$) and the spin connection ($\omega^a_{\ bc}$), rather than the metric tensor. Consequently, the notion of symmetry in TPG extends beyond isometries (metric symmetries) to encompass affine frame symmetries. These symmetries must preserve not only the metric but also the torsion tensor and its covariant derivatives, as captured by Equation (5) in the context.

Torsion and Curvature: TPG attributes gravity to the torsion of spacetime while maintaining zero curvature. This contrasts with GR, where gravity is a manifestation of curvature. Modified theories like $f(R)$ gravity modify the curvature sector, while others like scalar-tensor theories introduce additional fields. These modifications directly impact the symmetry properties of solutions. For instance, the presence of a scalar field can impose additional constraints on the Killing vectors, potentially reducing the symmetry group compared to GR.

Teleparallel Equivalent of General Relativity (TEGR): The provided text highlights that for vanishing torsion scalar ($T=0$), the field equations of TPG reduce to those of TEGR, which are equivalent to GR. This implies that in this specific case, the symmetry properties of pp-waves in TPG would coincide with those in GR. However, for more general TPG theories with non-zero torsion scalar, the symmetry groups might differ.
In summary, the frame-based nature of TPG and its reliance on torsion introduce unique aspects to symmetry considerations. A direct comparison with other modified gravity theories would require a detailed analysis of their respective field equations and symmetry constraints. The specific form of modifications in each theory would dictate how the symmetry properties of solutions, including pp-waves, deviate from GR and from each other.

Could the newly discovered solutions in teleparallel gravity have observable consequences in astrophysical phenomena?

The text mentions the discovery of "two overlooked solutions" permitted in both TPG and GR. However, it doesn't provide explicit details about the physical interpretation or potential astrophysical consequences of these solutions.
To assess their observable consequences, we need to consider several factors:

Nature of the Solutions: The text only mentions that the solutions were found but doesn't elaborate on their specific properties. Are these vacuum solutions, or do they involve matter fields? What are their asymptotic behaviors? Answering these questions is crucial to understanding if these solutions represent physically realistic scenarios.

Deviations from GR: Even if the solutions are permitted in both TPG and GR, their observational consequences might differ. TPG, being a modified theory of gravity, could predict subtle deviations from GR in the strong-field regime or at cosmological scales. These deviations might manifest as differences in the gravitational wave signatures of merging black holes or in the evolution of the early Universe.

Observational Tests: To determine if these newly found solutions have observable consequences, we need to identify specific astrophysical phenomena where TPG's predictions deviate significantly from GR. This could involve searching for deviations in the gravitational wave signals detected by LIGO/Virgo or looking for signatures in the cosmic microwave background radiation.
In conclusion, while the discovery of new solutions in TPG is intriguing, it's premature to claim observable consequences without a thorough investigation of their physical properties and potential deviations from GR. Further research is needed to explore their implications for astrophysical phenomena and to devise observational tests that could distinguish TPG from GR.

What are the implications of these findings for our understanding of the relationship between geometry and gravity?

The exploration of pp-wave solutions and their symmetries in TPG offers valuable insights into the intricate relationship between geometry and gravity, particularly in the context of modified theories. Here are some key implications:

Alternative Descriptions of Gravity: The fact that TPG, a theory fundamentally different from GR in its description of gravity (torsion vs. curvature), can still harbor solutions like pp-waves, which are well-known in GR, underscores the possibility of having alternative, yet viable, geometric representations of gravity. This challenges the notion of a unique geometric description and suggests a richer interplay between geometry and gravity than captured solely by GR.

Role of Torsion: The findings emphasize the significance of torsion, a geometric property often neglected in standard GR, in shaping gravitational phenomena. The constraints imposed by torsion on the symmetry properties of pp-waves in TPG highlight its non-trivial role in determining the spacetime structure and dynamics.

Beyond Riemannian Geometry: TPG, by relying on torsion, ventures beyond the realm of Riemannian geometry, which underpins GR. The study of solutions in TPG, therefore, provides a window into the broader landscape of possible geometries that could potentially harbor gravitational theories. This expands our understanding of the geometric foundations of gravity.

Modified Theories and Symmetries: The analysis of symmetry groups in TPG, particularly the potential reduction in symmetry compared to GR, highlights a crucial aspect of modified gravity theories. Modifications to GR often come with altered symmetry properties, which can have profound implications for the nature of solutions and the predicted physical phenomena.
In essence, the exploration of pp-waves in TPG enriches our understanding of the relationship between geometry and gravity by:

Revealing the possibility of alternative geometric formulations of gravity.
Highlighting the role of torsion as a fundamental geometric property in gravity.
Expanding the scope of geometries relevant for gravitational theories.
Emphasizing the impact of modifications to GR on the symmetry properties of solutions.

These findings motivate further investigation into the broader implications of TPG and other modified theories for our understanding of the deep connection between geometry and gravity.