arXiv:2409.08344v1 Announce Type: new
Abstract: We study the exterior solution for a static, spherically symmetric source in Weyl conformal gravity in terms of the Newman–Penrose formalism. We first show that both the static, uncharged black hole solution of Mannheim and Kazanas and the static, charged Reissner–Nordstr”{o}m-like solution can be found more easily in this formalism than in the traditional coordinate-basis approach, where the metric tensor components are taken as the basic variables. Second, we show that the Newman-Penrose formalism offers a particularly convenient framework that is well suited for the discussion of conformal gravity solutions corresponding to Petrov ”type-D” spacetimes. This is illustrated with a two-parameter class of wormhole solutions that includes the Ellis–Bronnikov wormhole solution of Einstein’s gravity as a limiting case. Other salient issues, such as the gauge equivalence of solutions and the inclusion of the electromagnetic field are also discussed.
Introduction
In this article, we explore the exterior solution for a static, spherically symmetric source in Weyl conformal gravity using the Newman-Penrose formalism. We highlight the advantages of this formalism over the traditional coordinate-based approach and discuss its applications in the study of conformal gravity solutions.
Advantages of the Newman-Penrose Formalism
We demonstrate that the Newman-Penrose formalism provides a more straightforward method for finding both the static, uncharged black hole solution and the static, charged Reissner-Nordström-like solution as compared to the traditional coordinate-basis approach. By utilizing the metric tensor components as basic variables, we simplify the computation process.
Applications in Conformal Gravity
We illustrate how the Newman-Penrose formalism offers a convenient framework for analyzing conformal gravity solutions corresponding to Petrov “type-D” spacetimes. We present a specific class of wormhole solutions that includes the Ellis-Bronnikov wormhole solution of Einstein’s gravity as a limiting case. This demonstrates the potential for utilizing conformal gravity to achieve wormhole solutions with interesting properties.
Other Salient Issues
We also address additional significant topics in this article. We discuss the gauge equivalence of solutions in the Newman-Penrose formalism, highlighting the importance of considering different gauge choices to obtain a complete understanding of the physics involved. Additionally, we explore the inclusion of the electromagnetic field and its impact on the conformal gravity solutions.
Future Roadmap, Challenges, and Opportunities
Roadmap
- Further explore the Newman-Penrose formalism for other types of solutions in Weyl conformal gravity
- Investigate the physical implications and potential applications of the two-parameter class of wormhole solutions
- Study the gauge equivalence of various solutions and its consequences
- Examine the effects of electromagnetic fields on conformal gravity solutions in more detail
Challenges
One of the main challenges in future research is to extend the use of the Newman-Penrose formalism to more complex systems and solutions in Weyl conformal gravity. This may require developing new mathematical techniques and computational tools to handle the increased complexity.
Opportunities
Exploring the two-parameter class of wormhole solutions and their properties opens up opportunities for applications in areas such as faster-than-light travel and exotic matter. Additionally, studying the gauge equivalence of solutions and the role of electromagnetic fields may lead to a deeper understanding of the fundamental physics involved in conformal gravity.
Conclusion
The Newman-Penrose formalism offers a more straightforward approach to find solutions in Weyl conformal gravity, particularly for static, spherically symmetric sources. By utilizing this framework, we have demonstrated the ease of obtaining black hole and wormhole solutions. The inclusion of the electromagnetic field and the study of gauge equivalence adds further depth to the analysis of conformal gravity solutions. Future research should focus on expanding the use of the Newman-Penrose formalism and exploring the implications and applications of wormhole solutions, while addressing challenges that arise with increased complexity.