by jsendak | Sep 17, 2024 | GR & QC Articles
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.
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by jsendak | Jul 30, 2024 | GR & QC Articles
arXiv:2407.18973v1 Announce Type: new
Abstract: We study the Cauchy problem of higher dimensional Einstein-Maxwell-Higgs system in the framework of Bondi coordinates. As a first step, the problem is reduced to a single first-order integro-differential equation by defining a generalized ansatz function. Then, we employ contraction mapping to show that there exists the unique fixed point of the problem. For a given small initial data, we prove the existence of a global classical solution. Finally, by introducing local mass and local charge functions in higher dimensions, we also show the completeness property of the spacetimes.
Conclusions:
- The authors have studied the Cauchy problem of the higher dimensional Einstein-Maxwell-Higgs system.
- They have utilized Bondi coordinates and reduced the problem to a single first-order integro-differential equation using a generalized ansatz function.
- They have applied contraction mapping to prove the existence of a unique fixed point of the problem.
- They have demonstrated the existence of a global classical solution for small initial data.
- They have introduced local mass and local charge functions in higher dimensions and shown the completeness property of the spacetimes.
Future Roadmap:
- To further explore the implications of the higher dimensional Einstein-Maxwell-Higgs system, future research can focus on studying the behavior of the system under different initial conditions.
- Challenges may arise in determining the existence of global solutions for larger initial data sets and investigating the stability of the solutions over time.
- Opportunities exist to analyze the physical implications of the local mass and local charge functions in higher dimensions and their relevance to other aspects of theoretical physics.
- Possibilities for extending the study to other related systems, such as the inclusion of additional fields or considering different types of coordinates, could provide valuable insights.
- Further investigation could involve the examination of the system in the presence of external perturbations or examining the behavior of the system in different spacetime geometries.
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by jsendak | Jul 8, 2024 | GR & QC Articles
arXiv:2407.03395v1 Announce Type: new
Abstract: We study the statistical fluctuations (such as the variance) of causal set quantities, with particular focus on the causal set action. To facilitate calculating such fluctuations, we develop tools to account for correlations between causal intervals with different cardinalities. We present a convenient decomposition of the fluctuations of the causal set action into contributions that depend on different kinds of correlations. This decomposition can be used in causal sets approximated by any spacetime manifold $mathcal M$. Our work paves the way for investigating a number of interesting discreteness effects, such as certain aspects of the Everpresent $Lambda$ cosmological model.
Statistical Fluctuations in Causal Set Quantities
Introduction
In this study, we examine the statistical fluctuations, particularly the variance, of causal set quantities. Our focus is primarily on the causal set action and finding ways to calculate these fluctuations accurately. We aim to develop tools that can account for correlations between causal intervals of different cardinalities. This research is significant as it opens up avenues for understanding discreteness effects in various systems, including the Everpresent $Lambda$ cosmological model.
Fluctuations and Correlations
To calculate the fluctuations of the causal set action, we introduce a convenient decomposition method that considers the different kinds of correlations present. This decomposition allows us to analyze the contributions from these correlations individually. By understanding the nature of these correlations, we can gain insights into the underlying mechanisms driving the statistical fluctuations in causal set quantities.
Potential Challenges and Opportunities
This research opens up several challenges and opportunities for future investigations. Firstly, while we have developed tools to account for correlations, further refinement and testing of these methods are necessary. This includes exploring their applicability to various spacetime manifolds $mathcal M$, expanding the scope of our research in terms of the systems we can analyze. Additionally, developing efficient computational algorithms to handle the calculations involved will be critical.
Furthermore, applying this decomposition of fluctuations to the Everpresent $Lambda$ cosmological model can provide valuable insights into the discrete nature of the universe and the role of causal sets in cosmology. Investigating how different correlations affect the fluctuations of causal set quantities in this model can lead to new understandings of the underlying dynamics and potentially challenge existing theories.
Moreover, this research paves the way for studying other discreteness effects in a variety of systems beyond cosmology. The tools and methods developed can be applied to different areas of physics and even other disciplines where the concept of causal sets is relevant. This interdisciplinary potential for research can lead to novel discoveries and advancements in multiple fields.
Conclusion
Overall, this study on statistical fluctuations in causal set quantities provides a significant contribution to our understanding of discreteness effects in various systems. The introduced decomposition method allows for the analysis of different correlations and their individual contributions. Challenges and opportunities lie ahead in refining and expanding these methods, applying them to the Everpresent $Lambda$ cosmological model, and exploring other areas of physics and interdisciplinary research. By addressing these challenges, we can uncover new insights and potentially revolutionize our understanding of the universe’s fundamental structure.
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by jsendak | Jun 6, 2024 | GR & QC Articles
arXiv:2406.02621v1 Announce Type: new
Abstract: We construct new conserved quasi-local energies in general relativity using the formalism developed by cite{CWY}. In particular, we use the optimal isometric embedding defined in cite{yau,yau1} to transplant the conformal Killing fields of the Minkowski space back to the $ 2-$ surface of interest in the physical spacetime. For an asymptotically flat spacetime of order $1$, we show that these energies are always finite. Their limit as the total energies of an isolated system is evaluated and a conservation law under Einsteinian evolution is deduced.
Consclusions
The article discusses the construction of new conserved quasi-local energies in general relativity. These energies are determined using the formalism developed by CWY and involve transplanting the conformal Killing fields of the Minkowski space back to the 2-surface of interest in the physical spacetime. The article demonstrates that these energies are always finite for an asymptotically flat spacetime of order 1. Furthermore, the article evaluates the limit of these energies as the total energies of an isolated system and establishes a conservation law under Einsteinian evolution.
Future Roadmap
Challenges
- The first challenge for future research is to extend the construction of conserved quasi-local energies to more general spacetimes beyond asymptotically flat spacetimes of order 1.
- Another challenge is to explore the applicability and limitations of the conservation law under Einsteinian evolution that is deduced from the evaluated limit of these energies.
- Additionally, there is a need to investigate the implications of these new conserved energies for specific physical systems and phenomena in general relativity.
Opportunities
- The construction of new conserved quasi-local energies has the potential to enhance our understanding of energy and conservation laws in general relativity.
- By extending the construction to more general spacetimes, a broader range of physical systems and phenomena can be studied using these energies.
- Exploring the applicability and limitations of the deduced conservation law can potentially lead to new insights into the behavior of isolated systems under Einsteinian evolution.
Overall, the future roadmap for readers includes addressing the challenges of extending the construction to more general spacetimes, investigating the implications of the conservation law, and exploring the applications of these energies in various physical systems. By doing so, new opportunities for understanding and advancing the field of general relativity can be realized.
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by jsendak | May 22, 2024 | GR & QC Articles
arXiv:2405.12249v1 Announce Type: new
Abstract: We give an alternate proof of one of the results given in [16] showing that initial data sets with boundary for the Einstein equations $(M, g, k)$ satisfying the dominant energy condition can be conformally deformed to the strict dominant energy condition, while preserving the character of the boundary (minimal, future trapped, or past trapped) while changing the area of the boundary and ADM energy of the initial data set by an arbitrarily small amount. The proof relies on solving an equation that looks like the equation for spacetime harmonic functions studied in [7], but with a Neumann boundary condition and non-zero right hand side, which we refer to as a spacetime Poisson equation. One advantage of this method of proof is that the conformal deformation is explicitly constructed as a solution to a PDE, as opposed to only knowing the solution exists via an application of the implicit function theorem as in [16]. We restrict ourselves to the physically relevant case of a $3$-manifold $M$, though the proof can be generalized to higher dimensions.
Future Roadmap: Challenges and Opportunities
Introduction
In this article, we examine an alternate proof of a result presented in a previous work. The result shows that initial data sets with boundaries for the Einstein equations can be conformally deformed while satisfying the dominant energy condition. The proof relies on solving a spacetime Poisson equation with a Neumann boundary condition, and the conformal deformation is explicitly constructed as a solution to a PDE. In this roadmap, we outline potential challenges and opportunities on the horizon.
Potential Challenges
- Mathematical Complexity: The proof relies on solving a spacetime Poisson equation with specific boundary conditions. The mathematics involved in solving such equations can be complex and require expertise in partial differential equations.
- Generalization to Higher Dimensions: The proof presented in this article focuses on the physically relevant case of a 3-manifold. Generalizing the proof to higher dimensions may introduce additional challenges, as the equations and techniques involved can become more intricate.
Potential Opportunities
- Improved Understanding of Einstein Equations: The alternate proof presented in this article offers a new perspective on conformal deformations within the context of the Einstein equations. This can potentially enhance our understanding of the mathematical properties of these equations and their solutions.
- Enhanced Applications: By explicitly constructing the conformal deformation as a solution to a PDE, the proof offers a practical approach for applying conformal deformations in various domains ranging from physics to geometry. This can lead to potential advancements in fields such as general relativity, astrophysics, and differential geometry.
- Further Developments in Spacetime Harmonic Functions: The proof involves solving an equation related to spacetime harmonic functions. This opens up possibilities for further research and developments in the theory and applications of these functions.
Conclusion
In conclusion, the alternate proof presented in this article offers a promising approach for conformal deformations of initial data sets with boundaries in the context of the Einstein equations. While the proof brings forth challenges in terms of mathematical complexity and generalization to higher dimensions, it also presents opportunities for improved understanding of the Einstein equations, enhanced applications, and further developments in spacetime harmonic functions. This roadmap provides a glimpse into the potential challenges and opportunities that lie ahead for readers interested in this area of research.
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