In this work, we delve into the model of the shift symmetric and
parity-preserving Beyond Horndeski theory in all its generality. We present an
explicit algorithm to extract static and spherically symmetric black holes with
primary scalar charge adhering to the conservation of the Noether current
emanating from the shift symmetry. We show that when the functionals $G_2$ and
$G_4$ of the theory are linearly dependent, analytic homogeneous black-hole
solutions exist, which can become regular by virtue of the primary charge
contribution. Such geometries can easily enjoy the preservation of the Weak
Energy Conditions, elevating them into healthier compact objects than most
hairy black holes in modified theories of gravity. Finally, we revisit the
concept of disformal transformations as a solution-generating mechanism and
discuss the case of generic $G_2$ and $G_4$ functionals.
Future Roadmap:
1. Introduction
Begin the article by introducing the topic of the shift symmetric and parity-preserving Beyond Horndeski theory. Explain why this theory is important and relevant in the study of black holes and modified theories of gravity.
2. Model Description
Provide a detailed description of the model, including its mathematical formulation and the role of the Noether current in conserving the primary scalar charge. Explain the significance of the linear dependence between the functionals G_2 and G_4.
3. Extraction of Static and Spherically Symmetric Black Holes
Present the explicit algorithm developed in this work to extract static and spherically symmetric black hole solutions in the model. Discuss the conditions under which these solutions exist and demonstrate how they adhere to the conservation of the Noether current.
4. Regularization of Black Hole Solutions
Highlight the regularity of the black hole solutions obtained through the primary charge contribution. Discuss how these solutions can preserve the Weak Energy Conditions, making them healthier compact objects compared to other hairy black holes in modified theories of gravity. Present evidence or examples supporting this conclusion.
5. Revisiting Disformal Transformations
Revisit and explore the concept of disformal transformations as a solution-generating mechanism in the model. Discuss how generic G_2 and G_4 functionals influence or contribute to this mechanism. Provide insights or examples to illustrate this concept.
6. Conclusion
Summarize the key findings and conclusions of the study, emphasizing the potential of the shift symmetric and parity-preserving Beyond Horndeski theory in understanding and studying black holes in modified theories of gravity. Highlight any implications or future directions for research based on the results obtained.
Challenges and Opportunities:
- One potential challenge in further exploring this model is the complexity of the mathematical formulation. Research and development of more efficient algorithms or computational methods may be necessary to extract and analyze a larger variety of black hole solutions.
- Another challenge is the validation and verification of the regularity and preservation of Weak Energy Conditions in the obtained black hole solutions. Further theoretical analysis and numerical simulations could help address these challenges.
- An opportunity lies in investigating the physical properties and astrophysical implications of the regularized black hole solutions in the model. Understanding how these solutions behave under various conditions or in the presence of other objects or forces could lead to valuable insights and potential applications in astrophysics and cosmology.
- The concept of disformal transformations presents an interesting avenue for future research. Exploring different types of functionals, their effects on solution generation, and their physical interpretations could uncover new possibilities and deepen our understanding of black hole physics.