We investigate possible manifolds characterizing traversable wormholes in the
presence of a scalar field, which is minimally coupled to gravity and has both
kinetic and potential energy. The feature of traversability requires the
violation of the null energy condition, which, in turn, signals the existence
of exotic matter with negative energy density. For this reason, we impose a
hypothetical Casimir apparatus with plates positioned at a distance either
parametrically fixed or radially varying. The main feature of all the derived
solutions is the conservation of the Stress Energy Tensor. Such a conservation
though requires the introduction of an auxiliary field, which we interpret as a
gravitational response of the Traversable Wormhole to the original source.
Interestingly, the only case that seems to avoid the necessity for such an
auxiliary field, is the one involving a scalar field with a potential, in
combination with a Casimir device with fixed plates.
Future Roadmap: Challenges and Opportunities
1. Exploring Manifolds Characterizing Traversable Wormholes
One of the primary areas of focus for future research should be a deeper investigation into the possible manifolds characterizing traversable wormholes in the presence of a scalar field. By understanding the nature of these manifolds, we can gain insights into the underlying physics governing these structures.
2. Overcoming the Null Energy Condition Violation
The feature of traversability in wormholes necessitates the violation of the null energy condition, indicating the existence of exotic matter with negative energy density. Finding ways to overcome this challenge and potentially discovering alternative mechanisms for traversability would be a significant breakthrough in this field of study.
3. Casimir Apparatus as a Key Element
The use of a hypothetical Casimir apparatus with plates positioned at various distances plays a crucial role in exploring the properties of traversable wormholes. Investigating different configurations and setups of this Casimir device can provide valuable insights into the behavior and characteristics of these wormholes.
4. Conservation of Stress Energy Tensor
An essential aspect of the derived solutions is the conservation of the Stress Energy Tensor, which indicates a preservation of physical quantities within the wormhole system. Exploring the implications and consequences of this conservation law can shed light on the dynamics and stability of traversable wormholes.
5. The Role of Auxiliary Fields
The presence of an auxiliary field is necessary to maintain the conservation of the Stress Energy Tensor, except in the case involving a scalar field with a potential combined with a Casimir apparatus with fixed plates. Further investigation into the role and behavior of this auxiliary field can provide valuable insights into the nature of the gravitational response of the Traversable Wormhole.
Conclusion
Understanding the complexities and characteristics of traversable wormholes in the presence of a scalar field is a field of study with immense potential. Overcoming challenges such as the violation of the null energy condition and exploring the role of auxiliary fields can pave the way for groundbreaking discoveries in this area. By further investigating manifolds, Casimir apparatus configurations, and the conservation of physical quantities, researchers can unravel the mysteries surrounding traversable wormholes and potentially revolutionize our understanding of the fabric of spacetime.