We study transformations of the dynamical fields – a metric, a flat affine
connection and a scalar field – in scalar-teleparallel gravity theories. The
theories we study belong either to the general teleparallel setting, where no
further condition besides vanishing curvature is imposed on the affine
connection, or the symmetric or metric teleparallel gravity, where one also
imposes vanishing torsion or nonmetricity, respectively. For each of these
three settings, we find a general class of scalar-teleparallel action
functionals which retain their form under the aforementioned field
transformations. This is achieved by generalizing the constraint of vanishing
torsion or nonmetricity to non-vanishing, but algebraically constrained torsion
or nonmetricity. We find a number of invariant quantities which characterize
these theories independently of the choice of field variables, and relate these
invariants to analogues of the conformal frames known from scalar-curvature
gravity. Using these invariants, we are able to identify a number of physically
relevant subclasses of scalar-teleparallel theories. We also generalize our
results to multiple scalar fields, and speculate on further extended theories
with non-vanishing, but algebraically constrained curvature.
Conclusions:
The study examines transformations of the dynamical fields in scalar-teleparallel gravity theories. It analyzes theories under different settings, including general teleparallel, symmetric teleparallel, and metric teleparallel gravity. The study finds a general class of scalar-teleparallel action functionals that remain unchanged under the field transformations by imposing algebraically constrained torsion or nonmetricity. Several invariant quantities are identified that characterize these theories independently of the choice of field variables, and are related to conformal frames in scalar-curvature gravity. The study also extends the results to multiple scalar fields and speculates on further extended theories with algebraically constrained curvature.
Future Roadmap:
The findings of this study open up several potential opportunities and challenges for further exploration in the field of scalar-teleparallel gravity theories.
Potential Challenges:
- Validation: One of the challenges is the validation of these theories through experiments or observations. Further research is needed to test the predictions made by these scalar-teleparallel theories and compare them with existing gravitational theories.
- Mathematical Complexity: The introduction of algebraically constrained torsion or nonmetricity adds complexity to the mathematical formulations. Finding exact solutions and performing calculations in these extended theories may pose challenges.
- Consistency with Other Theories: It is important to investigate the consistency of these scalar-teleparallel theories with other established theories, such as general relativity. The compatibility and agreement between different theoretical frameworks should be explored.
Potential Opportunities:
- Extension to Multiple Scalar Fields: The study suggests that the results can be generalized to include multiple scalar fields. This opens up possibilities for studying interactions and dynamics between multiple scalar fields in the context of scalar-teleparallel gravity.
- Identification of Physically Relevant Subclasses: The identification of physically relevant subclasses of scalar-teleparallel theories provides a foundation for further investigations. These subclasses can be studied in greater detail to explore their implications for various physical phenomena.
- Exploration of Extended Theories: The speculation about extended theories with algebraically constrained curvature presents an opportunity for future research. Investigating the consequences and implications of such extended theories can provide new insights into the nature of gravity.
Conclusion:
The findings of this study lay the groundwork for further exploration in the field of scalar-teleparallel gravity theories. While challenges such as validation and mathematical complexity exist, there are also opportunities to extend the theories to multiple scalar fields, identify physically relevant subclasses, and explore extended theories with constrained curvature. Future research in these directions can significantly contribute to our understanding of gravity and its fundamental properties.