We show that the seemingly different methods used to derive non-Lorentzian
(Galilean and Carrollian) gravitational theories from Lorentzian ones are
equivalent. Specifically, the pre-nonrelativistic and the pre-ultralocal
parametrizations can be constructed from the gauging of the Galilei and Carroll
algebras, respectively. Also, the pre-ultralocal approach of taking the
Carrollian limit is equivalent to performing the ADM decomposition and then
setting the signature of the Lorentzian manifold to zero. We use this
uniqueness to write a generic expansion for the curvature tensors and construct
Galilean and Carrollian limits of all metric theories of gravity of finite
order ranging from the $f(R)$ gravity to a completely generic higher derivative
theory, the $f(g_{munu},R_{munusigma rho},nabla_{mu})$ gravity. We
present an algorithm for calculation of the $n$-th order of the Galilean and
Carrollian expansions that transforms this problem into a constrained
optimization problem. We also derive the condition under which a gravitational
theory becomes a modification of general relativity in both limits
simultaneously.
Conclusions
The article concludes that the methods used to derive non-Lorentzian gravitational theories from Lorentzian ones (Galilean and Carrollian) are equivalent. The pre-nonrelativistic and pre-ultralocal parametrizations can be constructed from the gauging of the Galilei and Carroll algebras respectively. Additionally, the pre-ultralocal approach of taking the Carrollian limit is equivalent to performing the ADM decomposition and setting the signature of the Lorentzian manifold to zero.
The article also presents a generic expansion for the curvature tensors and constructs Galilean and Carrollian limits of all metric theories of gravity of finite order, ranging from $f(R)$ gravity to a completely generic higher derivative theory, the $f(g_{munu},R_{munusigma rho},nabla_{mu})$ gravity.
An algorithm for calculating the $n$-th order of the Galilean and Carrollian expansions is presented, transforming the problem into a constrained optimization problem. Additionally, the condition under which a gravitational theory becomes a modification of general relativity in both limits simultaneously is derived.
Future Roadmap
The future roadmap for readers can be outlined as follows:
1. Further Exploration of Equivalence
Readers can explore the concept of equivalence between different methods in deriving non-Lorentzian gravitational theories from Lorentzian ones. This exploration can involve diving deeper into the gauging of Galilei and Carroll algebras and understanding the connection between the pre-nonrelativistic and pre-ultralocal parametrizations.
2. Understanding the Generic Expansion
There is an opportunity to delve into the generic expansion for curvature tensors presented in the article. This expansion encompasses a wide range of metric theories of gravity, from $f(R)$ gravity to higher derivative theories. Readers can study and analyze this expansion to gain a deeper understanding of its implications for gravitational theories.
3. Algorithm Implementation
The article introduces an algorithm for calculating the $n$-th order of the Galilean and Carrollian expansions. Readers can implement and test this algorithm to verify its effectiveness and usefulness in practical applications. Challenges may arise in dealing with the constrained optimization problem that the algorithm transforms into, so readers should be prepared to tackle these challenges and find appropriate solutions.
4. Modification of General Relativity
The condition under which a gravitational theory becomes a modification of general relativity in both Galilean and Carrollian limits simultaneously is derived in the article. Readers can explore this condition further and analyze its implications for understanding modifications to the fundamental theory of gravity. Opportunities for research and experimentation in this area may arise.
In conclusion, the article provides a foundation for further exploration and understanding of non-Lorentzian gravitational theories. By examining the equivalence of different methods, exploring the generic expansion, implementing the algorithm, and investigating modifications of general relativity, readers can contribute to the advancement of this field and potentially uncover new insights and applications.