Studying ADM Formalism for Gravity Theories

[marie]

Dear All
I am interested in studying the ADM formalism for gravity theories. After finding the momentum and getting rid of the time derivative variables ,One of the steps is to write the hamiltonian as sum of 2 parts of coefficients the lapse and shift functions repectively H= N H_t + N_i H_i.
My question is that what if the Hamiltonian is not linear in N and N_i.?
\Thank you

 

[Ambitwistor]

for your interest in studying the ADM formalism for gravity theories. The Hamiltonian in the ADM formalism is typically written as a sum of two parts, the lapse and shift functions, as you have mentioned. However, it is possible for the Hamiltonian to not be linear in these functions.

In such cases, the ADM formalism may not be applicable and alternative approaches may need to be considered. One potential solution is to use a different set of variables that can capture the nonlinearity of the Hamiltonian.

Another option is to explore different formalisms for gravity theories, such as the Palatini formalism or the Hamilton-Jacobi formalism, which may be better suited for non-linear Hamiltonians.

In any case, it is important to carefully consider the implications of a non-linear Hamiltonian and to thoroughly analyze its consequences for the theory being studied. I wish you the best of luck in your research.