Abstract
After reformulating $F($Riemann$)$ gravity theory as a second derivative theory by introducing two auxiliary fields to the bulk action, we derive the surface term as well as the corner term supplemented to the bulk action for a generic non-smooth boundary such that the variational principle is well posed. We also introduce the counter term to make the boundary term invariant under the reparametrization for the null segment. Then as a demonstration of the power of our formalism, not only do we apply our expression for the full action to evaluate the corresponding action growth rate of the Wheeler-DeWitt patch in the Schwarzchild anti-de Sitter black hole for the $F(R)$ gravity and critical gravity, where the corresponding late time behavior recovers the previous one derived by other approaches, but also in the asymptotically Anti-de Sitter black hole for the critical Einsteinian cubic gravity, where the late time growth rate vanishes but still saturates the Lloyd bound.
Highlights
In order to make the variational principle well posed for gravity theories, one is required to add the surface term to the bulk action
As a demonstration of the power of our formalism, do we apply our expression for the full action to evaluate the corresponding action growth rate of the Wheeler-DeWitt patch in the Schwarzchild anti–de Sitter black hole for the FðRÞ gravity and critical gravity, where the corresponding late time behavior recovers the previous one derived by other approaches, and in the asymptotically anti–de Sitter black hole for the critical Einsteinian cubic gravity, where the late time growth rate vanishes but still saturates the Lloyd bound
We have presented a complete discussion of the variational problem for FðRiemannÞ gravity with a nonsmooth boundary
Summary
In order to make the variational principle well posed for gravity theories, one is required to add the surface term to the bulk action. We shall focus exclusively on this situation and formulate the well-posed variational principle for more general circumstances, where the boundary is not necessarily required to be non-null or smooth Another motivation to evaluate the full action with a nonsmooth boundary including null segments comes from the “complexity equals action” (CA) conjecture [25,26]. It is generically difficult to find the corresponding boundary term, if any, for the bulk action (2) This problem can be circumvented by introducing two auxiliary fields ψabcd and φabcd, which allows us to recast the original bulk action (2) into the following form [23]. We shall derive the boundary term for a more general boundary by requiring this new action have a well-posed variational principle
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
