On the canonical formalism of $f(R)$-type gravity using Lie derivatives

TL;DR

This paper develops a more concise canonical formalism for $f(R)$ gravity by thoroughly employing Lie derivatives and spacetime foliation, improving upon previous mixed-derivative approaches.

Contribution

It introduces a refined formalism for $f(R)$ gravity using Lie derivatives and spacetime foliation, simplifying the canonical structure compared to earlier methods.

Findings

More concise and clear formalism for $f(R)$ gravity

Elimination of mixed derivatives in the formalism

Enhanced mathematical structure and simplicity

Abstract

We present a canonical formalism of the $f(R)$-type gravity using the Lie derivatives instead of the time derivatives by refining the formalism of our group. The previous formalism is a direct generalization of the Ostrogradski's formalism. However the use of the Lie derivatives was not sufficient in that Lie derivatives and time derivatives are used in a mixed way, so that the expressions are somewhat complicated. In this paper, we use the Lie derivatives and foliation structure of the spacetime thoroughly, which makes the procedure and the expressions far more concise.