Brans-Dicke Galileon and the Variational Principle

TL;DR

This paper provides a detailed, pedagogical derivation of the equations of motion in the Brans-Dicke Galileon theory, highlighting the variational principle and the cancellation of higher derivatives to yield second-order equations.

Contribution

It offers a comprehensive, step-by-step derivation of the Brans-Dicke Galileon equations, emphasizing the variational approach and algebraic details, useful for students studying modified gravity theories.

Findings

Derivation of second-order motion equations from higher-derivative Lagrangians.

Clarification of the variational principle in scalar-tensor theories.

Practical tips for deriving field equations in modified gravity theories.

Abstract

This paper is aimed at a (mostly) pedagogical exposition of the derivation of the motion equations of certain modifications of general relativity. Here we derive in all detail the motion equations in the Brans-Dicke theory with the cubic self-interaction. This is a modification of the Brans-dicke theory by the addition of a term in the Lagrangian which is non-linear in the derivatives of the scalar field: it contains second-order derivatives. This is the basis of the so-called Brans-Dicke Galileon. We pay special attention to the variational principle and to the algebraic details of the derivation. It is shown how higher order derivatives of the fields appearing in the intermediate computations cancel out leading to second order motion equations. The reader will find useful tips for the derivation of the field equations of modifications of general relativity such as the scalar-tensor theories and $f(R)$ theories, by means of the (stationary action) variational principle. The content of this paper is specially recommended to those graduate and postgraduate students who are interested in the study of the mentioned modifications of general relativity.