Palatini Variational Principle for $N$-Dimensional Dilaton Gravity

TL;DR

This paper explores the Palatini variational principle applied to general N-dimensional torsion-free dilaton gravity, analyzing equations of motion, invariance constraints, and special cases like N=2, including a comparison with the Hilbert principle.

Contribution

It provides a detailed derivation of equations of motion for N-dimensional dilaton gravity using the Palatini approach and identifies conditions where it aligns with the Hilbert variational principle.

Findings

Derivation of equations of motion for N-dimensional dilaton gravity.

Identification of conditions where Palatini and Hilbert variations coincide.

Analysis of the role of conformal transformations in generalized Brans-Dicke theories.

Abstract

We consider a Palatini variation on a general $N$-Dimensional second order, torsion-free dilaton gravity action and determine the resulting equations of motion. Consistency is checked by considering the restraint imposed due to invariance of the matter action under simple coordinate transformations, and the special case of N=2 is examined. We also examine a sub-class of theories whereby a Palatini variation dynamically coincides with that of the "ordinary" Hilbert variational principle; in particular we examine a generalized Brans-Dicke theory and the associated role of conformal transformations.