Second-Order, Biconformally Invariant Scalar-Tensor Field Theories in a Four-Dimensional Space

TL;DR

This paper develops a comprehensive framework for second-order scalar-tensor field theories in four-dimensional space that are invariant under biconformal transformations, identifying conditions for their invariance and constructing all such theories.

Contribution

It introduces necessary and sufficient conditions for biconformal invariance and constructs all second-order invariant scalar-tensor theories in four dimensions.

Findings

All second-order biconformally invariant theories can be derived from a linear combination of two Lagrangians.

The invariance condition is both necessary and sufficient for these theories.

The field equations are obtainable from a finite set of Lagrangians with constant coefficients.

Abstract

In this paper I shall consider field theories in a space of four-dimensions which have field variables consisting of the components of a metric tensor and scalar field. The field equations of these scalar-tensor field theories will be derivable from a variational principle using a Lagrange scalar density which is a concomitant of the field variables and their derivatives of arbitrary, but finite, order. I shall consider biconformal transformations of the field variables, which are conformal transformations which affect both the metric tensor and scalar field. A necessary and sufficient condition will be developed to determine when the Euler-Lagrange tensor densities are biconformally invariant. This condition will be employed to construct all of the second-order biconformally invariant scalar-tensor field theories in a space of four-dimensions. It turns out that the field equations of these theories can be derived from a linear combination of (at most) two second-order Lagrangians, with the coefficients in that linear combination being real constants.