Second-Order Bi-Scalar-Tensor Field Equations in a Space of Four-Dimensions

TL;DR

This paper derives the most general second-order bi-scalar-tensor field equations in four-dimensional space, revealing a set of eleven Lagrangians with specific constraints and connections to wave equations in Minkowski space.

Contribution

It constructs the complete set of second-order Euler-Lagrange tensor densities for two scalar fields coupled with a metric in four dimensions, identifying their structure and constraints.

Findings

Eleven fundamental Lagrangians generate all second-order tensor densities.

Ten coefficient functions satisfy linear PDEs related to 2+1D wave equations.

Six Lagrangians do not involve second derivatives of the metric.

Abstract

Lagrange scalar densities which are concomitants of two scalar fields, a pseudo-Riemannian metric tensor, and their derivatives of arbitrary differential order are investigated in a space of four-dimensions. I construct the most general second-order Euler-Lagrange tensor densities derivable from such a Lagrangian. It is demonstrated that all such second-order Euler-Lagrange tensor densities can be derived from a set of eleven Lagrangians which are at most of second-order. Of these eleven Lagrangians six do not involve the second derivatives of the metric tensor, and are algebraically at most of first degree in the second derivatives of the scalar fields. Each of the eleven Lagrangians will have a scalar coefficient which is a concomitant of five variables: the two scalar fields, and the three inner products of the gradients of the two scalar fields. Of these eleven coefficient functions only one is arbitrary, while the other ten must satisfy linear partial differential equations. Surprisingly these partial differential equations are related to the wave equation in 2+1 dimensional Minkowski space. The paper concludes with a few observations on the form that a Lagrangian which yields the most general second-order, multi-scalar-tensor field equations in a space of arbitrary dimension can assume.