Exploring Second-Order Bi-Scalar-Tensor Field Equations in Four-Dimensional Space

A recent paper titled "Second-Order Bi-Scalar-Tensor Field Equations in a Space of Four-Dimensions" by Gregory W. Horndeski explores the construction of second-order Euler-Lagrange tensor densities derived from Lagrange scalar densities involving two scalar fields and a pseudo-Riemannian metric tensor. The research investigates these tensor densities in a four-dimensional space, aiming to establish the most general second-order equations that can be formed from a specific set of Lagrangians.

The findings reveal that a total of six scalar coefficients can be derived from four Lagrangians, with only one of these coefficients being arbitrary. The remaining five coefficients must adhere to linear partial differential equations. These coefficients are categorized into three groups, with varying combinations appearing across different Lagrangians. Notably, each of the five functions leads to solutions for the wave equation in three-dimensional Minkowski space.

Horndeski concludes by presenting twelve distinct second-order bi-scalar-tensor Lagrangians, which involve eighteen coefficient functions. He suggests that these Lagrangians could potentially be utilized to construct all possible second-order bi-scalar-tensor field equations in four-dimensional space. This work contributes to the broader understanding of scalar-tensor theories in the context of general relativity, which may have implications for future theoretical developments and applications in cosmology and gravitational physics.