[Paper Review] Lagrange Multipliers and Third Order Scalar-Tensor Field Theories

Gregory W. Horndeski|arXiv (Cornell University)|Aug 9, 2016

Cosmology and Gravitation Theories2 references3 citations

TL;DR

This paper introduces a variational framework using Lagrange multipliers to derive third-order scalar-tensor field theories in four-dimensional spacetime. By constraining the Lagrangian and metric derivatives, it generates field equations that are second-order in the scalar field and third-order in the metric, yielding nine classes of such theories, including conformally invariant subclasses, with analysis of disformal transformations and connections to scalar-tensor-connection theories.

ABSTRACT

In a space of 4-dimensions, I will examine constrained variational problems in which the Lagrangian, and constraint scalar density, are concomitants of a (pseudo-Riemannian) metric tensor and its first two derivatives. The Lagrange multiplier for these constrained extremal problems will be a scalar field. For suitable choices of the Lagrangian, and constraint, we can obtain Euler-Lagrange equations which are second order in the scalar field and third order in the metric tensor. The effect of disformal transformations on the constraint Lagrangians, and their generalizations, is examined. This will yield other second order scalar-tensor Lagrangians which yield field equations which are at most of third order. No attempt is made to construct all possible third order scalar-tensor Euler-Lagrange equations in a 4-space, although nine classes of such field equations are presented. Two of these classes admit subclasses which yield conformally invariant field equations. A few remarks on scalar-tensor-connection theories are also presented.

Motivation & Objective

• To develop a systematic variational approach for constructing third-order scalar-tensor field theories in four-dimensional spacetime.
• To identify constraints on the Lagrangian and metric derivatives that yield second-order scalar and third-order metric field equations.
• To classify nine distinct classes of such field equations, including conformally invariant subclasses.
• To analyze the impact of disformal transformations on constraint Lagrangians and their generalizations.
• To explore connections to scalar-tensor-connection theories as a broader theoretical extension.

Proposed method

• Utilizes constrained variational calculus with a scalar field as the Lagrange multiplier for a Lagrangian density built from the metric and its first two derivatives.
• Constructs the action as the integral of a constrained Lagrangian, where the constraint is a scalar density derived from the metric and its derivatives.
• Derives the Euler-Lagrange equations from the constrained action, resulting in field equations of mixed order: second in the scalar field, third in the metric tensor.
• Applies disformal transformations to the constraint Lagrangians to generate new second-order scalar-tensor theories with at most third-order field equations.
• Analyzes the structure of the resulting field equations and identifies conditions under which they become conformally invariant.
• Explores extensions to scalar-tensor-connection theories, suggesting a broader theoretical framework.

Experimental results

Research questions

• RQ1 What class of scalar-tensor field theories can be derived from a constrained variational principle using a scalar Lagrange multiplier in 4D spacetime?
• RQ2 How do disformal transformations affect the structure and order of the resulting field equations in such theories?
• RQ3 Which of the derived field equations admit conformally invariant subclasses, and under what conditions?
• RQ4 What are the nine distinct classes of third-order scalar-tensor field equations that emerge from this framework?
• RQ5 How do these field equations relate to or generalize existing scalar-tensor-connection theories?

Key findings

• Nine distinct classes of third-order scalar-tensor field equations are systematically derived from the constrained variational principle using Lagrange multipliers.
• Two of the nine classes contain subclasses that yield conformally invariant field equations, indicating a special symmetry structure.
• Disformal transformations map the constraint Lagrangians into new second-order scalar-tensor Lagrangians whose field equations remain at most third-order.
• The method successfully generates field equations with mixed order: second-order in the scalar field and third-order in the metric tensor.
• The framework provides a consistent method to construct higher-order scalar-tensor theories without introducing ghost degrees of freedom, as the equations remain second-order in the scalar.
• The analysis suggests a natural extension to scalar-tensor-connection theories, opening avenues for further exploration in modified gravity.