Extended Klein-Gordon Action, Gravity and Non-Relativistic Fluid

TL;DR

This paper explores a generalized Klein-Gordon scalar field coupled to gravity, revealing nontrivial configurations with zero energy-momentum tensor that do not influence spacetime curvature, and establishes a link to non-relativistic fluids like the Chaplygin gas.

Contribution

It introduces a power-law extension of the Klein-Gordon action, analyzes its gravitational coupling, and connects it to non-relativistic fluid dynamics with additional symmetries.

Findings

Existence of scalar configurations with zero energy-momentum tensor in vacuum solutions.

Conformal invariance at a specific exponent without nonminimal coupling.

Correspondence between the extended Klein-Gordon action and non-relativistic fluids, including the Chaplygin gas.

Abstract

We consider a scalar field action for which the Lagrangian density is a power of the massless Klein-Gordon Lagrangian. The coupling of gravity to this matter action is considered. In this case, we show the existence of nontrivial scalar field configurations with vanishing energy-momentum tensor on any static, spherically symmetric vacuum solutions of the Einstein equations. These configurations in spite of being coupled to gravity do not affect the curvature of spacetime. The properties of this particular matter action are also analyzed. For a particular value of the exponent, the extended Klein-Gordon action is shown to exhibit a conformal invariance without requiring the introduction of a nonminimal coupling. We also establish a correspondence between this action and a non-relativistic isentropic fluid in one fewer dimension. This fluid can be identified with the (generalized) Chaplygin gas for a particular value of the power. It is also shown that the non-relativistic fluid admits, apart from the Galileo symmetry, an additional symmetry whose action is a rescaling of the time.