A Galileon Primer

TL;DR

This work surveys elementary Galileon models, constructing a general determinant-based formalism that yields a hierarchy of second-order field equations from higher-derivative actions. It develops universal field equations, Legendre dualities, and Legendre-self-dual structures, and derives explicit and implicit classical solutions in flat and curved spacetimes. The analysis includes mixtures of Lagrangians, self-couplings to the energy-momentum trace, and minimal coupling to gravity, uncovering static spherically symmetric solutions with either naked curvature singularities or horizons depending on boundary data, as well as shock-front conjectures. The findings highlight rich nonlinear scalar dynamics with potential gravitational signatures, motivate stability and UV-completion studies, and suggest observable consequences across astrophysical scales while pointing to open questions about naked singularities and geon-like configurations.

Abstract

Elementary features of galileon models are discussed at an introductory level. Following a simple example, a general formalism leading to a hierarchy of field equations and Lagrangians is developed for flat spacetimes. Legendre duality is discussed. Implicit and explicit solutions are then constructed and analyzed in some detail. Galileon shock fronts are conjectured to exist. Finally, some interesting general relativistic effects are studied for galileons coupled minimally to gravity. Spherically symmetric galileon and metric solutions with naked curvature singularities are obtained and are shown to be separated from solutions which exhibit event horizons by a critical curve in the space of boundary data.