Noether symmetries as a geometric criterion to select theories of gravity

Abstract

We review the Noether Symmetry Approach as a geometric criterion to select theories of gravity. Specifically, we deal with Noether Symmetries to solve the field equations of given gravity theories. The method allows to find out exact solutions, but also to constrain arbitrary functions in the action. Specific cosmological models are taken into account.a Noether point symmetries are a subclass of Lie symmetries applied to dynamical systems that are described by a point Lagrangian and leave invariant the action integral.

Spherical and cylindrical solutions in f(T) gravity by Noether symmetry approach

“…They are useful both for simplification of differential equations and determination of integrable systems. In our context, the Noether Symmetry Approach is often used in modified and extended gravity theories for constraining the functional form of the Lagrangian densities f (R), f (T ), f (R, G ) etc., as well as finding solutions of the field equations from such point-like Lagrangians [7,[20][21][22][23][24][25].…”

Exact Spherically Symmetric Solutions in Modified Gauss–Bonnet Gravity from Noether Symmetry Approach

“…, 5. Then the Noether symmetry condition (25) for this Lagrangian yields an overdetermined system of 60 partial differential equations. We will now solve these differential equations, which will fix the symmetry generator X and the form of function f (R, G) as in the following possibilities.…”

“…It is well known that Noether symmetries (i.e. invariance under point transformations) can be used as a geometric criterion to discriminate between alternative theories of gravity [25]. The existence of symmetries means that the dynamical system has some conserved quantities that, in many cases, are related to physical/observed quantities.…”

“…The existence of symmetries means that the dynamical system has some conserved quantities that, in many cases, are related to physical/observed quantities. Hence, this renders the possibility to use it as a theoretical constraint in order to classify different models, based on the geometric symmetries of the spacetime [25]. In addition, these symmetries could also help us in reducing the dynamics of a system and thus find exact solutions more easily.…”

“…The successful realization of the dark energy and of the inflationary era is studied for some classes of f (R, G) models [38]. Both the vacuum and the non-vacuum theories of f (R, G) gravity admitting Noether symmetries are worked by [25,39,40], and they found some exact cosmological solutions. Obviously, apart from Noether symmetries, one could use contact symmetries, Cartan symmetries [41], etc to studying the invariance of a system of differential equations under specific types of transformations.…”