New Exact Spherically Symmetric Solutions in f(R,ϕ,X)f(R,\phi,X) gravity by Noether's symmetry approach

The exact solutions of spherically symmetric space-times are explored by using Noether symmetries in $f(R,\phi,X)$ gravity with $R$ the scalar curvature, $\phi$ a scalar field and $X$ the kinetic term of $\phi$. Some of these solutions can represent new black holes solutions in this extended theory of gravity. The classical Noether approach is particularly applied to acquire the Noether symmetry in $f(R,\phi,X)$ gravity. Under the classical Noether theorem, it is shown that the Noether symmetry in $f(R,\phi,X)$ gravity yields the solvable first integral of motion. With the conservation relation obtained from the Noether symmetry, the exact solutions for the field equations can be found. The most important result in this paper is that, without assuming $R=\textrm{constant}$, we have found new spherically symmetric solutions in different theories such as: power-law $f(R)=f_0 R^n$ gravity, non-minimally coupling models between the scalar field and the Ricci scalar $f(R,\phi,X)=f_0 R^n \phi^m+f_1 X^q-V(\phi)$, non-minimally couplings between the scalar field and a kinetic term $f(R,\phi,X)=f_0 R^n +f_1\phi^mX^q$ , and also in extended Brans-Dicke gravity $f(R,\phi,X)=U(\phi,X)R$. It is also demonstrated that the approach with Noether symmetries can be regarded as a selection rule to determine the potential $V(\phi)$ for $\phi$, included in some class of the theories of $f(R,\phi,X)$ gravity.Comment: Accepted for publication in JCAP. The title and the abstract have been changed and new exact solutions have been added with more cases. The existence of horizons of our solutions has been discussed to

Physical Significance of Noether Symmetries

In this paper, we will trace the development of the use of symmetry in discussing the theory of motion initiated by Emmy Noether in 1918. Though it started with its use in classical mechanics, and has been heavily used in engineering applications of mechanics, it came into its own in relativity, and quantum theory and their applications in particle physics and field theory. It will be beyond the scope of this article to explain the quantum field theory applications in any detail, but the base for understanding it will be provided here. We will also go on to discuss an insight from some more mathematical developments related to Noether symmetry

Conformal Symmetries of the Energy–Momentum Tensor of Spherically Symmetric Static Spacetimes

Conformal matter collineations of the energy–momentum tensor of a general spherically symmetric static spacetime are studied. The general form of these collineations is found when the energy–momentum tensor is non-degenerate, and the maximum number of independent conformal matter collineations is 15. In the degenerate case of the energy–momentum tensor, it is found that these collineations have infinite degrees of freedom. In some subcases of degenerate energy– momentum, the Ricci tensor is non-degenerate, that is, there exist non-degenerate Ricci inheritance collineations

Exact Spherically Symmetric Solutions in Modified Teleparallel gravity

Finding spherically symmetric exact solutions in modified gravity is usually a difficult task. In this paper we use the Noether's symmetry approach for a modified Teleparallel theory of gravity labelled as $f(T,B)$ gravity where $T$ is the scalar torsion and $B$ the boundary term. Using the Noether's theorem, we were able to find exact spherically symmetric solutions for different forms of the function $f(T,B)$ coming from the Noether's symmetries.Comment: 22 pages; Matches published version in Symmetr

Scalar-Tensor Teleparallel Wormholes by Noether Symmetries

A gravitational theory of a scalar field non-minimally coupled with torsion and boundary term is considered with the aim to construct Lorentzian wormholes. Geometrical parameters including shape and redshift functions are obtained for these solutions. We adopt the formalism of Noether Gauge Symmetry Approach in order to find symmetries, Lie brackets and invariants (conserved quantities). Furthermore by imposing specific forms of potential function, we are able to calculate metric coefficients and discuss their geometrical behavior.Comment: Slightly updated version. Accepted for publication in PR

Exact spherically symmetric solutions in modified Gauss-Bonnet gravity from Noether symmetry approach

It is broadly known that Lie point symmetries and their subcase, Noether symmetries, can be used as a geometric criterion to select alternative theories of gravity. Here, we use Noether symmetries as a selection criterion to distinguish those models of $f(R,G)$ theory, with $R$ and $G$ being the Ricci and the Gauss-Bonnet scalars respectively, that are invariant under point transformations in a spherically symmetric background. In total, we find ten different forms of $f$ that present symmetries and calculate their invariant quantities, i.e Noether vector fields. Furthermore, we use these Noether symmetries to find exact spherically symmetric solutions in some of the models of $f(R,G)$ theory.Comment: 17 pages. Accepted for Publication in Symmetries in the special issue "Noether's symmetry approach in gravity and cosmology

Conformal Ricci collineations of static spherically symmetric spacetimes

Conformal Ricci collineations of static spherically symmetric spacetimes are studied. The general form of the vector fields generating conformal Ricci collineations is found when the Ricci tensor is non-degenerate, in which case the number of independent conformal Ricci collineations is \emph{fifteen}; the maximum number for 4-dimensional manifolds. In the degenerate case it is found that the static spherically symmetric spacetimes always have an infinite number of conformal Ricci collineations. Some examples are provided which admit non-trivial conformal Ricci collineations, and perfect fluid source of the matter

Conformal Symmetries of the Energy–Momentum Tensor of Spherically Symmetric Static Spacetimes

Conformal matter collineations of the energy–momentum tensor of a general spherically symmetric static spacetime are studied. The general form of these collineations is found when the energy–momentum tensor is non-degenerate, and the maximum number of independent conformal matter collineations is 15. In the degenerate case of the energy–momentum tensor, it is found that these collineations have infinite degrees of freedom. In some subcases of degenerate energy–momentum, the Ricci tensor is non-degenerate, that is, there exist non-degenerate Ricci inheritance collineations

On dark matter as a geometric effect in the galactic halo

We obtain more straightforwardly some features of dark matter distribution in the halos of galaxies by considering the spherically symmetric space-time, which satisfies the flat rotational curve condition, and the geometric equation of state resulting from the modified gravity theory. In order to measure the equation of state for dark matter in the galactic halo, we provide a general formalism taking into account the modified f(X) gravity theories. Here, f(X) is a general function of X∈ { R, G, T} , where R, G and T are the Ricci scalar, the Gauss-Bonnet scalar and the torsion scalar, respectively. These theories yield that the flat rotation curves appear as a consequence of the additional geometric structure accommodated by those of modified gravity theories. Constructing a geometric equation of state wX≡ pX/ ρX and inspiring by some values of the equation of state for the ordinary matter, we infer some properties of dark matter in galactic halos of galaxies