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

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

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

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

Ricci collineations in Bianchi II spacetime

Abdus Salam ICTP; Higher Education Commission (HEC); National University of Sciences and Technology (NUST)12th Regional Conference on Mathematical Physics -- 27 March 2006 through 1 April 2006 -- Islamabad -- 102759In this study we classify the Ricci collineations (RCs) of Bianchi II spacetime according to the degenerate and non-degenerate cases of the Ricci tensor. It is shown that we have thirteen possibilities to be considered for the degenerate case, and found that there are generally infinitely many RCs whereas some cases give finite dimensional Lie algebras of the RCs which have three, four or five RCs. For the non-degenerate Ricci tensor cases, the Lie algebra of the obtained RCs are finite dimensional, in which the number of RCs is also three, four or five. Copyright © 2007 by World Scientific Publishing Co. Pvt. Ltd

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 15, the maximum number for four-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