This paper is devoted to investigate the Killing and Noether symmetries of static plane symmetric spacetime. For this purpose, five different cases have been discussed. The Killing and Noether symmetries of Minkwoski spacetime in cartesian coordinates are calculated as a special case and it is found that Lie algebra of the Lagrangian is 10 and 17 dimensional respectively. The symmetries of Taub's universe, anti-deSitter universe, self similar solutions of infinite kind for parallel perfect fluid case and self similar solutions of infinite kind for parallel dust case are also explored. In all the cases, the Noether generators are calculated in the presence of gauge term. All these examples justify the conjecture that Killing symmetries form a subalgebra of Noether symmetries [1].
We investigate the approximate Noether gauge symmetries of the geodesic Lagrangian for the Bardeen spacetime model. This is accommodated by a set of new approximate Noether gauge symmetry relations for the perturbed geodesic Lagrangian in the spacetime. A detailed analysis to the spacetime of Bardeen model up to third-order approximate Noether gauge symmetries is presented.
Noether symmetries from geodetic Lagrangian for time-conformal plane symmetric spacetime are presented. Here, time conformal factor is used to find the approximate Noether symmetries. This is a generalization of the idea discussed by I. Hussain and A. Noether symmetries from geodetic Lagrangian for time-conformal plane symmetric spacetime are presented. Here, time conformal factor is used to find the approximate Noether symmetries. This is a generalization of the idea discussed by I. Hussain and A. Qadir [3,4], where they obtained approximate Noether symmetries from Lagrangian for a particular…
Through this article, I will overview the use of Noether symmetry approach in discussing the integration of geodesic equations for the geodesic Lagrangians of spacetimes. I will also give some examples to reveal the efficiency of Noether symmetry approach by finding the first integrals related for the geodesic Lagrangians of the Gödel-type, Schwarzschild, Reissner-Nordström and Kerr spacetimes. After obtaining the approximate Noether symmetries of the Schwarzschild, Reissner-Nordström and Kerr spacetimes, the first integrals associated with each of approximate Noether symmetries have been inte…
The Noether symmetry approach is useful tool to restrict the arbitrariness in a gravity theory when the equations of motion are underdetermined due to the high number of functions to be determined in the ansatz. We consider two scalar-coupled theories of gravity, one motivated by induced gravity, the other more standard; in Bianchi I, Bianchi III and Kantowski-Sachs cosmological models. For these models, we present a full set of Noether gauge symmetries, which are more general than those obtained by the strict Noether symmetry approach in our recent work. Some exact solutions are derived using…
When spacetime torsion is present, geodesics and autoparallels generically do not coincide. In this work, the well-known method that uses Killing vectors to solve the geodesic equations is generalized for autoparallels. The main definition is that of T-Killing vectors: vector fields such that, when their index is lowered with the metric, have vanishing symmetric derivative when acted with a torsionfull and metric-compatible derivative. The main property of T-Killing vectors is that their contraction with the autoparallels' tangents are constant along these curves. As an example, in a static an…
We show that Belinfante construction of an improved energy-momentum tensor can be carried over to curved backgrounds, in analogy to the case of flat spacetime. The results hold irrespective of the background being dynamical or a fixed, non-backreacting one. It turns out that the analogous would-be canonical energy-momentum tensor is not covariantly conserved in general, but its Belinfante "improvement" is. We relate this last tensor with the Hilbert tensor obtained by functionally derivating the Lagrangian with respect to the metric. When the background in non-dynamical, we discuss some issues…
In this paper, a complete classification of Friedmann-Robertson-Walker (FRW) spacetime by using approximate Noether approach is presented. Considered spacetime is discussed for three different types of universe i.e. flat, open and closed. Different forms of cosmic scale factor $a$ with respect to the nature of the universe, which posses the nontrivial Noether gauge symmetries (NGS) are reported. The perturbed Lagrangian corresponding to FRW metric in the Noether equation is used to get Noether operators. For different types of universe minimal and maximal set of Noether operators are reported.…
We investigate the Noether symmetries of the Lagrangian for the stationary rotating BTZ-type three-dimensional spacetimes in $f(R)$ theory of gravity. A detailed analysis of Noether symmetries of (2+1)-dimensional rotating BTZ-type black hole spacetime model is presented. Applying the Noether symmetry approach, the first integrals (constants of motion) for each of Noether symmetries are obtained to look for the exact solutions. After solving the first integral equations depending on the form of the function $f(R)$, we derived some new (2+1)-dimensional rotating BTZ-type black hole solutions. W…
This is a short communication in reference to the comments in (arXiv: 1605.05147) of our paper in Int. J. Mod. Phys. D 25, 1650051 (2016). We have tried to answer to the comments or criticism in the paper.
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.
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 Noet…
In this paper, we have presented the Noether symmetries of flat FRW spacetime in the context of a new action in Teleparallel Gravity which we construct it based on $f(R)$ version. This modified action contains a coupling between scalar field potential and magnetism. Also, we introduce an innovative approach (B.N.S. Approach) for exact solutions which carry more conserved currents than Noether approach. By data analysis of the exact solutions, obtained from Noether approach, late time acceleration and phase crossing are realized, and some deep connections with observational data such as the age…
We discuss an extended Teleparallel gravity models comprising functions of scalar invariants constructed by torsion, torsion Gauss-Bonnet and boundary terms. We adopt the Noether Symmetry Approach to select the functional forms, the first integrals and, eventually, the exact solutions of the dynamics in the context of flat Friedman-Robertson-Walker cosmology. Standard perfect fluid matter, minimally coupled to geometry, is considered. Several physically consistent models are derived with related exact solutions.
In the framework of teleparallel gravity, the Friedman-Robertson-Walker cosmological model with scalar tensor theory where scalar field is non-minimally coupled to both the torsion scalar and boundary term is studied. Utilizing the Noether symmetry approach in such a theory, we obtain the explicit forms of the couplings and potential as a function the scalar field. We present some important cosmological solutions for the modified field equations using these functions getting via the Noether symmetry approach. Finally, the interesting cosmological properties of these solutions are discussed in …
The $f(R)$ theory is considered for static cylindrically symmetric and plane-symmetric spacetimes. In order to find solutions to the field equations of these models, the Noether symmetry method is used. First, we examine the GR case for cylindrically symmetrical space-time with the $w=-1$ dark energy state. Then, with the assumption of $f(R) = f_0R^n$, cases with matter and non-matter are examined and general solutions are determined for both space-times. Thus, it is shown that inclusive new solutions are obtained, considering the Noether symmetric conditions. In addition, the GR limit for eac…
This paper is devoted to investigate the recently proposed modified Gauss-Bonnet $f(\mathcal{G},T)$ gravity, with $\mathcal{G}$, the Gauss-Bonnet term, coupled with ${T}$, the trace of energy-momentum tensor. We have used the Noether symmetry methodology to discuss some cosmologically important $f(\mathcal{G},T)$ gravity models with anisotropic background. In particular, the Noether symmetry equations for modified $f(\mathcal{G},T)$ gravity are reported for locally rotationally symmetric Bianchi type $I$ universe. Explicitly, two models have been proposed to explore the exact solutions and the…
This work aims to investigate some possible emergence of relativistic compact stellar objects in modified $f(\mathcal{G})$ gravity using Noether symmetry approach. For this purpose, we assume static spherically symmetric spacetime in the presence of isotropic matter distribution. We construct Noether symmetry generators along with associated conserved quantities by considering the standard choice of viable $f(\mathcal{G})$ gravity model i.e. $f(\mathcal{G})= α\mathcal{G}^{n}$, where $α$ is the model parameter. In particular, we use conservation relation acquired from the classical Noether appr…
Benefiting from the index spinorial formalism, the Killing spinor equation is integrated in six-dimensional spacetimes. The integrability conditions for the existence of a Killing spinor are worked out and the Killing spinors are classified in two algebraic types, in the first type the scalar curvature of the spacetime must be negative, while in the second type the spacetime must be an Einstein manifold. In addition, the equations that define Killing-Yano (KY) and closed conformal Killing-Yano (CCKY) tensors are expressed in the index notation and, as consequence, all non-vanishing KY and CCKY…
It is an open question whether fluctuations at the Planck scale in a non-perturbative theory of quantum gravity behave in such a way that the resulting semi-classical geometry can be modelled by a space that admits (approximate) Killing symmetries. We have investigated whether the notion of approximate Killing vector fields is suitable to address this question in lattice theories of quantum gravity, such as (Causal) Dynamical Triangulations. We show that it is possible to construct quantum observables related to approximate Killing vector fields using the framework of Discrete Exterior Calculu…
We investigate the hidden symmetries and existence of Killing spinors of D=5 minimal gauged supergravity solutions which admit a Killing-Maxwell system in the sense of Carter, such as the Chong-Cveti$\check{c}$-Lü-Pope (CCLP) Kerr-dS black hole spacetime and a 5-dimensional minimal gauged supergravity solution endowed with a Sasaki structure deformed by torsion, a limiting case of the generalization of the 5-dimensional toric Sasaki-Einstein $L^{a,b,c}$ spacetime. We note that when an electromagnetic tensor is present and an associated Killing-Maxwell system can be constructed in the sense of …
We present a novel family of slowly rotating black hole solutions in four, and higher dimensions, that extend the well known Lense-Thirring spacetime and solve the field equations to linear order in rotation parameter. As "exact metrics" in their own right, the new (non-vacuum) spacetimes feature the following two remarkable properties: i) near the black hole horizon they can be cast in the, manifestly regular, Painlevé-Gullstrand form and ii) they admit exact Killing tensor symmetries. We show these symmetries are inherited from the principal Killing-Yano tensor of the exact rotating black ho…
We discuss non-minimally coupled cosmologies involving different geometric invariants. Specifically, actions containing a non-minimally coupled scalar field to gravity described, in turn, by curvature, torsion and Gauss--Bonnet scalars are considered. We show that couplings, potentials and kinetic terms are determined by the existence of Noether symmetries which, moreover, allows to reduce and solve dynamics. The main finding of the paper is that different non-minimally coupled theories, presenting the same Noether symmetries, are dynamically equivalent. In other words, Noether symmetries are …
We investigate the hidden symmetries of the Kerr-Newman, of the D=5 minimal gauged supergravity which admits a Killing-Maxwell system in the sense of Carter and of higher dimensional spacetimes in the presence of Killing-Yano torsion. We note that when an electromagnetic tensor is present and an associated Killing-Maxwell system can be constructed in the sense of Carter, the Killing- Maxwell field becomes a PCKY (primary conformal Killing-Yano) tensor or a PGCKY (primary generalized conformal Killing-Yano) tensor, the latter in the presence of torsion. We point out the structure of the Dirac-t…
We apply the Noether Symmetry Approach to point-like teleparallel Lagrangians in view to derive minisuperspaces suitable for Quantum Cosmology. Adopting the Arnowitt-Deser-Misner formalism, we find out related Wave Functions of the Universe. Specifically, by means of appropriate changes of variables suggested by the existence of Noether symmetries, it is possible to obtain the cosmological Hamiltonians whose solutions are classical trajectories interpretable as observable universes.
In this paper, based on the works of Capozziello et al., we have studied the Noether symmetry approach in the cosmological model with scalar and gauge fields proposed recently by Soda et al. The correct Noether symmetries and related Lie algebra are given according to the minisuperspace quantum cosmological model. The Wheeler-De Witt (WDW) equation is presented after quantization and the classical trajectories are then obtained in the semi-classical limit. The oscillating features of the wave function in the cosmic evolution recover the so-called Hartle criterion, and the selection rule in min…
In this paper we give a complete classification of spherically symmetric static space-times by their Noether symmetries. The determining equations for Noether symmetries are obtained by using the usual Lagrangian of a general spherically symmetric static spacetime which are integrated for each case. In particular we observe that spherically symmetric static spacetimes are categorized into six distinct classes corresponding to Noether algebra of dimensions 5, 6, 7, 9, 11 and 17. Using Noether`s theorem we also write down the first integrals for each class of such spacetimes corresponding to the…
A didactic and systematic derivation of Noether point symmetries and conserved currents is put forward in special relativistic field theories, without a priori assumptions about the transformation laws. Given the Lagrangian density, the invariance condition develops as a set of partial differential equations determining the symmetry transformation. The solution is provided in the case of real scalar, complex scalar, free electromagnetic, and charged electromagnetic fields. Besides the usual conservation laws, a less popular symmetry is analyzed: the symmetry associated with the linear superpos…
We develop a new approach to find asymptotic symmetries in general relativity as a modification of the Lie-algebra-based approach proposed in T. Tomitsuka et al. [Classical Quantum Gravity 38, 225007 (2021)]. Those authors proposed an algorithmic protocol to investigate asymptotic symmetries. In particular, their guiding principle helps us to find a non-vanishing charge that generates an infinitesimal diffeomorphism. However, in order to check the integrability condition for the charges, it is necessary to solve differential equations to identify the integral curve of vector fields, which is u…
Non-local gravity cosmologies are considered under the standard of Noether Symmetry Approach. In particular, we focus on non-local theories whose gravitational actions depend on curvature and Gauss-Bonnet scalar invariants. Specific functional forms of the related point-like Lagrangians are selected by Noether symmetries and we solve the corresponding field equations finding out exact cosmological solutions.
