Second-order Symmetries Research Articles

Gravitational mass and approximate Lie symmetries for the charged Hayward black hole

Here we investigate the gravitational mass (energy) of the charged Hayward black hole utilizing the notion of the approximate version of the Lie symmetries. To investigate the approximate symmetries of the second-order perturbed geodesic equations and the gravitational mass for the charged Hayward black hole, we consider the black hole mass M and the charge Q as a small perturbation parameter ϵ. We recover the exact Lie symmetries as trivial second-order approximate Lie symmetries of the perturbed geodesic equations. The existence of the trivial second-order approximate symmetries of the perturbed geodesic equations provides the re-scaling of the arc-length parameter s, indicating that the gravitational energy in the charged Hayward black hole spacetime must be re-scaled by a factor that depends on the mass M, charge Q, coordinate r and the parameter l appears in the spacetime metric of charged Hayward black hole. We compare this energy re-scaling factor of the charged Hayward black hole to the energy re-scaling factor of the Reissner–Nordström black hole (Hussain et al., 2007). We notice that due to the presence of the Hayward parameter l, the gravitational energy in this black hole spacetime decreases.

Effect of magnetic field on the gravitational energy of static and charged black holes

We investigate the effect of the magnetic field on the gravitational energy of the magnetized Reissner–Nordström black hole (RN-BH) carrying a magnetic charge and an electric charge, by applying approximate Lie symmetry methods. For this purpose, second-order perturbed geodesic equations are constructed by considering the mass, charge, and magnetic field in terms of a small perturbation parameter ϵ. It is shown that in the process of calculating the second-order approximate Lie symmetries, energy must be rescaled by some factor. The obtained energy-rescaling factors depend on the parameters of the black hole and the magnetic field. The rescaling factors obtained here are compared with the energy rescaling factor of the RN-BH investigated in Hussain et al. (2007) and the effects of the magnetic field on it are discussed. It is observed that the effect of the magnetic field results in increasing the gravitational energy of the underlying static and charged black holes.

Rigorous Modal Characterization of First- and Second-Order Symmetric Waveguides Using Specular Periodic Boundary Conditions in 2D-FEM

Symmetries are widely exploited in the electromagnetic characterization of millimeter-wave devices and antennas, as they usually bring an improvement in resource usage and accuracy in full-wave simulations. Given the rising relevance of higher order symmetric devices, a challenge of integrating many different symmetries into compact formulations appears, with the objective of finding synergies and combining the strengths each one separately brings. In this work, we present a 2-D finite element method (FEM) formulation that fulfills this task for arbitrary waveguide cross sections presenting at the same time specular first-order and rotational second-order symmetries. It is based on newly introduced specular periodic boundary conditions (SPBCs), which effectively exploit the best case simulation domain for this problem. Furthermore, they avoid the need of using a mesh matching or projection procedure for periodic boundaries, while also bringing an improvement to mode classification.

Symmetry classes in piezoelectricity from second-order symmetries

The piezoelectricity law is a constitutive model that describes how mechanical andelectric fields are coupled within a material. In its linear formulation this law comprises threeconstitutive tensors of increasing order: the second order permittivity tensor S, the third orderpiezoelectricity tensor P and the fourth-order elasticity tensor C. In a first part of the paper,the symmetry classes of the piezoelectricity tensor alone are investigated. Using a new approachbased on the use of the so-called clips operations, we establish the 16 symmetry classes of thistensor and provide their associated normal forms. Second order orthogonal transformations(plane symmetries and $\pi$-angle rotations) are then used to characterize and classify directly 11out of the 16 symmetry classes of the piezoelectricity tensor. An additional step to distinguishthe remaining classes is proposed

Fast In Vivo Imaging of SHG Nanoprobes with Multiphoton Light-Sheet Microscopy.

Two-photon light-sheet microscopy (2P-SPIM) provides a unique combination of advantages for fast and deep fluorescence imaging in live tissues. Detecting coherent signals such as second-harmonic generation (SHG) in 2P-SPIM in addition to fluorescence would open further imaging opportunities. However, light-sheet microscopy involves an orthogonal configuration of illumination and detection that questions the ability to detect coherent signals. Indeed, coherent scattering from micron-sized structures occurs predominantly along the illumination beam. By contrast, point-like sources such as SHG nanocrystals can efficiently scatter light in multiple directions and be detected using the orthogonal geometry of a light-sheet microscope. This study investigates the suitability of SHG light-sheet microscopy (SHG-SPIM) for fast imaging of SHG nanoprobes. Parameters that govern the detection efficiency of KTiOPO4 and BaTiO3 nanocrystals using SHG-SPIM are investigated theoretically and experimentally. The effects of incident polarization, detection numerical aperture, nanocrystal rotational motion, and second-order susceptibility tensor symmetries on the detectability of SHG nanoprobes in this specific geometry are clarified. Guidelines for optimizing SHG-SPIM imaging are established, enabling fast in vivo light-sheet imaging combining SHG and two-photon excited fluorescence. Finally, microangiography was achieved in live zebrafish embryos by SHG imaging at up to 180 frames per second and single-particle tracking of SHG nanoprobes in the blood flow.

The Second-Order Correction to the Energy and Momentum in Plane Symmetric Gravitational Waves Like Spacetimes

This research provides second-order approximate Noether symmetries of geodetic Lagrangian of time-conformal plane symmetric spacetime. A time-conformal factor is of the form e ϵ f ( t ) which perturbs the plane symmetric static spacetime, where ϵ is small a positive parameter that produces perturbation in the spacetime. By considering the perturbation up to second-order in ϵ in plane symmetric spacetime, we find the second order approximate Noether symmetries for the corresponding Lagrangian. Using Noether theorem, the corresponding second order approximate conservation laws are investigated for plane symmetric gravitational waves like spacetimes. This technique tells about the energy content of the gravitational waves.

Effect of a quintessence model on the energy of the Reissner–Nordstrom black hole

The energy content of the Reissner-Nordstrom black hole surrounded by quintessence is investigated using approximate Lie symmetry methods. It is mainly done by assuming mass and charge of the black hole as small quantities ($\epsilon$), and by retaining its second power in the perturbed geodesic equations for such black hole while neglecting its higher powers. Due to the presence of trivial second-order approximate Lie symmetries of these perturbed geodesic equations, a rescaling of the geodetic parameter gives a rescaling of the energy in this black hole. Interestingly we obtain an explicit relation of the rescaling factor that depends on the square of the charge to mass ratio of the black hole, the normalization factor $\alpha$, which is related to the state parameter of the quintessence matter, and the coordinate $r$. A comparison of this rescaling factor with that of the Reissner-Nordstrom black hole (Hussain et. al SIGMA, 2007), without quintessence is given. It is observed that the presence of the quintessence field reduces the energy in this black hole spacetime. Further it is found that there exists a point outside the event horizon of this black hole where the effect of quintessence balances the energy content in this black hole without quintessence, and where the total energy of the underlying spacetime becomes zero.

Energy in the spacetime field of the charged rotating BTZ black hole via approximate Lie symmetries

Using Lie approximate symmetry methods for differential equations, second-order approximate symmetries of the geodesic equations for the charged rotating Bañados, Teitelboim and Zanelli black hole are investigated. For consistency of the trivial second-order approximate symmetries of the geodesic equations, a rescaling factor for energy in this spacetime is obtained.

Re-scaling of energy in stringy charged black hole solutions using approximate symmetries

This paper examines the energy problem in General Relativity using approximate Lie symmetry methods for differential equations. We evaluate second-order approximate symmetries of the geodesic equations for the stringy charged black hole solutions. We conclude that energy must be re-scaled by some factor in the second-order approximation.

Energy of the Kerr–Newman-AdS black hole by using approximate Lie symmetries

Using Lie approximate symmetry methods for differential equations second-order approximate symmetries of the geodesic equations for the Kerr–Newman-AdS (KN-AdS) spacetime are investigated. For this purpose the KN-AdS metric is considered as a second perturbation of the AdS metric. A rescaling of the arc length parameter for consistency of the trivial second-order approximate symmetries of the geodesic equations indicates that the energy in the KN-AdS spacetime has to be rescaled. There is an extra contribution to the energy of the KN-AdS spacetime due to the cosmological constant. This energy expression is compared with that for the Kerr–Newman (KN) spacetime.

BI-HAMILTONIAN REPRESENTATION, SYMMETRIES AND INTEGRALS OF MIXED HEAVENLY AND HUSAIN SYSTEMS

In the recent paper by one of the authors (MBS) and A. A. Malykh on the classification of second-order PDEs with four independent variables that possess partner symmetries [1], mixed heavenly equation and Husain equation appear as closely related canonical equations admitting partner symmetries. Here for the mixed heavenly equation and Husain equation, formulated in a two-component form, we present recursion operators, Lax pairs of Olver–Ibragimov–Shabat type and discover their Lagrangians, symplectic and bi-Hamiltonian structure. We obtain all point and second-order symmetries, integrals and bi-Hamiltonian representations of these systems and their symmetry flows together with infinite hierarchies of nonlocal higher symmetries.

Approximate Noether symmetries of the geodesic equations for the charged-Kerr spacetime and rescaling of energy

Using approximate symmetry methods for differential equations we have investigated the exact and approximate symmetries of a Lagrangian for the geodesic equations in the Kerr spacetime. Taking Minkowski spacetime as the exact case, it is shown that the symmetry algebra of the Lagrangian is 17 dimensional. This algebra is related to the 15 dimensional Lie algebra of conformal isometries of Minkowski spacetime. First introducing spin angular momentum per unit mass as a small parameter we consider first-order approximate symmetries of the Kerr metric as a first perturbation of the Schwarzschild metric. We then consider the second-order approximate symmetries of the Kerr metric as a second perturbation of the Minkowski metric. The approximate symmetries are recovered for these spacetimes and there are no non- trivial approximate symmetries. A rescaling of the arc length parameter for consistency of the trivial second-order approximate symmetries of the geodesic equations indicates that the energy in the charged-Kerr metric has to be rescaled and the rescaling factor is r-dependent. This re-scaling factor is compared with that for the Reissner–Nordstrom metric.

Second-Order Approximate Symmetries of the Geodesic Equations for the Reissner-Nordström Metric and Re-Scaling of Energy of a Test Particle

Following the use of approximate symmetries for the Schwarzschild spacetime by A.H. Kara, F.M. Mahomed and A. Qadir (Nonlinear Dynam., to appear), we have in- vestigated the exact and approximate symmetries of the system of geodesic equations for the Reissner-Nordstrspacetime (RN). For this purpose we are forced to use second order approximate symmetries. It is shown that in the second-order approximation, energy must be rescaled for the RN metric. The implications of this rescaling are discussed.

SUSY approach to Pauli Hamiltonians with an axial symmetry

A two-dimensional Pauli Hamiltonian describing the interaction of a neutral spin-1/2 particle with a magnetic field having axial and second-order symmetries is considered. After separation of variables, the one-dimensional matrix Hamiltonian is analysed from the point of view of supersymmetric quantum mechanics. Attention is paid to the discrete symmetries of the Hamiltonian and also to the Hamiltonian hierarchies generated by intertwining operators. The spectrum is studied by means of the associated matrix shape invariance. The relation between the intertwining operators and the second-order symmetries is established, and the full set of ladder operators that complete the dynamical algebra is constructed.

Second-order superintegrable systems in conformally flat spaces. I. Two-dimensional classical structure theory

This paper is the first in a series that lays the groundwork for a structure and classification theory of second-order superintegrable systems, both classical and quantum, in conformally flat spaces. Many examples of such systems are known, and lists of possible systems have been determined for constant curvature spaces in two and three dimensions, as well as few other spaces. Observed features of these systems are multiseparability, closure of the quadratic algebra of second-order symmetries at order 6, use of representation theory of the quadratic algebra to derive spectral properties of the quantum Schrödinger operator, and a close relationship with exactly solvable and quasi-exactly solvable systems. Our approach is, rather than focus on particular spaces and systems, to use a general theoretical method based on integrability conditions to derive structure common to all systems. In this first paper we consider classical superintegrable systems on a general two-dimensional Riemannian manifold and uncover their common structure. We show that for superintegrable systems with nondegenerate potentials there exists a standard structure based on the algebra of 2×2 symmetric matrices, that such systems are necessarily multiseparable and that the quadratic algebra closes at level 6. Superintegrable systems with degenerate potentials are also analyzed. This is all done without making use of lists of systems, so that generalization to higher dimensions, where relatively few examples are known, is much easier.

Lie symmetries and superintegrability in quantum mechanics

Starting from the structure of the higher order Lie symmetries of the Schrodinger equation in the Euclidean plane E2, we establish, in the case of first-and second-order symmetries, the relations between separation of variables and superintegrable systems in quantum mechanics.

On superintegrable symmetry-breaking potentials in N-dimensional Euclidean space

We give a graphical prescription for obtaining and characterizing all separable coordinates for which the Schr?dinger equation admits separable solutions for one of the superintegrable potentials Here xn+1 is a distinguished Cartesian variable. The algebra of second-order symmetries of the resulting Schr?dinger equation is given and, for the first potential, the closure relations of the corresponding quadratic algebra. These potentials are particularly interesting because they occur in all dimensions n ? 1, the separation of variables problem is highly nontrivial for them, and many other potentials are limiting cases.

Initial-value problems for evolutionary partial differential equations and higher-order conditional symmetries

We suggest a new approach to the problem of dimensional reduction of initial/boundary value problems for evolution equations in one spatial variable. The approach is based on higher-order (generalized) conditional symmetries of the equations involved. It is shown that reducibility of an initial value problem for an evolution equation to a Cauchy problem for a system of ordinary differential equations can be fully characterized in terms of conditional symmetries which leave invariant the equation in question. We also give some examples of the solution of initial value problems for second- and third-order nonlinear differential equations by reduction by their conditional symmetries. We give a systematic classification of general second-order partial differential equations admitting second-order conditional symmetries, based on Lie’s classification of invariant second-order ordinary differential equations. This yields five classes of principally new initial value problems for nonlinear evolution equations which admit no Lie symmetries and are reducible via second-order conditional symmetries.

Recursion Operators, Higher-order Symmetries and Superintegrability in Quantum Mechanics

A connection between the theory of superintegrable quantum-mechanical systems, which admit a maximal number of integrals of motion, and the standard Lie group theory is established. It is shown that the flows generated by first- and second-order Lie symmetries of the bidimensional Schrodinger equation can be classified and interpreted as quantum-mechanical operators which commute with integrable or superintegrable Hamiltonians. In this way, all known superintegrable potentials in the plane are naturally obtained and slightly more general integrals of motion are found.

Approximate first integrals of nonlinear oscillators with one-degree of freedom

Exact symmetries of the unperturbed (linear) part of the dynamical systems are determined. Resonance conditions which lead to the symmetry-breaking of the symmetries of the unperturbed part are obtained. The second-order approximate symmetries of the one degree of freedom, damped-driven oscillators are found. By employing an approximate version of Noether’s theorem, second-order approximate first integrals are obtained for undamped oscillators. The results are discussed on the contour plots of the first integrals.