Spacetime Encodings III - Second Order Killing Tensors

This paper explores the Petrov type D, stationary axisymmetric vacuum (SAV) spacetimes that were found by Carter to have separable Hamilton-Jacobi equations, and thus admit a second-order Killing tensor. The derivation of the spacetimes presented in this paper borrows from ideas about dynamical systems, and illustrates concepts that can be generalized to higher- order Killing tensors. The relationship between the components of the Killing equations and metric functions are given explicitly. The origin of the four separable coordinate systems found by Carter is explained and classified in terms of the analytic structure associated with the Killing equations. A geometric picture of what the orbital invariants may represent is built. Requiring that a SAV spacetime admits a second-order Killing tensor is very restrictive, selecting very few candidates from the group of all possible SAV spacetimes. This restriction arises due to the fact that the consistency conditions associated with the Killing equations require that the field variables obey a second-order differential equation, as opposed to a fourth-order differential equation that imposes the weaker condition that the spacetime be SAV. This paper introduces ideas that could lead to the explicit computation of more general orbital invariants in the form of higher-order Killing Tensors.Comment: 9 page

Finite volume and pseudo-spectral schemes for the fully nonlinear 1D Serre equations

After we derive the Serre system of equations of water wave theory from a generalized variational principle, we present some of its structural properties. We also propose a robust and accurate finite volume scheme to solve these equations in one horizontal dimension. The numerical discretization is validated by comparisons with analytical, experimental data or other numerical solutions obtained by a highly accurate pseudo-spectral method.Comment: 28 pages, 16 figures, 75 references. Other author's papers can be downloaded at http://www.denys-dutykh.com

Scaling Without Conformal Invariants in the NonLocal Relativistic Quantum Systems in Living Cells

Since the 1948 the mathematical description of the so-called Casimir world as a part of the physical observed space-time in the relativistic sense is to be considered by the help of the Hamiltonian quantum field’s theory and furthermore even it is based on the fine play between the continuity and the discrete too. The axiomatic-physical methods of the local quantum fields theory has given us the other possibility than the Lagrange quantum field’s theory and precisely on this rigorously mathematical way to understand the singularities and the black holes, also the dark energy and the dark matter from one uniformly point of view. Aside from this, the essential difference is that external forces other than gravity, e.g. such as Casimir force, play a major role in the phenomena, i.e. there is not observed in our seeing world a local classical relativistic electromagnetic field potential Aμ(x). And also it is possible to describe the fundamental interactions between anyone concrete fundamental relativistic quantum field system with someone other or with the external and innerness material objects as a additional boundary conditions by the proving the fulfilling of the causality conditions and consider they as an external classical fields, and everyone internal background fields. At the first in his famous work “To the Electrodynamics of moved bodies”, Leipzig, 1905, Einstein has proved the possibility to understand the nature from the relativistic point of view in the classical physics. By the living cells as an object of the fundamental cryobiological researches i.e. in this case the metabolisms is minimal so that by the help of the axiomatic-physical methods by the relativistic theory of quantum fields systems considering as a Micro fields it is possible to be taken in the account the problem of a “time’s arrow” at the microscopic level by the contemporary considerations of the quantum vacuum in the Casimir world as a ground state of anyone relativistic quantum system becomes a fixture by the lyophilized elementary living cells. So the possibility to understand the Hilbert space with indefinite metric for the further considerations the word elementary understands a one structure idealization of the living cells. Also the many miracle properties of so defined living cells apparent enchanting by consideration of his functions yet are putting besides in the molecules but in the fundamental quantum field interactions between the quantum vacuum of anyone quantum fields system in the Microsoft matter and the molecules but taken in the Minkowsi space-time too. Moreover it can be represented the symmetrical selfadjoint Hamiltonian operator Φ taken by as for simplicity for the relativistic quantum scalar fields by definition obtained as virtual (potential) element in the Hilbert functional space with indefinite metric. That is the quantum field operator obtained by everyone wave fields solution at the fixed time known as a virtual or “potential” quantum field operator acting on the virtual vacuum vector valued functional states of the Hilbert functional space with indefinite metric. So also it is realizable the possibility to be obtained the local or non local quantum force currents, i.e. whish interact minimal local or global by phase integration over the field potential with the field force carrier knowing as the so called virial current i.e. that impact near local or global at the distance of interactions with the classical neighborhood in the Microsoft matter in the Minkowski space-time. The probability interpretation of the spectral family give us the physical interpretation of the observed quantum entity by the relativistic quantum systems even for the dynamically (not thermodynamically) fine structure of the ground state as potential state also as virtual vector valued functional state, t.e. as the element of the Hilbert functional space with indefinite metric by the vacuum interactions in the Casimir world. It knows yet the Casimir force today is measured with exactness by 5 %. Precisely the impact of this force on the molecular biology (genetics) is still not clear, i.e. there is a new situation of the so called quantum cryobiology. The additional boundary conditions must be taken under account, e.g. in the cosmogony models it is not possible to consider additional boundary conditions. So also it is possible to understand better the molecules by the molecular biology as a classical object interacting with the ground state of the every one relativistic quantum field system. So also by definition it is considered the relevant operator valued functional Banach algebra or in the Schrödinger picture the vacuum wave functional as a solution of the impulse wave equation describing the some relativistic quantum system in the Minkowski space-time. With other words by the help of the so called S-matrix theory as in the non relativistic case where this theory is very gut proved we hope to understand better the nature under consideration.It knows the following fact that it is potentially force with a long-range action at the distance or with other words asserted every experiment in this genetics domain without clearness of the role of his impact on the neighborhoods in the living cell. Just therefore this is to be taken very good under account from the point of view of the nanophysics too. May be he is the cause for not observing of the so-called Goldstein massless bosons or the quarks as it is the case by the Coulomb force between the charged particles.Moreover the Casimir vacuum state of the relativistic quantum scalar field system may be not belonging in the operator definition’s functional domain of the field’s operators, but fulfill the additional causal and boundary conditions by the solving the boundary value problem for the carrier of the interaction force, the virtual fundamental scalar particles called by us scalars belonging to the domain structure of the Casimir world. So also the Casimir vacuum in the asymptotic past at the left of the one not moved perfectly conductor plate contains then from the micro-causal point of view propagation of the virtual particles for the initial observer understanding as referent system (a map). In the asymptotic future at the right of the same plate and the left of the second parallel moved perfectly conductor plate towards the plate at the rest with a constant velocity v the propagation of the see massive particles for the late-time observer, e.g. the Maxwell demon, and moreover at the right of the moved plate anew a propagation of the virtual relativistic quantum particles system. Precisely the scalar massless relativistic quantum field give us then that his local algebras are unitary equivalent in the bounded domains of the locally algebras by the matter field and also they have the same structure properties whish is from more great importance for the theory than the definiteness of the metric of the Hilbert functional space. So it is possible to be defined the double singularities which will be given by the ground state of local relativistic quantum scalar field system too. The symmetries and structure properties are mathematical described by the Banach algebra of the field’s operators defined in the Hilbert functional space with indefinite metric. Farther the ground state is defined over dies algebra but it can be negative too as remember from the indefinite metric of the Hilbert functional space. However then there are a number of additional properties generated from the physical distinctions by the massless systems: His scale i.e. the group of the scale transformations represented by the dilatations and special conformal transformations and conformal symmetries also obtained by the group of the conformal transformations give a double singularities of the quantum systems and the vacuum state, but scale invariance does not imply necessary a conformal invariance and as well the infrared effects leaded to manifest the global structure of the relativistic quantum systems and the vacuum state. Quantum Field Theory QFT and the Renormierungs groups theory RG-groups are classified by scale invariant, Infrared IR fixed point (Wilson’s philosophy). In the Doctor paper (Petrov, 1978) it is showed that the scaling behaviors of the some quantum entities are destroyed in longitudinal and conserved in the cross section’s direction by fulfilling the causality condition for non forward deep inelastic scattering of leptons and hadrons. Also the scale invariance is not from the same nature as the conformal invariance by the massless quantum fields and the scale invariance lead yet not necessarily to the conformal invariance. Furthermore the Hilbert functional space understands by means of the space of the test functions from his completion by anyone norm the possibility of the definition of the Casimir quantum vacuum state as well a ground state in the Schrödinger picture over the involutes Banach algebra of the field operators defined in the Hilbert functional space with indefinite metric. Then so one functional vector valued vacuum state can be negative as remember of the indefinite metric by definition but this is not from anyone significance for the theory. This question precisely spoken is a pure algebraically formulations of anyone relativistic quantum systems out of the Hilbert functional spaces with indefinite metric. The theoretical underpinnings of scale without the conformal invariance in relativistic quantum physics are given in the light of the results of the non local operator’s expansion on the light cone. Then the Casimir vacuum state of a given relativistic scalar quantum fields systems, precisely due to deep connectionsbetween scale-invariant theories and the recurrent scaling behaviors of the quantum entities in the Casimir world can be defined over the involutes Banach algebra of the field operators acting on the virtual vector valued state defined in the Hilbert functional space with indefinite metric. Furthermore the vacuum state in the Schrödinger picture defined over this algebra can be negative too as remembering of the indefinite metric but that is only a one algebraic problem. It can be shown that, on scaling-invariant time like paths of the virtual quantum particles, there is a redefinition of the dilatation current by the virial current that leads to virtual generators of dilatations operators. Also just that lead to the generations of the virtual vacuum fluctuations described by the relativistic quantum fields operators created an involutes Banach algebra of the quantum field operators out of the Hilbert space with indefinite metric. Finally, it can be develop a systematic algorithm by the Casimir world for the research of scaling-invariant non space like paths of virtual particles caused by virtual fluctuations of the vacuum with a zero point energy ZPE and broken scaling-invariant time like and non space like paths of see massive particles

A novel implementation of Petrov-Galerkin method to shallow water solitary wave pattern and superperiodic traveling wave and its multistability: generalized Korteweg-de Vries equation

This work deals with the constitute of numerical solutions of the generalized Korteweg-de Vries (GKdV) equation with Petrov-Galerkin finite element approach utilising a cubic B-spline function as the trial function and a quadratic function as the test function. Accurateness and effectiveness of the submitted methods are shown by employing propagation of single solitary wave. The L2, L∞error norms and I1, I2and I3invariants are used to validate the applicability and durability of our numerical algorithm. Implementing the Von-Neumann theory, it is manifested that the suggested method is marginally stable. Furthermore, supernonlinear traveling wave solution of the GKdV equation is presented using phase plots. It is seen that the GKdV equation supports superperiodic traveling wave solution only and it is significantly affected by velocity and nonlinear parameters. Also, considering a superficial periodic forcing multistability of traveling waves of perturbed GKdV equation is presented. It is found that the perturbed GKdV equation supports coexisting chaotic and various quasiperiodic features with same parametric values at different initial condition

Improvements and Analysis of Challenging Numerical Simulations of Binary Black Holes

We explore different gauge choices in the moving puncture formulation in order to improve the accuracy of a linear momentum measure evaluated on the horizon of the remnant black hole produced by the merger of a binary. In particular, motivated by the study of gauges in which the damping term in the shift m eta takes on a constant value, we design a gauge via a variable shift parameter m eta(r(t)). This parameter takes a low value asymptotically, 1/m, and then takes on a value of approximately 2 at the final hole horizon. This eta then follows the remnant black hole as it moves due to its net recoil velocity. We find that this choice keeps the accuracy of the binary evolution. Furthermore, if the asymptotic value of the parameter mis chosen about or below 1.0, it produces more accurate results for the recoil velocity than the corresponding evaluation of the radiated linear momentum at infinity, for typical numerical resolutions. Detailed studies of an unequal mass q = m1/m2 = 1/3 nonspinning binary are provided and then verified for other mass ratios (q = 1/2; 1/5) and spinning (q = 1) binary black hole mergers. We also use a position and black hole mass dependent damping term, eta[x1(t); x2(t);m1;m2], in the shift evolution, rather than a constant or conformal-factor dependent choice. We have found that this substantially reduces noise generation at the start of the numerical integration and keeps the numerical grid stable around both black holes, allowing for more accuracy with lower resolutions. We test our choices for this gauge in detail in a case study of a binary with a 7:1 mass ratio, and then use 15:1 and 32:1 binaries for a convergence study. Finally, we apply our new gauge to a 64:1 binary and a 128:1 binary to well cover the comparable and small mass ratio regimes. Finally, we perform an analytic study of two nonspinning binary systems with q = 1 and q = 1/3 that use Brill-Lindquist initial data. These spacetimes are rotated into a frame that is transverse, with two of the five Weyl scalars vanishing, and quasi-Kinnersley. We derive and evaluate an index D that, when used in conjunction with the Baker-Campanelli Specialty index S, allows us to analyze and classify these spacetimes into Petrov types in the strong-field regime and between the black holes. Also included is an appendix to be utilized in conjunction with the RIT Catalog. It provides scripts for generation of fitting coefficients for analytic formulae developed in [1], [2], and [3] for specific subsets of the full 777 waveform RIT Catalog. Finally, we use these scripts to generate fitting coefficients for all non-precessing binaries in the catalog

Killing Tensors in Koutras-McIntosh Spacetimes

This thesis is concerned with the (non)existence of Killing Tensors in Koutras-McIntosh spacetimes. Killing tensors are of particular interest in general relativity, because these correspond to conserved quantities for the geodesic motion. For instance, Carter found such a conserved quantity in the Kerr metric which he used to explicitly integrate the geodesic equations. The equation defining a Killing tensor is actually an overdetermined linear first order partial differential equation. We shall study the Killing equation using methods from the geometric theory of PDEs. More precisely, we use Cartan's prolongation method to prove the (non)existence of Killing tensors in several Koutras-McIntosh spacetimes. A subclass of the Koutras--McIntosh spacetimes are the conformally flat pp-waves. We show that a generic conf. flat pp-wave has an irreducible Killing 2-tensor, which reproves a result obtained by Keane and Tupper using a different method. Moreover, we prove in particular examples of pp-waves that all Killing tensors of degree 3 and 4 are reducible. We then study the Wils metric, another subclass of the Koutras-McIntosh spacetimes. This metric has a univariate function as its parameter. By using Cartan's prolongation method we deduce the explicit form of the function for which the Wils metric admits a Killing vector, and for which a Killing 2-tensor. This existence result for a Killing vector makes a statement by Koutras and McIntosh more precise. Finally, we show in particular examples of a Wils metric that all Killing 3- and 4-tensors are reducible