arXiv subjects

Y. Nutku

Publications and source records attributed to Y. Nutku.

At least 19 recordsLinked to original sources

arXiv2013

A family of heavenly metrics

This is a corrected and essentially extended version of the previous preprint arXiv:gr-qc/0105088 v4 (2002) by Y. Nutku and M. Sheftel containing new results. It is being published now in honor of Y. Nutku's memory. All responsibility for the additions and changes must be attributed to M. Sheftel. We present new anti-self-dual exact solutions of the Einstein field equations with Euclidean and neutral (ultra-hyperbolic) signatures that admit only one rotational Killing vector. Such solutions of the Einstein field equations are determined by non-invariant solutions of Boyer-Finley ($BF$) equation. For the case of Euclidean signature such a solution of the $BF$ equation was first constructed by Calderbank and Tod. Two years later, Martina, Sheftel and Winternitz applied the method of group foliation to the Boyer-Finley equation and reproduced the Calderbank-Tod solution together with new solutions for the neutral signature. In the case of Euclidean signature we obtain new metrics which asymptotically locally look like a flat space and have a non-removable singular point at the origin. In the case of ultra-hyperbolic signature there exist three inequivalent forms of metric. Only one of these can be obtained by analytic continuation from the Calderbank-Tod solution whereas the other two are new.

gr-qc↗

arXiv2008

Self-dual gravity is completely integrable

We discover multi-Hamiltonian structure of complex Monge-Ampere equation (CMA) set in a real first-order two-component form. Therefore, by Magri's theorem this is a completely integrable system in four real dimensions. We start with Lagrangian and Hamiltonian densities and obtain a symplectic form and the Hamiltonian operator that determines the Dirac bracket. We have calculated all point symmetries of two-component CMA system and Hamiltonians of the symmetry flows. We have found two new real recursion operators for symmetries which commute with the operator of a symmetry condition on solutions of the CMA system. These operators form two Lax pairs for the two-component system. The recursion operators, being applied to the first Hamiltonian operator, generate infinitely many real Hamiltonian structures. We show how to construct an infinite hierarchy of higher commuting flows together with the corresponding infinite chain of their Hamiltonians.

math-ph↗

arXiv2008

Hamiltonian structure of the complex Monge-Ampère equation

We discover Hamiltonian structure of the complex Monge-Amp`ere equation when written in a first order two-component form. We present Lagrangian and Hamiltonian functions, a symplectic form and the Hamiltonian operator that determines the Poisson bracket.

physics.class-ph↗

arXiv2007

Lift of noninvariant solutions of heavenly equations from three to four dimensions and new ultra-hyperbolic metrics

We demonstrate that partner symmetries provide a lift of noninvariant solutions of three-dimensional Boyer-Finley equation to noninvariant solutions of four-dimensional hyperbolic complex Monge-Ampere equation. The lift is applied to noninvariant solutions of the Boyer-Finley equation, obtained earlier by the method of group foliation, to yield noninvariant solutions of the hyperbolic complex Monge-Ampere equation. Using these solutions we construct new Ricci-flat ultra-hyperbolic metrics with non-zero curvature tensor that have no Killing vectors.

math-ph↗

arXiv2005

Multi-Hamiltonian structure of Plebanski's second heavenly equation

We show that Plebanski's second heavenly equation, when written as a first-order nonlinear evolutionary system, admits multi-Hamiltonian structure. Therefore by Magri's theorem it is a completely integrable system. Thus it is an example of a completely integrable system in four dimensions.

nlin.SI↗

arXiv2005

What is the present-day status of The Copernican Principle?

I point out that according to the Copernican principle our universe is not unique. The way to make sense out of this statement is for us to construct a gravitational instanton that will tunnel out of our vacuum into another, to form a universe other than our Hubble bubble.

gr-qc↗

arXiv2005

Electrovac $pp$-waves

New exact solutions of the Einstein-Maxwell field equations that describe $pp$-waves are presented.

gr-qc↗

arXiv2003

Anti-self-dual Riemannian metrics without Killing vectors, can they be realized on K3?

Explicit Riemannian metrics with Euclidean signature and anti-self dual curvature that do not admit any Killing vectors are presented. The metric and the Riemann curvature scalars are homogenous functions of degree zero in a single real potential and its derivatives. The solution for the potential is a sum of exponential functions which suggests that for the choice of a suitable domain of coordinates and parameters it can be the metric on a compact manifold. Then, by the theorem of Hitchin, it could be a class of metrics on $K3$, or on surfaces whose universal covering is $K3$.

gr-qc↗

arXiv2003

Partner symmetries of the complex Monge-Ampere equation yield hyper-Kahler metrics without continuous symmetries

We extend the Mason-Newman Lax pair for the elliptic complex Monge-Ampère equation so that this equation itself emerges as an algebraic consequence. We regard the function in the extended Lax equations as a complex potential. We identify the real and imaginary parts of the potential, which we call partner symmetries, with the translational and dilatational symmetry characteristics respectively. Then we choose the dilatational symmetry characteristic as the new unknown replacing the Kähler potential which directly leads to a Legendre transformation and to a set of linear equations satisfied by a single real potential. This enables us to construct non-invariant solutions of the Legendre transform of the complex Monge-Ampère equation and obtain hyper-Kähler metrics with anti-self-dual Riemann curvature 2-form that admit no Killing vectors.

math-ph↗

arXiv2003

Quantization with maximally degenerate Poisson brackets: The harmonic oscillator!

Nambu's construction of multi-linear brackets for super-integrable systems can be thought of as degenerate Poisson brackets with a maximal set of Casimirs in their kernel. By introducing privileged coordinates in phase space these degenerate Poisson brackets are brought to the form of Heisenberg's equations. We propose a definition for constructing quantum operators for classical functions which enables us to turn the maximally degenerate Poisson brackets into operators. They pose a set of eigenvalue problems for a new state vector. The requirement of the single valuedness of this eigenfunction leads to quantization. The example of the harmonic oscillator is used to illustrate this general procedure for quantizing a class of maximally super-integrable systems.

quant-ph↗

arXiv2001

Multi-Lagrangians for Integrable Systems

We propose a general scheme to construct multiple Lagrangians for completely integrable non-linear evolution equations that admit multi- Hamiltonian structure. The recursion operator plays a fundamental role in this construction. We use a conserved quantity higher/lower than the Hamiltonian in the potential part of the new Lagrangian and determine the corresponding kinetic terms by generating the appropriate momentum map. This leads to some remarkable new developments. We show that nonlinear evolutionary systems that admit $N$-fold first order local Hamiltonian structure can be cast into variational form with $2N-1$ Lagrangians which will be local functionals of Clebsch potentials. This number increases to $3N-2$ when the Miura transformation is invertible. Furthermore we construct a new Lagrangian for polytropic gas dynamics in $1+1$ dimensions which is a {\it local} functional of the physical field variables, namely density and velocity, thus dispensing with the necessity of introducing Clebsch potentials entirely. This is a consequence of bi-Hamiltonian structure with a compatible pair of first and third order Hamiltonian operators derived from Sheftel's recursion operator.

hep-th↗

arXiv2001

Superintegrable Calogero-type systems admit maximal number of Poisson structures

We present a general scheme for constructing the Poisson structure of super-integrable dynamical systems of which the rational Calogero-Moser system is one of the most interesting one. This dynamical system is $2N$ dimensional with $2N- 1$ first integrals and our construction yields $2N-1$ degenerate Poisson tensors that each admit $2(N-1)$ Casimirs. Our results are quite generally applicable to all super-integrable systems and form an alternative to the traditional bi-Hamiltonian approach.

nlin.SI↗

arXiv2000

New symplectic structure for KdV

Starting with Lagrangians, which turn out to be degenerate, the Hamiltonian operators for integrable systems can be constructed using Dirac's theory of constraints. We illustrate this by giving a systematic discussion of the first Hamiltonian structure of KdV. The first symplectic 2-form obtained from Dirac's theory is the time component of the covariant Witten-Zuckerman symplectic 2-form current. We derive the flux of the first symplectic 2-form. Then we turn to a new Lagrangian for KdV recently obtained by Pavlov and present the corresponding new covariant symplectic structure for KdV. This shows that the inverse of Magri's second Hamiltonian operator is also a Hamiltonian operator for KdV.

hep-th↗

arXiv2000

Impulsive spherical gravitational waves

Penrose's identification with warp provides the general framework for constructing the continuous form of impulsive gravitational wave metrics. We present the 2-component spinor formalism for the derivation of the full family of impulsive spherical gravitational wave metrics which brings out the power in identification with warp and leads to the simplest derivation of exact solutions. These solutions of the Einstein vacuum field equations are obtained by cutting Minkowski space into two pieces along a null cone and re-identifying them with warp which is given by an arbitrary non-linear holomorphic transformation. Using 2-component spinor techniques we construct a new metric describing an impulsive spherical gravitational wave where the vertex of the null cone lies on a world-line with constant acceleration.

gr-qc↗

arXiv2000

Topologically massive magnetic monopoles

We show that in the Maxwell-Chern-Simons theory of topologically massive electrodynamics the Dirac string of a monopole becomes a cone in anti-de Sitter space with the opening angle of the cone determined by the topological mass which in turn is related to the square root of the cosmological constant. This proves to be an example of a physical system, {\it a priory} completely unrelated to gravity, which nevertheless requires curved spacetime for its very existence. We extend this result to topologically massive gravity coupled to topologically massive electrodynamics in the framework of the theory of Deser, Jackiw and Templeton. These are homogeneous spaces with conical deficit. Pure Einstein gravity coupled to Maxwell-Chern-Simons field does not admit such a monopole solution.

gr-qc↗

arXiv2000

Covariant symplectic structure of the complex Monge-Ampère equation

The complex Monge-Ampère equation admits covariant bi-symplectic structure for complex dimension 3, or higher. The first symplectic 2-form is obtained from a new variational formulation of complex Monge- Ampère equation in the framework of the covariant Witten-Zuckerman approach to symplectic structure. We base our considerations on a reformulation of the Witten-Zuckerman theory in terms of holomorphic differential forms. The first closed and conserved Witten-Zuckerman symplectic 2-form for the complex Monge-Ampère equation is obtained in arbitrary dimension and for all cases elliptic, hyperbolic and homogeneous. The connection of the complex Monge-Ampère equation with Ricci-flat Kähler geometry suggests the use of the Hilbert action. However, we point out that Hilbert's Lagrangian is a divergence for Kähler metrics. Nevertheless, using the surface terms in the Hilbert Lagrangian we obtain the second Witten-Zuckerman symplectic 2-form for complex dimension>2.

hep-th↗

arXiv1998

Spinor formulation of topologically massive gravity

In the framework of real 2-component spinors in three dimensional space-time we present a description of topologically massive gravity (TMG) in terms of differential forms with triad scalar coefficients. This is essentially a real version of the Newman-Penrose formalism in general relativity. A triad formulation of TMG was considered earlier by Hall, Morgan and Perjes, however, due to an unfortunate choice of signature some of the spinors underlying the Hall-Morgan-Perjes formalism are real, while others are pure imaginary. We obtain the basic geometrical identities as well as the TMG field equations including a cosmological constant for the appropriate signature. As an application of this formalism we discuss the Bianchi Type $VIII - IX$ exact solutions of TMG and point out that they are parallelizable manifolds. We also consider various re-identifications of these homogeneous spaces that result in black hole solutions of TMG.

gr-qc↗

arXiv1998

A theorem on topologically massive gravity

We show that for three dimensional space-times admitting a hypersurface orthogonal Killing vector field Deser, Jackiw and Templeton's vacuum field equations of topologically massive gravity allow only the trivial flat space-time solution. Thus spin is necessary to support topological mass.

gr-qc↗