Stationary String Research Articles

Contagious Vibrations: Sympathetic Resonance as a Model for Disease Transmission in the Writings of Ficino, Fracastoro, and Cardano.

Contagious diseases were among the most vexing problems in ancient theories of health, which could not easily account for how a corruption of one person's humors could cause a similar corruption in another's. One useful explanatory concept for Renaissance doctors tackling this theoretical gap was the phenomenon of resonance or "sympathetic vibration" - where one stationary string begins to vibrate spontaneously when a similarly tuned string is plucked nearby - as both resonance and contagion involved some mysterious, insensible action at a distance between an agent and a patient. Tracing the writings of Marsilio Ficino, Girolamo Fracastoro, and Girolamo Cardano, this essay explores the relationships between the writers' accounts of sympathetic vibrations and their contagion theories. It argues that different conceptions of the acoustic phenomenon - either as a manifestation of a Neo-Platonic World-Soul that underpinned the universe or else as a physical effect - revealed the writers' cosmological views that, in turn, informed their accounts of the human body and disease.

Effect of sub-chronic ferrous sulfate treatment on motor skills, hematological and biochemical parameters in rats

ABSTRACTThis study investigated the effects of ferrous sulfate (FeSO4) on motor skills, hematological and biochemical parameters in rats. Adult rats were treated with dose of iron (280 mg/L, per os) for 15 consecutive days in drinking water. No significant difference was noticed for the motor skills in the stationary beam (p = 0.23) and suspended string tests (p = 0.48) between control and iron-treated rats. However, iron-treated rats showed a significant increase in white blood cells count (p = 0.01), mean corpuscular volume values (p = 0.02) and decrease in frequency of peristaltic contractions of the fragment of the intestine (in vitro) compared to control rats (p = 0.01). No significant difference in plasma iron level (p = 0.89) and transferrin amount were observed after iron treatment (p = 0.65). The findings indicate that iron treatment at 280 mg/L, per os for 15 consecutive days in adult rats induced increase of hematological parameters (sign of a potential inflammation), but not motor skills deficit.

Dynamic Stability of a Class of Second-Order Distributed Structural Systems With Sinusoidally Varying Velocities

Parametric instability in a system is caused by periodically varying coefficients in its governing differential equations. While parametric excitation of lumped-parameter systems has been extensively studied, that of distributed-parameter systems has been traditionally analyzed by applying Floquet theory to their spatially discretized equations. In this work, parametric instability regions of a second-order nondispersive distributed structural system, which consists of a translating string with a constant tension and a sinusoidally varying velocity, and two boundaries that axially move with a sinusoidal velocity relative to the string, are obtained using the wave solution and the fixed point theory without spatially discretizing the governing partial differential equation. There are five nontrivial cases that involve different combinations of string and boundary motions: (I) a translating string with a sinusoidally varying velocity and two stationary boundaries; (II) a translating string with a sinusoidally varying velocity, a sinusoidally moving boundary, and a stationary boundary; (III) a translating string with a sinusoidally varying velocity and two sinusoidally moving boundaries; (IV) a stationary string with a sinusoidally moving boundary and a stationary boundary; and (V) a stationary string with two sinusoidally moving boundaries. Unlike parametric instability regions of lumped-parameter systems that are classified as principal, secondary, and combination instability regions, the parametric instability regions of the class of distributed structural systems considered here are classified as period-1 and period-i (i>1) instability regions. Period-1 parametric instability regions are analytically obtained; an equivalent total velocity vector is introduced to express them for all the cases considered. While period-i (i>1) parametric instability regions can be numerically calculated using bifurcation diagrams, it is shown that only period-1 parametric instability regions exist in case IV, and no period-i (i>1) parametric instability regions can be numerically found in case V. Unlike parametric instability in a lumped-parameter system that is characterized by an unbounded displacement, the parametric instability phenomenon discovered here is characterized by a bounded displacement and an unbounded vibratory energy due to formation of infinitely compressed shock-like waves. There are seven independent parameters in the governing equation and boundary conditions, and the parametric instability regions in the seven-dimensional parameter space can be projected to a two-dimensional parameter plane if five parameters are specified. Period-1 parametric instability occurs in certain excitation frequency bands centered at the averaged natural frequencies of the systems in all the cases. If the parameters are chosen to be in the period-i (i≥1) parametric instability region corresponding to an integer k, an initial smooth wave will be infinitely compressed to k shock-like waves as time approaches infinity. The stable and unstable responses of the linear model in case I are compared with those of a corresponding nonlinear model that considers the coupled transverse and longitudinal vibrations of the translating string and an intermediate linear model that includes the effect of the tension change due to axial acceleration of the string on its transverse vibration. The parametric instability in the original linear model can exist in the nonlinear and intermediate linear models.

Stationary closed strings in five-dimensional flat spacetime

We investigate stationary rotating closed Nambu-Goto strings in five-dimensional flat spacetime. The stationary string is defined as a worldsheet that is tangent to a timelike Killing vector. Nambu-Goto equation of motion for the stationary string is reduces to the geodesic equation on the orbit space of the isometry group action generated by the Killing vector. We take a linear combination of a time-translation vector and space-rotation vectors as the Killing vector, and explicitly construct general solutions of stationary rotating closed strings in five-dimensional flat spacetime. We show a variety of their configurations and properties.

Three-point correlator of heavy vertex operators for circular winding strings in [formula omitted

We consider an extremal three-point correlator of three heavy vertex operators for the circular winding string state with one large spin and one winding number in AdS5 and one large spin and one winding number in S5. We use a Schwarz–Christoffel transformation to compute semiclassically the extremal three-point correlator on a stationary string trajectory which is mapped to the complex plane with three punctures. It becomes a 4d conformal invariant three-point correlator on the boundary. We discuss the marginality condition of vertex operator.

Simulation of nuclei morphologies for binary alloy

We study the critical nuclei morphologies of a binary alloy by the string method. The dynamic equation of the string, connecting the metastable phase (liquid) and stable phase (solid), is governed by Helmholtz free energy for the binary alloy system at a given temperature. The stationary string through the critical nucleus (saddle point) is obtained if the relaxation time of the string is sufficiently large. The critical nucleus radius and energy barrier to nucleation of a pure alloy with isotropic interface energy in two and three dimensions are calculated, which are consistent with the classical nucleation theory. The critical nuclei morphologies are sensitive to the anisotropy strength of interface energy and interface thickness of alloy in two and three dimensions. The critical nucleus and energy barrier to nucleation become smaller if the anisotropy strength of the interface energy is increased, which means that it is much easier to form a stable nucleus if the anisotropy of the interface energy is considered.

Asymptotic analysis on nonlinear vibration of axially accelerating viscoelastic strings with the standard linear solid model

Nonlinear parametric vibration of axially accelerating viscoelastic strings is investigated via an approximate analytical approach. The standard linear solid model using the material time derivative is employed to describe the string viscoelastic behaviors. A coordinate transformation is introduced to derive Mote’s model of transverse motion from the governing equation of the stationary string. Mote’s model leads to Kirchhoff’s model by replacing the tension with the averaged tension over the string. An asymptotic perturbation approach is proposed to study principal parametric resonance based on the two models. The amplitude and the existence conditions of the steady-state responses are determined by locating the nonzero fixed points in the modulation equations resulting from the solvability condition. Numerical results are presented to highlight the effects of the material parameters, the axial-speed fluctuation amplitude, and the initial stress on steady-state responses.

Stationary strings in the spacetime of rotating black holes in five-dimensional minimal gauged supergravity

We examine the separability properties of the equation of motion for a stationary string near a rotating charged black hole with two independent angular momenta in five-dimensional minimal gauged supergravity. It is known that the separability problem for the stationary string in a general stationary spacetime is reduced to that for the usual Hamilton-Jacobi equation for geodesics of its quotient space with one dimension fewer. Using this fact, we show that the ``effective metric'' of the quotient space does not allow the complete separability for the Hamilton-Jacobi equation, albeit such a separability occurs in the original spacetime of the black hole. We also show that only for two special cases of interest the Hamilton-Jacobi equation admits the complete separation of variables and therefore the integrability for the stationary string motion in the original background, namely, when the black hole has zero electric charge or it has an arbitrary electric charge but two equal angular momenta. We give the explicit expressions for the Killing tensors corresponding to these cases. However, for the general black hole spacetime the effective metric of the quotient space admits a conformal Killing tensor. We construct the explicit expression for this tensor.

Stationary strings and branes in the higher-dimensional Kerr-NUT-(A)dS spacetimes

We demonstrate complete integrability of the Nambu-Goto equations for a stationary string in the general Kerr-NUT-(A)dS spacetime describing the higher-dimensional rotating black hole. The stationary string in D dimensions is generated by a 1-parameter family of Killing trajectories and the problem of finding a string configuration reduces to a problem of finding a geodesic line in an effective (D-1)-dimensional space. Resulting integrability of this geodesic problem is connected with the existence of hidden symmetries which are inherited from the black hole background. In a spacetime with p mutually commuting Killing vectors it is possible to introduce a concept of a $\xi$-brane, that is a p-brane with the worldvolume generated by these fields and a 1-dimensional curve. We discuss integrability of such $\xi$-branes in the Kerr-NUT-(A)dS spacetime.

Spacelike strings and jet quenching from a Wilson loop

We investigate stationary string solutions with spacelike worldsheet in a five-dimensional AdS black hole background, and find that there are many branches of such solutions. Using a non-perturbative definition of the jet quenching parameter proposed by Liu et. al., hep-ph/0605178, we take the lightlike limit of these solutions to evaluate the jet quenching parameter in an N=4 super Yang-Mills thermal bath. We show that this proposed definition gives zero jet quenching parameter, independent of how the lightlike limit is taken. In particular, the minimum-action solution giving the dominant contribution to the Wilson loop has a leading behavior that is linear, rather than quadratic, in the quark separation.

No-drag string configurations for steadily moving quark-antiquark pairs in a thermal bath

We investigate the behavior of stationary string configurations on a five-dimensional AdS black hole background which correspond to quark-antiquark pairs steadily moving in an N=4 super Yang-Mills thermal bath. There are many branches of solutions, depending on the quark velocity and separation as well as on whether Euclidean or Lorentzian configurations are examined.

Computing the dynamic response of an axially moving continuum

The aim of this study was to apply an approximate Galerkin finite-element method to solve the initial boundary-value problem of a viscously damped axially moving pre-tensioned beam including arbitrary support excitations. In order to validate the Galerkin finite-element method the results were compared with exact solutions when available and with solutions obtained using the finite-difference and Galerkin methods. The energy dissipation of the system was considered in the form of an equivalent viscous-damping model. It was also shown that for certain values of the parameters, especially at high velocities, the Galerkin method using stationary string eigenfunctions can give a poor prediction of the dynamic response.

Stationary strings near a higher-dimensional rotating black hole

We study stationary string configurations in a space-time of a higher-dimensional rotating black hole. We demonstrate that the Nambu-Goto equations for a stationary string in the 5D Myers-Perry metric allow a separation of variables. We present these equations in the first-order form and study their properties. We prove that the only stationary string configuration which crosses the infinite red-shift surface and remains regular there is a principal Killing string. A worldsheet of such a string is generated by a principal null geodesic and a timelike at infinity Killing vector field. We obtain principal Killing string solutions in the Myers-Perry metrics with an arbitrary number of dimensions. It is shown that due to the interaction of a string with a rotating black hole there is an angular momentum transfer from the black hole to the string. We calculate the rate of this transfer in a spacetime with an arbitrary number of dimensions. This effect slows down the rotation of the black hole. We discuss possible final stationary configurations of a rotating black hole interacting with a string.

NONLINEAR DYNAMIC STABILITY OF A MOVING STRING BY HAMILTONIAN FORMULATION

In this paper, the stability behavior of an axially moving string is examined in the presence of parametric and combination resonances. The Galerkin discretization utilizing stationary string eigenfunctions is used to transform the partial differential equation governing transverse response into a set of coupled ordinary differential equations. Hamiltonian formulation and averaging method are used to yield a set of autonomous equations. The conditions of parametric and summed resonances are obtained over specific ranges between the natural and exciting frequencies. Explicit results of the stability boundaries for the first and secondary principal parametric and the first summation resonances and the bifurcation paths of the nontrivial amplitudes are obtained.

Statistical effects and the black-hole–D-brane correspondence

The horizon area and curvature of three-charge BPS black strings are studied in the D-brane ensemble for the stationary black string. The charge distributions along the string are used to translate the classical expressions for the horizon area and curvature of BPS black strings with waves into operators on the D-brane Hilbert space. Despite the fact that any `wavy' black string has smaller horizon area and divergent curvature, the typical values of the horizon area and effects of the horizon curvature in the D-brane ensemble deviate negligibly from those of the original stationary black string in the limit of large integer charges. Whether this holds in general will depend on certain properties of the quantum bound states.

How to create a two-dimensional black hole.

The interaction of a cosmic string with a four-dimensional stationary black hole is considered. If a part of an infinitely long string passes close to a black hole it can be captured. The final stationary configurations of such captured strings are investigated. A uniqueness theorem is proved, namely it is shown that the minimal 2-D surface $\Sigma$ describing a captured stationary string coincides with a {\it principal Killing surface}, i.e. a surface formed by Killing trajectories passing through a principal null ray of the Kerr-Newman geometry. Geometrical properties of principal Killing surfaces are investigated and it is shown that the internal geometry of $\Sigma$ coincides with the geometry of a 2-D black or white hole ({\it string hole}). The equations for propagation of string perturbations are shown to be identical with the equations for a coupled pair of scalar fields 'living' in the spacetime of a 2-D string hole. Some interesting features of physics of 2-D string holes are described. In particular, it is shown that the existence of the extra dimensions of the surrounding spacetime makes interaction possible between the interior and exterior of a string black hole; from the point of view of the 2-D geometry this interaction is acausal. Possible application of this result to the information loss puzzle is briefly discussed.

THE EFFECT OF SPATIAL CURVATURE ON CLASSICAL AND QUANTUM STRINGS

We study the effects of spatial curvature on classical and quantum string dynamics. We find the general solution of the circular string motion in static Robertson–Walker space-times with closed or open sections. This is given closely and completely in terms of elliptic functions. The physical properties, string length, energy and pressure are computed and analyzed. We find the back-reaction effect of these strings on the space-time: the self-consistent solution to the Einstein equations is a spatially closed (K>0) space-time with a selected value of the curvature index K (the scale factor is normalized to unity). No self-consistent solutions with K≤0 exist. We semiclassically quantize the circular strings and find the mass m in each case. For K>0, the very massive strings, oscillating on the full hypersphere, have m2~Kn2(n∈N0)independent of α' and the level spacing grows with n, while the strings oscillating on one hemisphere (without crossing the equator) have m2α′~n and a finite number of states N~1/Kα′. For K<0, there are infinitely many string states with masses m log m ~ n, i.e. the level spacing grows slower than n. The stationary string solutions as well as the generic string fluctuations around the center of mass are also found and analyzed in closed form.

Stationary strings and principal Killing triads in 2 + 1 gravity

A new tool for the investigation of (2 + 1)-dimensional gravity is proposed. It is shown that in a stationary (2 + 1)-dimensional space-time, the eigenvectors of the covariant derivative of the timelike Killing vector form a rigid structure, the principal Killing triad. Two of the triad vectors are null, and in many respects they play the role similar to the principal null directions in the algebraically special 4D space-times. It is demonstrated that the principal Killing triad can be efficiently used for classification and study of stationary 2 + 1 space-times. One of the most interesting applications is a study of minimal surfaces in a stationary space-time. A principal Killing surface is defined as a surface formed by Killing trajectories passing through a null ray, which is tangent to one of the null vectors of the principal Killing triad. We prove that a principal Killing surface is minimal if and only if the corresponding null vector is geodesic. Furthermore, we prove that if the (2 + 1)-dimensional space-time contains a static limit, then the only regular stationary timelike minimal 2-surfaces that cross the static limit, are the minimal principal Killing surfaces. A timelike minimal surface is a solution to the Nambu-Goto equations of motion and hence it describes a cosmic string configuration. A stationary string interacting with a (2 + 1)-dimensional rotating black hole is discussed in detail.

Stationary strings and 2-D black holes

A general description of string excitations in stationary spacetimes is developed. If a stationary string passes through the ergosphere of a 4-dimensional black hole, its world-sheet describes a 2-dimensional black (or white) hole with horizon coinciding with the static limit of the 4-dimensional black hole. Mathematical results for 2-dimensional black holes can therefore be applied to physical objects, say cosmic strings in the vicinity of Kerr black holes. An immediate general result is that the string modes are thermally excited. The string excitations are determined by a coupled system of scalar field equations in the world-sheet metric. In the special case of excitations propagating along a stationary string in the equatorial plane of the Kerr-Newman black hole, they reduce to the s-wave scalar field equations in the 4-dimensional Reissner-Nordström black hole. We briefly discuss possible applications of our results to the black hole information puzzle.

Propagation of perturbations along strings

A covariant formalism for physical perturbations propagating along a string in an arbitrary curved space-time is developed. In the case of a stationary string in a static background the propagation of the perturbations is described by a wave equation with a potential consisting of two terms: The first term describing the time dilation and the second is connected with the curvature of space. As applications of the developed approach the propagation of perturbations along a stationary string in Rindler, de Sitter, Schwarzschild and Reissner-Nordström space -times are investigated.