Embedding Coordinates Research Articles

CANONICAL ANALYSIS OF NONCOMMUTATIVITY OF OPEN BOSONIC STRINGS

Generally, in the presence of an antisymmetric tensor background field, the open string moving with mixed Neumann and Dirichlet boundary conditions will make the spacetime coordinates noncommutative. In this paper, comparing to the flat and static D-brane case studied in the literature, we first generalize the D-branes to be dynamical. And then by using the Hamiltonian approach, through a calculation of the classical Poisson brackets among the Fourier mode components of the string embedding coordinates and a consistent quantization procedure, we show that the spacetime coordinates of the string endpoints in the directions parallel and perpendicular to the D-branes both become noncommutative.

Does a black hole rotate in Chern-Simons modified gravity?

Rotating black hole solutions in the $(3+1)$-dimensional Chern-Simons modified gravity theory are discussed by taking account of perturbation around the Schwarzschild solution. The zenith-angle dependence of a metric function related to the frame-dragging effect is determined from a constraint equation independently of a choice of the embedding coordinate. We find that at least within the framework of the first-order perturbation method, the black hole cannot rotate for finite black hole mass if the embedding coordinate is taken to be a timelike vector. However, the rotation can be permitted in the limit of $M/r\ensuremath{\rightarrow}0$ (where $M$ is the black hole mass and $r$ is the radius). For a spacelike vector, the rotation can also be permitted for any value of the black hole mass.

Noncommutative BTZ black hole and discrete time

We search for all Poisson brackets for the BTZ black hole which are consistent with the geometry of the commutative solution and are of lowest order in the embedding coordinates. For arbitrary values of the angular momentum we obtain two two-parameter families of contact structures. We obtain the symplectic leaves, which characterize the irreducible representations of the corresponding noncommutative theory. The requirement that they be invariant under the action of the isometry group restricts to symplectic leaves which are topologically , where is associated with the Schwarzschild time. Quantization may then lead to a discrete spectrum for the time operator.

The Universe Accelerated Expansion using Extra-dimensions with Metric Components Found by a New Equivalence Principle

Curved multi-dimensional space-times (5D and higher) are constructed by embedding them in one higher-dimensional flat space. The condition that the embedding coordinates have a separable form, plus the demand of an orthogonal resulting space-time, implies that the curved multi-dimensional space-time has 4D de-Sitter subspaces (for constant extra-dimensions) in which the 3D subspace has an accelerated expansion. A complete determination of the curved multi-dimensional spacetime geometry is obtained provided we impose a new type of “equivalence principle”, meaning that there is a geodesic which from the embedding space has a rectliniar motion. According to this new equivalence principle, we can find the extra-dimensions metric components, each curved multi-dimensional spacetime surface’s equation, the energy-momentum tensors and the extra-dimensions as functions of a scalar field. The generic geodesic in each 5D spacetime are studied: they include solutions where particle’s motion along the extra-dimension is periodic and the 3D expansion factor is inflationary (accelerated expansion). Thus, the 3D subspace has an accelerated expansion.

A Riemannian approach to graph embedding

In this paper, we make use of the relationship between the Laplace–Beltrami operator and the graph Laplacian, for the purposes of embedding a graph onto a Riemannian manifold. To embark on this study, we review some of the basics of Riemannian geometry and explain the relationship between the Laplace–Beltrami operator and the graph Laplacian. Using the properties of Jacobi fields, we show how to compute an edge-weight matrix in which the elements reflect the sectional curvatures associated with the geodesic paths on the manifold between nodes. For the particular case of a constant sectional curvature surface, we use the Kruskal coordinates to compute edge weights that are proportional to the geodesic distance between points. We use the resulting edge-weight matrix to embed the nodes of the graph onto a Riemannian manifold. To do this, we develop a method that can be used to perform double centring on the Laplacian matrix computed from the edge-weights. The embedding coordinates are given by the eigenvectors of the centred Laplacian. With the set of embedding coordinates at hand, a number of graph manipulation tasks can be performed. In this paper, we are primarily interested in graph-matching. We recast the graph-matching problem as that of aligning pairs of manifolds subject to a geometric transformation. We show that this transformation is Pro-crustean in nature. We illustrate the utility of the method on image matching using the COIL database.

Formulating LLE using alignment technique

LLE is a well-known method to nonlinear dimensionality reduction. In this short paper, we present an alternative way to formulate LLE. The alignment technique is exploited to align the local coordinates on the local patches of manifolds to be the global ones. The efficient computation of embedding coordinates of LLE automatically appears in the proposed framework.

Aspects of Dirichlet S-branes

In this Letter, we discuss some features of the Dirichlet S-brane, defined as a Dirichlet boundary condition on a time-like embedding coordinate of open strings. We analyze the Euclidean theory on the S-brane world-volume, and trace its instability to the infinite fine-tuning of the initial conditions required to produce an infinitely extended space-like defect. Using their equivalence under T-duality with D-branes with supercritical electric field, we argue that under generic perturbation, S-branes turn into D-brane/anti-D-branes. We extract the imaginary part of the cylinder amplitude, and interpret its inverse as a “decay length”, beyond which a pair of S-branes annihilates. Finally, we reconsider the boundary state of the Dirichlet S-brane and find that it is either a solution of type II string theory with imaginary R–R fields, or a solution of type II∗ with real fields. This leaves the non-BPS S-branes as potentially physical solutions of type II string theory.

Observability of multivariate differential embeddings

The present paper extends some results recently developed for the analysis of observability in nonlinear dynamical systems. The aim of the paper is to address the problem of embedding an attractor using more than one observable. A multivariate nonlinear observability matrix is proposed which includes the monovariable nonlinear and linear observability matrices as particular cases. Using the developed framework and a number of worked examples, it is shown that the choice of embedding coordinates is critical. Moreover, in some cases, to reconstruct the dynamics using more than one observable could be worse than to reconstruct using a scalar measurement. Finally, using the developed framework it is shown that increasing the embedding dimension, observability problems diminish and can even be eliminated. This seems to be a physically meaningful interpretation of the Takens embedding theorem.

Chern-Simons modification of general relativity

General relativity is extended by promoting the three-dimensional gravitational Chern-Simons term to four dimensions. This entails choosing an embedding coordinate ${v}_{\ensuremath{\mu}}$---an external quantity, which we fix to be a nonvanishing constant in its time component. The theory is identical to one in which the embedding coordinate is itself a dynamical variable, rather than a fixed, external quantity. Consequently diffeomorphism symmetry breaking is hidden in the modified theory: the Schwarzschild metric is a solution; gravitational waves possess two polarizations, each traveling at the velocity of light; a conserved energy-momentum (pseudo)tensor can be constructed. The modification is visible in the intensity of gravitational radiation: the two polarizations of a gravity wave carry intensities that are suppressed or enhanced by the extension.

ON SCHWARZSCHILD'S TOPOLOGY IN BRANE-WORLDS

The topological structure of Schwazschild's space-time and its maximal analytic extension are investigated in the context of brane worlds. Using the embedding coordinates, these geometries are seen as different states of the evolution of a single brane-world. Comparing the topologies and the embeddings it is shown that this evolution must be followed by a signature change in the bulk.

Extracting messages masked by chaotic signals of time-delay systems.

We show how to extract messages masked by a chaotic signal of a time-delay system with very high dimensions and many positive Lyapunov exponents. Using a special embedding coordinate, the infinite-dimensional phase space of the time-delay system is projected onto a special three-dimensional space, which enables us to identify the time delay of the system from the transmitted signal and reconstruct the chaotic dynamics to unmask the hidden message successfully. The message extraction procedure is illustrated by simulations with the Mackey-Glass time-delay system for two types of masking schemes and different kinds of messages.

The Kaluza-Klein monopole in a massive IIA background

We construct the effective action of the KK monopole in a massive Type IIA background. We follow two approaches. First we construct a massive M-theory KK monopole from which the IIA monopole is obtained by double dimensional reduction. This eleven-dimensional monopole contains two isometries: one under translations of the Taub-NUT coordinate and the other under massive transformations of the embedding coordinates. Secondly, we construct the massive T-duality rules that map the Type IIB NS5-brane onto the massive Type IIA KK monopole. This provides a check of the action constructed from eleven dimensions.

Summing over world-sheet boundaries

The moduli associated with boundaries in a Riemann surface are parametrized by the positions and strengths of electric charges. This suggests a method for summing over orientable Riemann surfaces with Dirichlet boundary conditions on the embedding coordinates. A light-cone parameterization of such boundaries is also discussed.

Remarks on the quantum mechanics of the relativistic string

The quantum mechanics of the free relativistic string is studied by hamiltonian methods. It is shown that if the string is coupled to the local scale of its world surface the scale field plays the role of an extra embedding coordinate. That extra coordinate contributes by a continuously adjustable amount to the central charge in the gauge algebra, which permits to go below the critical spacetime dimension D=26. A different model is exhibited in which each of the original embedding coordinates is endowed with an adjustable central charge. In that model one can go below D=26 without bringing in the local scale field. The extension to the supersymmetric case is also indicated.