Anti-de Sitter Metrics Research Articles

Casimir forces on deformed fermionic chains

We characterize the Casimir forces for the Dirac vacuum on free-fermionic chains with smoothly varying hopping amplitudes, which correspond to (1+1)D curved spacetimes with a static metric in the continuum limit. The first-order energy potential for an obstacle on that lattice corresponds to the Newtonian potential associated to the metric, while the finite-size corrections are described by a curved extension of the conformal field theory predictions, including a suitable boundary term. We show that, for weak deformations of the Minkowski metric, Casimir forces measured by a local observer at the boundary are metric-independent. We provide numerical evidence for our results on a variety of (1+1)D deformations: Minkowski, Rindler, anti-de Sitter (the so-called rainbow system) and sinusoidal metrics.

Symplectic Wick rotations between moduli spaces of 3-manifolds

Given a closed hyperbolic surface $S$, let $\cQF$ denote the space of quasifuchsian hyperbolic metrics on $S\times\R$ and $\cGH_{-1}$ the space of maximal globally hyperbolic anti-de Sitter metrics on $S\times\R$. We describe natural maps between (parts of) $\cQF$ and $\cGH_{-1}$, called "Wick rotations", defined in terms of special surfaces (e.g. minimal/maximal surfaces, CMC surfaces, pleated surfaces) and prove that these maps are at least $C^1$ smooth and symplectic with respect to the canonical symplectic structures on both $\cQF$ and $\cGH_{-1}$. Similar results involving the spaces of globally hyperbolic de Sitter and Minkowski metrics are also described. These 3-dimensional results are shown to be equivalent to purely 2-dimensional ones. Namely, consider the double harmonic map $\cH:T^*\cT\to\cTT$, sending a conformal structure $c$ and a holomorphic quadratic differential $q$ on $S$ to the pair of hyperbolic metrics $(m_L,m_R)$ such that the harmonic maps isotopic to the identity from $(S,c)$ to $(S,m_L)$ and to $(S,m_R)$ have, respectively, Hopf differentials equal to $i q$ and $-i q$, and the double earthquake map $\cE:\cT\times\cML\to\cTT$, sending a hyperbolic metric $m$ and a measured lamination $l$ on $S$ to the pair $(E_L(m,l), E_R(m,l))$, where $E_L$ and $E_R$ denote the left and right earthquakes. We describe how such 2-dimensional double maps are related to 3-dimensional Wick rotations and prove that they are also $C^1$ smooth and symplectic.

PARABOLIC CLASSICAL CURVATURE FLOWS

We consider classical curvature flows: 1-parameter families of convex embeddings of the 2-sphere into Euclidean 3-space, which evolve by an arbitrary (nonhomogeneous) function of the radii of curvature (RoC). We determine conditions for parabolic flows that ensure the boundedness of various geometric quantities and investigate some examples. As a new tool, we introduce the RoC diagram of a surface and its hyperbolic or anti-de Sitter metric. The relationship between the RoC diagram and the properties of Weingarten surfaces is also discussed.

Astronomical time-of-flight photon speedometer.

A dual-band, fiber-optic, photon time-of-flight instrument was developed. Its design was optimized for measuring the velocity of visible photons emanating from relatively dim astronomical sources (apparent magnitude m>12), such as distant galaxies and quasars. We report the first direct photon group velocity measurements for extragalactic objects. The photon group velocity is found to be 3.00±0.03×108 ms-1 and is invariant, within experimental error, over the range of redshifts measured (0≤z≤1.33). This measurement provides additional validation of general relativity and is consistent with the Friedmann-Lemaître-Robertson-Walker and hyperbolic anti-de Sitter metrics but not with the elliptical de Sitter metric.

Yet another family of diagonal metrics for de Sitter and anti–de Sitter spacetimes

In this work we present and analyze a new class of coordinate representations of de Sitter and anti-de Sitter spacetimes for which the metrics are diagonal and (typically) static and axially symmetric. Contrary to the well-known forms of these fundamental geometries, that usually correspond to a 1+3 foliation with the 3-space of a constant spatial curvature, the new metrics are adapted to a 2+2 foliation, and are warped products of two 2-spaces of constant curvature. This new class of (anti-)de Sitter metrics depends on the value of cosmological constant Lambda and two discrete parameters +1,0,-1 related to the curvature of the 2-spaces. The class admits 3 distinct subcases for Lambda>0 and 8 subcases for Lambda<0. We systematically study all these possibilities. In particular, we explicitly present the corresponding parametrizations of the (anti-)de Sitter hyperboloid, visualize the coordinate lines and surfaces within the global conformal cylinder, investigate their mutual relations, present some closely related forms of the metrics, and give transformations to standard de Sitter and anti-de Sitter metrics. Using these results, we also provide a physical interpretation of B-metrics as exact gravitational fields of a tachyon.

On Perturbations of the Schwarzschild Anti-De Sitter Spaces of Positive Mass

In this paper we prove the Penrose inequality for metrics that are small perturbations of the Schwarzschild anti-de Sitter metrics of positive mass. We use the existence of a global foliation by weakly stable constant mean curvature spheres and the monotonicity of the Hawking mass.

Information metric from a linear sigma model

The idea that a space-time metric emerges as a Fisher-Rao "information metric" of instanton moduli space has been examined in several field theories, such as the Yang-Mills theories and nonlinear σ models. In this paper, we report that the flat Euclidean or Minkowskian metric, rather than an anti-de Sitter metric that generically emerges from instanton moduli spaces, can be obtained as the Fisher-Rao metric from a nontrivial solution of the massive Klein-Gordon field (a linear σ model). This realization of the flat space from the simple field theory would be useful to investigate the ideas that relate the space-time geometry with the information geometry.

Exact solutions of [formula omitted]-dimensional Yang–Mills equations in curved space–time

In the context of a semiclassical approach where vectorial gauge fields can be considered as classical fields, we obtain exact static solutions of the SU(N) Yang–Mills equations in an (n+1)-dimensional curved space–time, for the cases n=1,2,3. As an application of the results obtained for the case n=3, we consider the solutions for the anti-de Sitter and Schwarzschild metrics. We show that these solutions have a confining behavior and can be considered as a first step in the study of the corrections of the spectra of quarkonia in a curved background. Since the solutions that we find in this work are valid also for the group U(1), the case n=2 is a description of the (2+1) electrodynamics in the presence of a point charge. For this case, the solution has a confining behavior and can be considered as an application of the planar electrodynamics in a curved space–time. Finally we find that the solution for the case n=1 is invariant under a parity transformation and has the form of a linear confining solution.

Dynamical holographic QCD with area-law confinement and linear Regge trajectories

We construct a new solution of five-dimensional gravity coupled to a dilaton which encodes essential features of holographic QCD backgrounds dynamically. In particular, it implements linear confinement, i.e., the area-law behavior of the Wilson loop, by means of a dynamically deformed anti-de Sitter metric. The predicted square masses of the light-flavored natural-parity mesons and their excitations lie on linear trajectories of an approximately universal slope with respect to both radial and spin quantum numbers and are in satisfactory agreement with experimental data.

Non-monotonic orbital velocity profiles around rapidly rotating Kerr–(anti-)de Sitter black holes

It has been recently demonstrated that the orbital velocity profile around Kerr black holes in the equatorial plane as observed in the locally non-rotating frame exhibits a non-monotonic radial behaviour. We show here that this unexpected minimum–maximum feature of the orbital velocity remains if the Kerr vacuum is generalized to the Kerr–de Sitter or Kerr–anti-de Sitter metric. This is a new general relativity effect in Kerr spacetimes with a non-vanishing cosmological constant. Assuming that the profile of the orbital velocity is known, this effect constrains the spacetime parameters.

Classical boundary-value problem in Riemannian quantum gravity and Taub–Bolt–anti-de Sitter geometries

For an SU(2)× U(1)-invariant S 3 boundary the classical Dirichlet problem of Riemannian quantum gravity is studied for positive-definite regular solutions of the Einstein equations with a negative cosmological constant within biaxial Bianchi-IX metrics containing bolts, i.e., within the family of Taub–Bolt–anti-de Sitter (Taub–Bolt–AdS) metrics. Such metrics are obtained from the two-parameter Taub–NUT–anti-de Sitter family. The condition of regularity requires them to have only one free parameter ( L) and constrains L to take values within a narrow range; the other parameter is determined as a double-valued function of L and hence there is a bifurcation within the family. We found that any axially symmetric S 3-boundary can be filled in with at least one solution coming from each of these two branches despite the severe limit on the permissible values of L. The number of infilling solutions can be one, three or five and they appear or disappear catastrophically in pairs as the values of the two radii of S 3 are varied. The solutions occur simultaneously in both branches and hence the total number of independent infillings is two, six or ten. We further showed that when the two radii are of the same order and large the number of solutions is two. In the isotropic limit this holds for small radii as well. These results are to be contrasted with the one-parameter self-dual Taub–NUT–AdS infilling solutions of the same boundary-value problem studied previously.

Using algebraic programming to teach general relativity

The authors' department considers general relativity (GR) teaching a necessity for midlevel graduate physics students, so we introduce a GR course at the third-year undergraduate level. During the last two years, we have used several packages in the course (Reduce plus Excalc for algebraic programming, Mathematica and Maple for graphic visualization). Even when the students were real beginners in computer manipulation, we obtained visibly good results. Algebraic programming systems (such as Reduce) that contain differential geometry packages can become an important tool for learning. Using a computer, students can quickly and comfortably learn the important notions of differential geometry, tensor calculus, and exterior calculus (for example, using Excalc). We also introduce some aspects of GR using computer algebra systems. For example, we can easily obtain the Schwarzschild solution and the Reissner-Nordstrom solution as exact solutions of the Einstein equations. We also use computer algebra to find and treat other exact homogeneous solutions of the Einstein equations containing the cosmological constant (de Sitter and anti-de Sitter metrics). Using some simple Excalc procedures, the article illustrates how to teach Riemannian geometry as well as how to use computer algebra to obtain some exact solutions of Einstein equations.

Geometric models of (d+1)-dimensional relativistic rotating oscillators

Geometric models of quantum relativistic rotating oscillators in arbitrary dimensions are defined on backgrounds with deformed anti-de Sitter metrics. It is shown that these models are analytically solvable, deriving the formulas of the energy levels and corresponding normalized energy eigenfunctions. An important property is that all these models have the same nonrelativistic limit, namely the usual harmonic oscillator.

Gödel metric as a squashed anti-de Sitter geometry

We show that the non-flat factor of the Gödel metric belongs to a one-parameter family of (2 + 1)-dimensional geometries that also includes the anti-de Sitter metric. The elements of this family allow a generalization à la Kaluza-Klein of the usual (3 + 1)-dimensional Gödel metric. Their lightcones can be viewed as deformations of the anti-de Sitter ones, involving tilting and squashing. This provides a simple geometric picture of the causal structure of these spacetimes, anti-de Sitter geometry appearing as the boundary between causally safe and causally pathological spaces. Furthermore, we construct a global algebraic isometric embedding of these metrics in (4 + 3)- or (3 + 4)-dimensional flat spaces, thereby illustrating in another way the occurrence of the closed timelike curves.

Formula omitted] M-branes and AdSp+2 geometries

We discuss the class of BPS saturated M-branes that are in one-to-one correspondence with the Freund-Rubin compactifications of M-theory on either AdS 4 × G/ H or AdS 7 × G/ H, where G/ H is any of the seven (or four) dimensional Einstein coset manifolds with Killing spinors classified long ago in the context of Kaluza-Klein supergravity. These G/ H M-branes, whose existence was previously pointed out in the literature, are solitons that interpolate between flat space at infinity and the old Kaluza-Klein compactifications at the horizon. They preserve N/2 supersymmetries where N is the number of Killing spinors of the AdS × G/ H vacuum. A crucial ingredient in our discussion is the identification of a solvable Lie algebra parametrization of the Lorentzian noncompact coset SO(2, p + 1)/ SO(1, p + 1) corresponding to anti-de Sitter space AdS p + 2. The solvable coordinates are those naturally emerging from the near horizon limit of the G/ H p-brane and correspond to the Bertotti-Robinson form of the anti-de Sitter metric. The pull-back of anti-de Sitter isometries on the p-brane world-volume contain, in particular, the recently found broken conformal transformations

The Energy Spectra of the Relativistic Rotating Oscillators

We study, by using numerical computation methods, the structure of the energy spectra of the quantum relativistic rotating oscillators we have recently proposed, starting with a set of one-parameter deformations of the static (3+1) de Sitter or anti-de Sitter metrics.1

Geometric Models of the Quantum Relativistic Rotating Oscillator

A family of geometric models of quantum relativistic rotating oscillator is defined by using a set of one-parameter deformations of the static (3+1) de Sitter or anti-de Sitter metrics. It is shown that all these models lead to the usual isotropic harmonic oscillator in the nonrelativistic limit, even though their relativistic behavior is different. As in the case of the (1+1) models,1 these will have even countable energy spectra or mixed ones, with a finite discrete sequence and a continuous part. In addition, all these spectra, except that of the pure anti-de Sitter model, will have a fine-structure, given by a rotator-like term.

Symmetries of static, spherically symmetric space-times

In this paper it is shown that reduction from maximal to minimal static, spherical symmetry of a space-time occurs in only one step reducing the number of independent Killing vector fields from 10 to 4. Maximal symmetry corresponds only to the de Sitter, anti-de Sitter, and Minkowski metrics, without reference to the Einstein field equations.

Stability of gravity with a cosmological constant

The stability properties of Einstein theory with a cosmological constant Λ are investigated. For Λ > 0, stability is established for small fluctuations, about the de Sitter background, occurring inside the event horizon and semiclassical stability is analyzed. For Λ < 0, stability is demonstrated for all asymptotically anti-de Sitter metrics. The analysis is based on the general construction of conserved flux-integral expressions associated with the symmetries of a chosen background. The effects of an event horizon, which lead to Hawking radiation, are expressedfor general field hamiltonians. Stability for Λ < 0 is proved, using supergravity techniques, in terms of the graded anti-de Sitter algebra with spinorial charges also expressed as flux integrals.