A Note on Real Tunneling Geometries

In the Hartle-Hawking ``no boundary'' approach to quantum cosmology, a real tunneling geometry is a configuration that represents a transition from a compact Riemannian spacetime to a Lorentzian universe. I complete an earlier proof that in three spacetime dimensions, such a transition is ``probable,'' in the sense that the required Riemannian geometry yields a genuine maximum of the semiclassical wave function.Comment: 5 page

Cosmology, cohomology, and compactification

Ashtekar and Samuel have shown that Bianchi cosmological models with compact spatial sections must be of Bianchi class A. Motivated by general results on the symmetry reduction of variational principles, we show how to extend the Ashtekar-Samuel results to the setting of weakly locally homogeneous spaces as defined, e.g., by Singer and Thurston. In particular, it is shown that any m-dimensional homogeneous space G/K admitting a G-invariant volume form will allow a compact discrete quotient only if the Lie algebra cohomology of G relative to K is non-vanishing at degree m.Comment: 6 pages, LaTe

Measuring Topological Chaos

The orbits of fluid particles in two dimensions effectively act as topological obstacles to material lines. A spacetime plot of the orbits of such particles can be regarded as a braid whose properties reflect the underlying dynamics. For a chaotic flow, the braid generated by the motion of three or more fluid particles is computed. A ``braiding exponent'' is then defined to characterize the complexity of the braid. This exponent is proportional to the usual Lyapunov exponent of the flow, associated with separation of nearby trajectories. Measuring chaos in this manner has several advantages, especially from the experimental viewpoint, since neither nearby trajectories nor derivatives of the velocity field are needed.Comment: 4 pages, 6 figures. RevTeX 4 with PSFrag macro

Holography for the Lorentz Group Racah Coefficients

A known realization of the Lorentz group Racah coefficients is given by an integral of a product of 6 ``propagators'' over 4 copies of the hyperbolic space. These are ``bulk-to-bulk'' propagators in that they are functions of two points in the hyperbolic space. It is known that the bulk-to-bulk propagator can be constructed out of two bulk-to-boundary ones. We point out that there is another way to obtain the same object. Namely, one can use two bulk-to-boundary and one boundary-to-boundary propagator. Starting from this construction and carrying out the bulk integrals we obtain a realization of the Racah coefficients that is ``holographic'' in the sense that it only involves boundary objects. This holographic realization admits a geometric interpretation in terms of an ``extended'' tetrahedron.Comment: 12 pages, 2 figures; v2: minor changes; v3: "extended" tetrahedron interpretation adde

Quantum creation of an Inhomogeneous universe

In this paper we study a class of inhomogeneous cosmological models which is a modified version of what is usually called the Lema\^itre-Tolman model. We assume that we have a space with 2-dimensional locally homogeneous spacelike surfaces. In addition we assume they are compact. Classically we investigate both homogeneous and inhomogeneous spacetimes which this model describe. For instance one is a quotient of the AdS$_4$ space which resembles the BTZ black hole in AdS$_3$. Due to the complexity of the model we indicate a simpler model which can be quantized easily. This model still has the feature that it is in general inhomogeneous. How this model could describe a spontaneous creation of a universe through a tunneling event is emphasized.Comment: 21 pages, 5 ps figures, REVTeX, new subsection include

Light scattering and phase behavior of Lysozyme-PEG mixtures

Measurements of liquid-liquid phase transition temperatures (cloud points) of mixtures of a protein (lysozyme) and a polymer, poly(ethylene glycol) (PEG) show that the addition of low molecular weight PEG stabilizes the mixture whereas high molecular weight PEG was destabilizing. We demonstrate that this behavior is inconsistent with an entropic depletion interaction between lysozyme and PEG and suggest that an energetic attraction between lysozyme and PEG is responsible. In order to independently characterize the lysozyme/PEG interactions, light scattering experiments on the same mixtures were performed to measure second and third virial coefficients. These measurements indicate that PEG induces repulsion between lysozyme molecules, contrary to the depletion prediction. Furthermore, it is shown that third virial terms must be included in the mixture's free energy in order to qualitatively capture our cloud point and light scattering data. The light scattering results were consistent with the cloud point measurements and indicate that attractions do exist between lysozyme and PEG.Comment: 5 pages, 2 figures, 1 tabl

Invariant Peano curves of expanding Thurston maps

We consider Thurston maps, i.e., branched covering maps $f\colon S^2\to S^2$ that are postcritically finite. In addition, we assume that $f$ is expanding in a suitable sense. It is shown that each sufficiently high iterate $F=f^n$ of $f$ is semi-conjugate to $z^d\colon S^1\to S^1$, where $d$ is equal to the degree of $F$. More precisely, for such an $F$ we construct a Peano curve $\gamma\colon S^1\to S^2$ (onto), such that $F\circ \gamma(z) = \gamma(z^d)$ (for all $z\in S^1$).Comment: 63 pages, 12 figure

Poincare ball embeddings of the optical geometry

It is shown that optical geometry of the Reissner-Nordstrom exterior metric can be embedded in a hyperbolic space all the way down to its outer horizon. The adopted embedding procedure removes a breakdown of flat-space embeddings which occurs outside the horizon, at and below the Buchdahl-Bondi limit (R/M=9/4 in the Schwarzschild case). In particular, the horizon can be captured in the optical geometry embedding diagram. Moreover, by using the compact Poincare ball representation of the hyperbolic space, the embedding diagram can cover the whole extent of radius from spatial infinity down to the horizon. Attention is drawn to advantages of such embeddings in an appropriately curved space: this approach gives compact embeddings and it distinguishes clearly the case of an extremal black hole from a non-extremal one in terms of topology of the embedded horizon.Comment: 16 pages, 8 figures; CQG accepte

Warped compactification on curved manifolds

The characterization of a six- (or seven)-dimensional internal manifold with metric as having positive, zero or negative curvature is expected to be an important aspect of warped compactifications in supergravity. In this context, Douglas and Kallosh recently pointed out that a compact internal space with negative curvature could help to construct four-dimensional de Sitter solutions only if the extra dimensions are strongly warped or there are large stringy corrections. That is, the problem of finding 4-dimensional de Sitter solutions is well posed, if all extra dimensions are physically compact, which is called a no-go theorem. Here, we show that the above conclusion does not extend to a general class of warped compactifications in classical supergravity that allow a non-compact direction or cosmological solutions for which the internal space is asymptotic to a cone over a product of compact Einstein spaces or spheres. For clarity, we present classical solutions that compactify higher-dimensional spacetime to produce a Robertson--Walker universe with de Sitter-type expansion plus one extra non-compact direction. Such models are found to admit both an effective four-dimensional Newton constant that remains finite and a normalizable zero-mode graviton wavefunction. We also exhibit the possibility of obtaining 4D de Sitter solutions by including the effect of fluxes (p-form field strengths).Comment: 24 pages, 1 figure; v5 significant changes in the presentation, published (journal) versio

Quantization and spacetime topology

We consider classical and quantum dynamics of a free particle in de Sitter's space-times with different topologies to see what happens to space-time singularities of removable type in quantum theory. We find analytic solution of the classical dynamics. The quantum dynamics is solved by finding an essentially self-adjoint representation of the algebra of observables integrable to the unitary representations of the symmetry group of each considered gravitational system. The dynamics of a massless particle is obtained in the zero-mass limit of the massive case. Our results indicate that taking account of global properties of space-time enables quantization of particle dynamics in all considered cases.Comment: Class. Quantum Grav. 20 (2003) 2491-2507; no figures, RevTeX