223 research outputs found
Classification of unit-vector fields in convex polyhedra with tangent boundary conditions
A unit-vector field n on a convex three-dimensional polyhedron P is tangent
if, on the faces of P, n is tangent to the faces. A homotopy classification of
tangent unit-vector fields continuous away from the vertices of P is given. The
classification is determined by certain invariants, namely edge orientations
(values of n on the edges of P), kink numbers (relative winding numbers of n
between edges on the faces of P), and wrapping numbers (relative degrees of n
on surfaces separating the vertices of P), which are subject to certain sum
rules. Another invariant, the trapped area, is expressed in terms of these. One
motivation for this study comes from liquid crystal physics; tangent
unit-vector fields describe the orientation of liquid crystals in certain
polyhedral cells.Comment: 21 pages, 2 figure
\bbbc P^2 and \bbbc P^{1} Sigma Models in Supergravity: Bianchi type IX Instantons and Cosmologies
We find instanton/cosmological solutions with biaxial Bianchi-IX symmetry,
involving non-trivial spatial dependence of the \bbbc P^{1}- and \bbbc
P^{2}-sigma-models coupled to gravity. Such manifolds arise in N=1,
supergravity with supermatter actions and hence the solutions can be embedded
in supergravity. There is a natural way in which the standard coordinates of
these manifolds can be mapped into the four-dimensional physical space. Due to
its special symmetry, we start with \bbbc P^{2} with its corresponding scalar
Ansatz; this further requires the spacetime to be -invariant. The problem then reduces to a set of ordinary differential
equations whose analytical properties and solutions are discussed. Among the
solutions there is a surprising, special-family of exact solutions which owe
their existence to the non-trivial topology of \bbbc P^{2} and are in 1-1
correspondence with matter-free Bianchi-IX metrics. These solutions can also be
found by coupling \bbbc P^{1} to gravity. The regularity of these Euclidean
solutions is discussed -- the only possibility is bolt-type regularity. The
Lorentzian solutions with similar scalar Ansatz are all obtainable from the
Euclidean solutions by Wick rotation
THE UNIQUENESS THEOREM FOR ROTATING BLACK HOLE SOLUTIONS OF SELF-GRAVITATING HARMONIC MAPPINGS
We consider rotating black hole configurations of self-gravitating maps from
spacetime into arbitrary Riemannian manifolds. We first establish the
integrability conditions for the Killing fields generating the stationary and
the axisymmetric isometry (circularity theorem). Restricting ourselves to
mappings with harmonic action, we subsequently prove that the only stationary
and axisymmetric, asymptotically flat black hole solution with regular event
horizon is the Kerr metric. Together with the uniqueness result for
non-rotating configurations and the strong rigidity theorem, this establishes
the uniqueness of the Kerr family amongst all stationary black hole solutions
of self-gravitating harmonic mappings.Comment: 18 pages, latex, no figure
Type II Critical Collapse of a Self-Gravitating Nonlinear -Model
We report on the existence and phenomenology of type II critical collapse
within the one-parameter family of SU(2) -models coupled to gravity.
Numerical investigations in spherical symmetry show discretely self-similar
(DSS) behavior at the threshold of black hole formation for values of the
dimensionless coupling constant \ccbeta ranging from 0.2 to 100; at 0.18 we
see small deviations from DSS. While the echoing period of the
critical solution rises sharply towards the lower limit of this range, the
characteristic mass scaling has a critical exponent which is almost
independent of \ccbeta, asymptoting to at large
\ccbeta. We also find critical scaling of the scalar curvature for
near-critical initial data. Our numerical results are based on an
outgoing-null-cone formulation of the Einstein-matter equations, specialized to
spherical symmetry. Our numerically computed initial-data critical parameters
show 2nd order convergence with the grid resolution, and after
compensating for this variation in , our individual evolutions are
uniformly 2nd order convergent even very close to criticality.Comment: 23 pages, includes 10 postscript figure files, uses REVTeX, epsf,
psfrag, and AMS math fonts (amstex + amssymb); to appear in PRD15. Summary of
revisions from v2: fix wrong formula in figure 6 caption and y-axis label,
also minor wording changes and update publication status of refs 5-
Controllability on infinite-dimensional manifolds
Following the unified approach of A. Kriegl and P.W. Michor (1997) for a
treatment of global analysis on a class of locally convex spaces known as
convenient, we give a generalization of Rashevsky-Chow's theorem for control
systems in regular connected manifolds modelled on convenient
(infinite-dimensional) locally convex spaces which are not necessarily
normable.Comment: 19 pages, 1 figur
Perturbing gauge/gravity duals by a Romans mass
We show how to produce algorithmically gravity solutions in massive IIA (as
infinitesimal first order perturbations in the Romans mass parameter) dual to
assigned conformal field theories. We illustrate the procedure on a family of
Chern--Simons--matter conformal field theories that we recently obtained from
the N=6 theory by waiving the condition that the levels sum up to zero.Comment: 30 page
Calibrated Sub-Bundles in Non-Compact Manifolds of Special Holonomy
This paper is a continuation of math.DG/0408005. We first construct special
Lagrangian submanifolds of the Ricci-flat Stenzel metric (of holonomy SU(n)) on
the cotangent bundle of S^n by looking at the conormal bundle of appropriate
submanifolds of S^n. We find that the condition for the conormal bundle to be
special Lagrangian is the same as that discovered by Harvey-Lawson for
submanifolds in R^n in their pioneering paper. We also construct calibrated
submanifolds in complete metrics with special holonomy G_2 and Spin(7)
discovered by Bryant and Salamon on the total spaces of appropriate bundles
over self-dual Einstein four manifolds. The submanifolds are constructed as
certain subbundles over immersed surfaces. We show that this construction
requires the surface to be minimal in the associative and Cayley cases, and to
be (properly oriented) real isotropic in the coassociative case. We also make
some remarks about using these constructions as a possible local model for the
intersection of compact calibrated submanifolds in a compact manifold with
special holonomy.Comment: 20 pages; for Revised Version: Minor cosmetic changes, some
paragraphs rewritten for improved clarit
The classification of static vacuum space-times containing an asymptotically flat spacelike hypersurface with compact interior
We prove non-existence of static, vacuum, appropriately regular,
asymptotically flat black hole space-times with degenerate (not necessarily
connected) components of the event horizon. This finishes the classification of
static, vacuum, asymptotically flat domains of outer communication in an
appropriate class of space-times, showing that the domains of outer
communication of the Schwarzschild black holes exhaust the space of
appropriately regular black hole exteriors.Comment: This version includes an addendum with a corrected proof of
non-existence of zeros of the Killing vector at degenerate horizons. A
problem with yet another Lemma is pointed out; this problem does not arise if
one assumes analyticity of the metric. An alternative solution, that does not
require analyticity, has been given in arXiv:1004.0513 [gr-qc] under
appropriate global condition
Numerical Investigation of Cosmological Singularities
Although cosmological solutions to Einstein's equations are known to be
generically singular, little is known about the nature of singularities in
typical spacetimes. It is shown here how the operator splitting used in a
particular symplectic numerical integration scheme fits naturally into the
Einstein equations for a large class of cosmological models and thus allows
study of their approach to the singularity. The numerical method also naturally
singles out the asymptotically velocity term dominated (AVTD) behavior known to
be characteristic of some of these models, conjectured to describe others, and
probably characteristic of a subclass of the rest. The method is first applied
to the unpolarized Gowdy T cosmology. Exact pseudo-unpolarized solutions
are used as a code test and demonstrate that a 4th order accurate
implementation of the numerical method yields acceptable agreement. For generic
initial data, support for the conjecture that the singularity is AVTD with
geodesic velocity (in the harmonic map target space) < 1 is found. A new
phenomenon of the development of small scale spatial structure is also
observed. Finally, it is shown that the numerical method straightforwardly
generalizes to an arbitrary cosmological spacetime on with one
spacelike U(1) symmetry.Comment: 37 pp +14 figures (not included, available on request), plain Te
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