480 research outputs found
Sufficient Conditions for Apparent Horizons in Spherically Symmetric Initial Data
We establish sufficient conditions for the appearance of apparent horizons in
spherically symmetric initial data when spacetime is foliated extrinsically.
Let and be respectively the total material energy and the total
material current contained in some ball of radius . Suppose that the
dominant energy condition is satisfied. We show that if then
the region must possess a future apparent horizon for some non -trivial closed
subset of such gauges. The same inequality holds on a larger subset of gauges
but with a larger constant of proportionality which depends weakly on the
gauge. This work extends substantially both our joint work on moment of time
symmetry initial data as well as the work of Bizon, Malec and \'O Murchadha on
a maximal slice.Comment: 16 pages, revtex, to appear in Phys. Rev.
Geometric Bounds in Spherically Symmetric General Relativity
We exploit an arbitrary extrinsic time foliation of spacetime to solve the
constraints in spherically symmetric general relativity. Among such foliations
there is a one parameter family, linear and homogeneous in the extrinsic
curvature, which permit the momentum constraint to be solved exactly. This
family includes, as special cases, the extrinsic time gauges that have been
exploited in the past. These foliations have the property that the extrinsic
curvature is spacelike with respect to the the spherically symmetric superspace
metric. What is remarkable is that the linearity can be relaxed at no essential
extra cost which permits us to isolate a large non - pathological dense subset
of all extrinsic time foliations. We identify properties of solutions which are
independent of the particular foliation within this subset. When the geometry
is regular, we can place spatially invariant numerical bounds on the values of
both the spatial and the temporal gradients of the scalar areal radius, .
These bounds are entirely independent of the particular gauge and of the
magnitude of the sources. When singularities occur, we demonstrate that the
geometry behaves in a universal way in the neighborhood of the singularity.Comment: 16 pages, revtex, submitted to Phys. Rev.
Necessary Conditions for Apparent Horizons and Singularities in Spherically Symmetric Initial Data
We establish necessary conditions for the appearance of both apparent
horizons and singularities in the initial data of spherically symmetric general
relativity when spacetime is foliated extrinsically. When the dominant energy
condition is satisfied these conditions assume a particularly simple form. Let
be the maximum value of the energy density and the radial
measure of its support. If is bounded from above by some
numerical constant, the initial data cannot possess an apparent horizon. This
constant does not depend sensitively on the gauge. An analogous inequality is
obtained for singularities with some larger constant. The derivation exploits
Poincar\'e type inequalities to bound integrals over certain spatial scalars. A
novel approach to the construction of analogous necessary conditions for
general initial data is suggested.Comment: 15 pages, revtex, to appear in Phys. Rev.
The Constraints in Spherically Symmetric General Relativity II --- Identifying the Configuration Space: A Moment of Time Symmetry
We continue our investigation of the configuration space of general
relativity begun in I (gr-qc/9411009). Here we examine the Hamiltonian
constraint when the spatial geometry is momentarily static (MS). We show that
MS configurations satisfy both the positive quasi-local mass (QLM) theorem and
its converse. We derive an analytical expression for the spatial metric in the
neighborhood of a generic singularity. The corresponding curvature singularity
shows up in the traceless component of the Ricci tensor. We show that if the
energy density of matter is monotonically decreasing, the geometry cannot be
singular. A supermetric on the configuration space which distinguishes between
singular geometries and non-singular ones is constructed explicitly. Global
necessary and sufficient criteria for the formation of trapped surfaces and
singularities are framed in terms of inequalities which relate appropriate
measures of the material energy content on a given support to a measure of its
volume. The strength of these inequalities is gauged by exploiting the exactly
solvable piece-wise constant density star as a template.Comment: 50 pages, Plain Tex, 1 figure available from the authors
Hamilton's equations for a fluid membrane: axial symmetry
Consider a homogenous fluid membrane, or vesicle, described by the
Helfrich-Canham energy, quadratic in the mean curvature. When the membrane is
axially symmetric, this energy can be viewed as an `action' describing the
motion of a particle; the contours of equilibrium geometries are identified
with particle trajectories. A novel Hamiltonian formulation of the problem is
presented which exhibits the following two features: {\it (i)} the second
derivatives appearing in the action through the mean curvature are accommodated
in a natural phase space; {\it (ii)} the intrinsic freedom associated with the
choice of evolution parameter along the contour is preserved. As a result, the
phase space involves momenta conjugate not only to the particle position but
also to its velocity, and there are constraints on the phase space variables.
This formulation provides the groundwork for a field theoretical generalization
to arbitrary configurations, with the particle replaced by a loop in space.Comment: 11 page
The Jang equation, apparent horizons, and the Penrose inequality
The Jang equation in the spherically symmetric case reduces to a first order
equation. This permits an easy analysis of the role apparent horizons play in
the (non)existence of solutions. We demonstrate that the proposed derivation of
the Penrose inequality based on the Jang equation cannot work in the
spherically symmetric case. Thus it is fruitless to apply this method, as it
stands, to the general case. We show also that those analytic criteria for the
formation of horizons that are based on the use of the Jang equation are of
limited validity for the proof of the trapped surface conjecture.Comment: minor misprints correcte
Helfrich-Canham bending energy as a constrained non-linear sigma model
The Helfrich-Canham bending energy is identified with a non-linear sigma
model for a unit vector. The identification, however, is dependent on one
additional constraint: that the unit vector be constrained to lie orthogonal to
the surface. The presence of this constraint adds a source to the divergence of
the stress tensor for this vector so that it is not conserved. The stress
tensor which is conserved is identified and its conservation shown to reproduce
the correct shape equation.Comment: 5 page
Descriptions of membrane mechanics from microscopic and effective two-dimensional perspectives
Mechanics of fluid membranes may be described in terms of the concepts of
mechanical deformations and stresses, or in terms of mechanical free-energy
functions. In this paper, each of the two descriptions is developed by viewing
a membrane from two perspectives: a microscopic perspective, in which the
membrane appears as a thin layer of finite thickness and with highly
inhomogeneous material and force distributions in its transverse direction, and
an effective, two-dimensional perspective, in which the membrane is treated as
an infinitely thin surface, with effective material and mechanical properties.
A connection between these two perspectives is then established. Moreover, the
functional dependence of the variation in the mechanical free energy of the
membrane on its mechanical deformations is first studied in the microscopic
perspective. The result is then used to examine to what extent different,
effective mechanical stresses and forces can be derived from a given, effective
functional of the mechanical free energy.Comment: 37 pages, 3 figures, minor change
Geometry of lipid vesicle adhesion
The adhesion of a lipid membrane vesicle to a fixed substrate is examined
from a geometrical point of view. This vesicle is described by the Helfrich
hamiltonian quadratic in mean curvature; it interacts by contact with the
substrate, with an interaction energy proportional to the area of contact. We
identify the constraints on the geometry at the boundary of the shared surface.
The result is interpreted in terms of the balance of the force normal to this
boundary. No assumptions are made either on the symmetry of the vesicle or on
that of the substrate. The strong bonding limit as well as the effect of
curvature asymmetry on the boundary are discussed.Comment: 7 pages, some major changes in sections III and IV, version published
in Physical Review
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