927 research outputs found
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 String Deviation Equation
The relative motion of many particles can be described by the geodesic
deviation equation. This can be derived from the second covariant variation of
the point particle's action. It is shown that the second covariant variation of
the string action leads to a string deviation equation.Comment: 18 pages, some small changes, no tables or diagrams, LaTex2
Self Similar Solutions of the Evolution Equation of a Scalar Field in an Expanding Geometry
We consider the functional Schrodinger equation for a self interacting scalar
field in an expanding geometry. By performing a time dependent scale
transformation on the argument of the field we derive a functional Schrodinger
equation whose hamiltonian is time independent but involves a time-odd term
associated to a constraint on the expansion current. We study the mean field
approximation to this equation and generalize in this case, for interacting
fields, the solutions worked out by Bunch and Davies for free fields.Comment: 8 pages, Latex, IPNO/TH 94-3
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
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
Axially symmetric membranes with polar tethers
Axially symmetric equilibrium configurations of the conformally invariant
Willmore energy are shown to satisfy an equation that is two orders lower in
derivatives of the embedding functions than the equilibrium shape equation, not
one as would be expected on the basis of axial symmetry. Modulo a translation
along the axis, this equation involves a single free parameter c.If c\ne 0, a
geometry with spherical topology will possess curvature singularities at its
poles. The physical origin of the singularity is identified by examining the
Noether charge associated with the translational invariance of the energy; it
is consistent with an external axial force acting at the poles. A one-parameter
family of exact solutions displaying a discocyte to stomatocyte transition is
described.Comment: 13 pages, extended and revised version of Non-local sine-Gordon
equation for the shape of axi-symmetric membrane
The isolation of gravitational instantons: Flat tori V flat R^4
The role of topology in the perturbative solution of the Euclidean Einstein
equations about flat instantons is examined.Comment: 15 pages, ICN-UNAM 94-1
A New Anomaly-Free Gauged Supergravity in Six Dimensions
We present a new anomaly-free gauged N=1 supergravity model in six
dimensions. The gauge group is E_7xG_2xU(1)_R, with all hyperinos transforming
in the product representation {56,14). The theory admits monopole
compactifications to R^4xS^2, leading to D=4 effective theories with broken
supersymmetry and massless fermions.Comment: 9 pages, RevTeX
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
Geometry of Deformations of Relativistic Membranes
A kinematical description of infinitesimal deformations of the worldsheet
spanned in spacetime by a relativistic membrane is presented. This provides a
framework for obtaining both the classical equations of motion and the
equations describing infinitesimal deformations about solutions of these
equations when the action describing the dynamics of this membrane is
constructed using {\it any} local geometrical worldsheet scalars. As examples,
we consider a Nambu membrane, and an action quadratic in the extrinsic
curvature of the worldsheet.Comment: 20 pages, Plain Tex, sign errors corrected, many new references
added. To appear in Physical Review
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