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 $\rho_{Max}$ be the maximum value of the energy density and $\ell$ the radial measure of its support. If $\rho_{Max}\ell^2$ 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