362 research outputs found
Stresses in lipid membranes
The stresses in a closed lipid membrane described by the Helfrich
hamiltonian, quadratic in the extrinsic curvature, are identified using
Noether's theorem. Three equations describe the conservation of the stress
tensor: the normal projection is identified as the shape equation describing
equilibrium configurations; the tangential projections are consistency
conditions on the stresses which capture the fluid character of such membranes.
The corresponding torque tensor is also identified. The use of the stress
tensor as a basis for perturbation theory is discussed. The conservation laws
are cast in terms of the forces and torques on closed curves. As an
application, the first integral of the shape equation for axially symmetric
configurations is derived by examining the forces which are balanced along
circles of constant latitude.Comment: 16 pages, introduction rewritten, other minor changes, new references
added, version to appear in Journal of Physics
Remarks on Conserved Quantities and Entropy of BTZ Black Hole Solutions. Part II: BCEA Theory
The BTZ black hole solution for (2+1)-spacetime is considered as a solution
of a triad-affine theory (BCEA) in which topological matter is introduced to
replace the cosmological constant in the model. Conserved quantities and
entropy are calculated via Noether theorem, reproducing in a geometrical and
global framework earlier results found in the literature using local
formalisms. Ambiguities in global definitions of conserved quantities are
considered in detail. A dual and covariant Legendre transformation is performed
to re-formulate BCEA theory as a purely metric (natural) theory (BCG) coupled
to topological matter. No ambiguities in the definition of mass and angular
momentum arise in BCG theory. Moreover, gravitational and matter contributions
to conserved quantities and entropy are isolated. Finally, a comparison of BCEA
and BCG theories is carried out by relying on the results obtained in both
theories.Comment: PlainTEX, 20 page
Second variation of the Helfrich-Canham Hamiltonian and reparametrization invariance
A covariant approach towards a theory of deformations is developed to examine
both the first and second variation of the Helfrich-Canham Hamiltonian --
quadratic in the extrinsic curvature -- which describes fluid vesicles at
mesoscopic scales. Deformations are decomposed into tangential and normal
components; At first order, tangential deformations may always be identified
with a reparametrization; at second order, they differ. The relationship
between tangential deformations and reparametrizations, as well as the coupling
between tangential and normal deformations, is examined at this order for both
the metric and the extrinsic curvature tensors. Expressions for the expansion
to second order in deformations of geometrical invariants constructed with
these tensors are obtained; in particular, the expansion of the Hamiltonian to
this order about an equilibrium is considered. Our approach applies as well to
any geometrical model for membranes.Comment: 20 page
Hamiltonian Analysis of non-chiral Plebanski Theory and its Generalizations
We consider non-chiral, full Lorentz group-based Plebanski formulation of
general relativity in its version that utilizes the Lagrange multiplier field
Phi with "internal" indices. The Hamiltonian analysis of this version of the
theory turns out to be simpler than in the previously considered in the
literature version with Phi carrying spacetime indices. We then extend the
Hamiltonian analysis to a more general class of theories whose action contains
scalars invariants constructed from Phi. Such theories have recently been
considered in the context of unification of gravity with other forces. We show
that these more general theories have six additional propagating degrees of
freedom as compared to general relativity, something that has not been
appreciated in the literature treating them as being not much different from
GR.Comment: 10 page
Hamiltonian dynamics of extended objects
We consider a relativistic extended object described by a reparametrization
invariant local action that depends on the extrinsic curvature of the
worldvolume swept out by the object as it evolves. We provide a Hamiltonian
formulation of the dynamics of such higher derivative models which is motivated
by the ADM formulation of general relativity. The canonical momenta are
identified by looking at boundary behavior under small deformations of the
action; the relationship between the momentum conjugate to the embedding
functions and the conserved momentum density is established. The canonical
Hamiltonian is constructed explicitly; the constraints on the phase space, both
primary and secondary, are identified and the role they play in the theory
described. The multipliers implementing the primary constraints are identified
in terms of the ADM lapse and shift variables and Hamilton's equations shown to
be consistent with the Euler-Lagrange equations.Comment: 24 pages, late
The one-loop elastic coefficients for the Helfrich membrane in higher dimensions
Using a covariant geometric approach we obtain the effective bending
couplings for a 2-dimensional rigid membrane embedded into a
-dimensional Euclidean space. The Hamiltonian for the membrane has three
terms: The first one is quadratic in its mean extrinsic curvature. The second
one is proportional to its Gaussian curvature, and the last one is proportional
to its area. The results we obtain are in agreement with those finding that
thermal fluctuations soften the 2-dimensional membrane embedded into a
3-dimensional Euclidean space.Comment: 9 page
The Generalised Raychaudhuri Equations : Examples
Specific examples of the generalized Raychaudhuri Equations for the evolution
of deformations along families of dimensional surfaces embedded in a
background dimensional spacetime are discussed. These include string
worldsheets embedded in four dimensional spacetimes and two dimensional
timelike hypersurfaces in a three dimensional curved background. The issue of
focussing of families of surfaces is introduced and analysed in some detail.Comment: 8 pages (Revtex, Twocolumn format). Corrected(see section on string
worldsheets), reorganised and shortened slightl
Hamiltonian dynamics for Einstein's action in G0 limit
The Hamiltonian analysis for the Einstein's action in limit is
performed. Considering the original configuration space without involve the
usual variables we show that the version for Einstein's action
is devoid of physical degrees of freedom. In addition, we will identify the
relevant symmetries of the theory such as the extended action, the extended
Hamiltonian, the gauge transformations and the algebra of the constraints. As
complement part of this work, we develop the covariant canonical formalism
where will be constructed a closed and gauge invariant symplectic form. In
particular, using the geometric form we will obtain by means of other way the
same symmetries that we found using the Hamiltonian analysis
Holographic Formulation of Quantum Supergravity
We show that supergravity with a cosmological constant can be
expressed as constrained topological field theory based on the supergroup
. The theory is then extended to include timelike boundaries with
finite spatial area. Consistent boundary conditions are found which induce a
boundary theory based on a supersymmetric Chern-Simons theory. The boundary
state space is constructed from states of the boundary supersymmetric
Chern-Simons theory on the punctured two sphere and naturally satisfies the
Bekenstein bound, where area is measured by the area operator of quantum
supergravity.Comment: 30 pages, no figur
Brst Cohomology and Invariants of 4D Gravity in Ashtekar Variables
We discuss the BRST cohomologies of the invariants associated with the
description of classical and quantum gravity in four dimensions, using the
Ashtekar variables. These invariants are constructed from several BRST
cohomology sequences. They provide a systematic and clear characterization of
non-local observables in general relativity with unbroken diffeomorphism
invariance, and could yield further differential invariants for four-manifolds.
The theory includes fluctuations of the vierbein fields, but there exits a
non-trivial phase which can be expressed in terms of Witten's topological
quantum field theory. In this phase, the descent sequences are degenerate, and
the corresponding classical solutions can be identified with the conformally
self-dual sector of Einstein manifolds. The full theory includes fluctuations
which bring the system out of this sector while preserving diffeomorphism
invariance.Comment: 15 page
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