305 research outputs found
Selfdual 2-form formulation of gravity and classification of energy-momentum tensors
It is shown how the different irreducibility classes of the energy-momentum
tensor allow for a Lagrangian formulation of the gravity-matter system using a
selfdual 2-form as a basic variable. It is pointed out what kind of
difficulties arise when attempting to construct a pure spin-connection
formulation of the gravity-matter system. Ambiguities in the formulation
especially concerning the need for constraints are clarified.Comment: title changed, extended versio
No New Symmetries of the Vacuum Einstein Equations
In this note we examine some recently proposed solutions of the linearized
vacuum Einstein equations. We show that such solutions are {\it not} symmetries
of the Einstein equations, because of a crucial integrability condition.Comment: 9 pages, Te
Constraints and Reality Conditions in the Ashtekar Formulation of General Relativity
We show how to treat the constraints and reality conditions in the
-ADM (Ashtekar) formulation of general relativity, for the case of a
vacuum spacetime with a cosmological constant. We clarify the difference
between the reality conditions on the metric and on the triad. Assuming the
triad reality condition, we find a new variable, allowing us to solve the gauge
constraint equations and the reality conditions simultaneously.Comment: LaTeX file, 12 pages, no figures; to appear in Classical and Quantum
Gravit
Hamiltonians for curves
We examine the equilibrium conditions of a curve in space when a local energy
penalty is associated with its extrinsic geometrical state characterized by its
curvature and torsion. To do this we tailor the theory of deformations to the
Frenet-Serret frame of the curve. The Euler-Lagrange equations describing
equilibrium are obtained; Noether's theorem is exploited to identify the
constants of integration of these equations as the Casimirs of the euclidean
group in three dimensions. While this system appears not to be integrable in
general, it {\it is} in various limits of interest. Let the energy density be
given as some function of the curvature and torsion, . If
is a linear function of either of its arguments but otherwise arbitrary, we
claim that the first integral associated with rotational invariance permits the
torsion to be expressed as the solution of an algebraic equation in
terms of the bending curvature, . The first integral associated with
translational invariance can then be cast as a quadrature for or for
.Comment: 17 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
On the Constant that Fixes the Area Spectrum in Canonical Quantum Gravity
The formula for the area eigenvalues that was obtained by many authors within
the approach known as loop quantum gravity states that each edge of a spin
network contributes an area proportional to sqrt{j(j+1)} times Planck length
squared to any surface it transversely intersects. However, some confusion
exists in the literature as to a value of the proportionality coefficient. The
purpose of this rather technical note is to fix this coefficient. We present a
calculation which shows that in a sector of quantum theory based on the
connection A=Gamma-gamma*K, where Gamma is the spin connection compatible with
the triad field, K is the extrinsic curvature and gamma is Immirzi parameter,
the value of the multiplicative factor is 8*pi*gamma. In other words, each edge
of a spin network contributes an area 8*pi*gamma*l_p^2*sqrt{j(j+1)} to any
surface it transversely intersects.Comment: Revtex, 7 pages, no figure
Real Ashtekar Variables for Lorentzian Signature Space-times
I suggest in this letter a new strategy to attack the problem of the reality
conditions in the Ashtekar approach to classical and quantum general
relativity. By writing a modified Hamiltonian constraint in the usual
Yang-Mills phase space I show that it is possible to describe space-times with
Lorentzian signature without the introduction of complex variables. All the
features of the Ashtekar formalism related to the geometrical nature of the new
variables are retained; in particular, it is still possible, in principle, to
use the loop variables approach in the passage to the quantum theory. The key
issue in the new formulation is how to deal with the more complicated
Hamiltonian constraint that must be used in order to avoid the introduction of
complex fields.Comment: 10 pages, LATEX, Preprint CGPG-94/10-
Conformally invariant bending energy for hypersurfaces
The most general conformally invariant bending energy of a closed
four-dimensional surface, polynomial in the extrinsic curvature and its
derivatives, is constructed. This invariance manifests itself as a set of
constraints on the corresponding stress tensor. If the topology is fixed, there
are three independent polynomial invariants: two of these are the
straighforward quartic analogues of the quadratic Willmore energy for a
two-dimensional surface; one is intrinsic (the Weyl invariant), the other
extrinsic; the third invariant involves a sum of a quadratic in gradients of
the extrinsic curvature -- which is not itself invariant -- and a quartic in
the curvature. The four-dimensional energy quadratic in extrinsic curvature
plays a central role in this construction.Comment: 16 page
A left-handed simplicial action for euclidean general relativity
An action for simplicial euclidean general relativity involving only
left-handed fields is presented. The simplicial theory is shown to converge to
continuum general relativity in the Plebanski formulation as the simplicial
complex is refined. This contrasts with the Regge model for which Miller and
Brewin have shown that the full field equations are much more restrictive than
Einstein's in the continuum limit. The action and field equations of the
proposed model are also significantly simpler then those of the Regge model
when written directly in terms of their fundamental variables.
An entirely analogous hypercubic lattice theory, which approximates
Plebanski's form of general relativity is also presented.Comment: Version 3. Adds current home address + slight corrections to
references of version 2. Version 2 = substantially clarified form of version
1. 29 pages, 4 figures, Latex, uses psfig.sty to insert postscript figures.
psfig.sty included in mailing, also available from this archiv
Alternative symplectic structures for SO(3,1) and SO(4) four-dimensional BF theories
The most general action, quadratic in the B fields as well as in the
curvature F, having SO(3,1) or SO(4) as the internal gauge group for a
four-dimensional BF theory is presented and its symplectic geometry is
displayed. It is shown that the space of solutions to the equations of motion
for the BF theory can be endowed with symplectic structures alternative to the
usual one. The analysis also includes topological terms and cosmological
constant. The implications of this fact for gravity are briefly discussed.Comment: 13 pages, LaTeX file, no figure
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