27,044 research outputs found
Lsdiff M and the Einstein Equations
We give a formulation of the vacuum Einstein equations in terms of a set of
volume-preserving vector fields on a four-manifold . These vectors
satisfy a set of equations which are a generalisation of the Yang-Mills
equations for a constant connection on flat spacetime.Comment: 5 pages, no figures, Latex, uses amsfonts, amssym.def and amssym.tex.
Note added on more direct connection with Yang-Mills equation
The ADHM construction and non-local symmetries of the self-dual Yang-Mills equations
We consider the action on instanton moduli spaces of the non-local symmetries
of the self-dual Yang-Mills equations on discovered by Chau and
coauthors. Beginning with the ADHM construction, we show that a sub-algebra of
the symmetry algebra generates the tangent space to the instanton moduli space
at each point. We explicitly find the subgroup of the symmetry group that
preserves the one-instanton moduli space. This action simply corresponds to a
scaling of the moduli space.Comment: AMSLatex, 19 pages, no figures. Some discussions clarified, and
citations made more accurate. I am grateful to the referee for detailed
comments. Version to appear in Communications in Mathematical Physic
A Spinorial Hamiltonian Approach to Gravity
We give a spinorial set of Hamiltonian variables for General Relativity in
any dimension greater than 2. This approach involves a study of the algebraic
properties of spinors in higher dimension, and of the elimination of
second-class constraints from the Hamiltonian theory. In four dimensions, when
restricted to the positive spin-bundle, these variables reduce to the standard
Ashtekar variables. In higher dimensions, the theory can either be reduced to a
spinorial version of the ADM formalism, or can be left in a more general form
which seems useful for the investigation of some spinorial problems such as
Riemannian manifolds with reduced holonomy group. In dimensions ,
the theory may be recast solely in terms of structures on the positive
spin-bundle , but such a reduction does not seem possible in
dimensions , due to algebraic properties of spinors in these
dimensions.Comment: 20 pages, Latex 2e. Published versio
Cosmic Strings and Chronology Protection
A space consisting of two rapidly moving cosmic strings has recently been
constructed by Gott that contains closed timelike curves. The global structure
of this space is analysed and is found that, away from the strings, the space
is identical to a generalised Misner space. The vacuum expectation value of the
energy momentum tensor for a conformally coupled scalar field is calculated on
this generalised Misner space. It is found to diverge very weakly on the
Chronology horizon, but more strongly on the polarised hypersurfaces. The
divergence on the polarised hypersurfaces is strong enough that when the proper
geodesic interval around any polarised hypersurface is of order the Planck
length squared, the perturbation to the metric caused by the backreaction will
be of order one. Thus we expect the structure of the space will be radically
altered by the backreaction before quantum gravitational effects become
important. This suggests that Hawking's `Chronology Protection Conjecture'
holds for spaces with non-compactly generated Chronology horizon.Comment: 15 pages, plain TeX, 2 figures (not included), DAMTP-R92/3
A positive mass theorem for low-regularity Riemannian metrics
We show that the positive mass theorem holds for continuous Riemannian
metrics that lie in the Sobolev space for manifolds of
dimension less than or equal to or spin-manifolds of any dimension. More
generally, we give a (negative) lower bound on the ADM mass of metrics for
which the scalar curvature fails to be non-negative, where the negative part
has compact support and sufficiently small norm. We show that a
Riemannian metric in for some with
non-negative scalar curvature in the distributional sense can be approximated
locally uniformly by smooth metrics with non-negative scalar curvature. For
continuous metrics in , there exist smooth approximating
metrics with non-negative scalar curvature that converge in for all
.Comment: 21 pages. The results on the positive mass theorem were announced in
arxiv:1205.1302, with a sketch of the proo
Optical bistability for two-level atoms in a standing-wave cavity
Observations of optical bistability are reported for a system composed of multiple atomic beams passing through a high-finesse optical cavity. Both the transmitted power and the intracavity fluorescent intensity have been recorded as functions of incident laser power for zero cavity and atomic detunings. A quantitative study has been made of the evolution of the steady-state switching intensities from well below the critical onset of bistability to well above this point. The results show reasonable agreement with a Gaussian-beam theory of optical bistability, but systematic departures are noted
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