4,653 research outputs found
Quantum Estimation of Parameters of Classical Spacetimes
We describe a quantum limit to measurement of classical spacetimes.
Specifically, we formulate a quantum Cramer-Rao lower bound for estimating the
single parameter in any one-parameter family of spacetime metrics. We employ
the locally covariant formulation of quantum field theory in curved spacetime,
which allows for a manifestly background-independent derivation. The result is
an uncertainty relation that applies to all globally hyperbolic spacetimes.
Among other examples, we apply our method to detection of gravitational waves
using the electromagnetic field as a probe, as in laser-interferometric
gravitational-wave detectors. Other applications are discussed, from
terrestrial gravimetry to cosmology.Comment: 23 pages. This article supersedes arXiv:1108.522
Nonlinear morphoelastic plates II: exodus to buckled states
Morphoelasticity is the theory of growing elastic materials. This theory is based on the multiple decomposition of the deformation gradient and provides a formulation of the deformation and stresses induced by growth. Following a companion paper, a general theory of growing nonlinear elastic Kirchhoff plate is described. First, a complete geometric description of incompatibility with simple examples is given. Second, the stability of growing Kirchhoff plates is analyzed
Optimal Transport and Ricci Curvature: Wasserstein Space Over the Interval
In this essay, we discuss the notion of optimal transport on geodesic measure
spaces and the associated (2-)Wasserstein distance. We then examine
displacement convexity of the entropy functional on the space of probability
measures. In particular, we give a detailed proof that the Lott-Villani-Sturm
notion of generalized Ricci bounds agree with the classical notion on smooth
manifolds. We also give the proof that generalized Ricci bounds are preserved
under Gromov-Hausdorff convergence. In particular, we examine in detail the
space of probability measures over the interval, equipped with the
Wasserstein metric . We show that this metric space is isometric to a
totally convex subset of a Hilbert space, , which allows for concrete
calculations, contrary to the usual state of affairs in the theory of optimal
transport. We prove explicitly that has vanishing Alexandrov
curvature, and give an easy to work with expression for the entropy functional
on this space. In addition, we examine finite dimensional Gromov-Hausdorff
approximations to this space, and use these to construct a measure on the limit
space, the entropic measure first considered by Von Renesse and Sturm. We
examine properties of the measure, in particular explaining why one would
expect it to have generalized Ricci lower bounds. We then show that this is in
fact not true. We also discuss the possibility and consequences of finding a
different measure which does admit generalized Ricci lower bounds.Comment: 47 pages, 9 figure
Nilpotent Approximations of Sub-Riemannian Distances for Fast Perceptual Grouping of Blood Vessels in 2D and 3D
We propose an efficient approach for the grouping of local orientations
(points on vessels) via nilpotent approximations of sub-Riemannian distances in
the 2D and 3D roto-translation groups and . In our distance
approximations we consider homogeneous norms on nilpotent groups that locally
approximate , and which are obtained via the exponential and logarithmic
map on . In a qualitative validation we show that the norms provide
accurate approximations of the true sub-Riemannian distances, and we discuss
their relations to the fundamental solution of the sub-Laplacian on .
The quantitative experiments further confirm the accuracy of the
approximations. Quantitative results are obtained by evaluating perceptual
grouping performance of retinal blood vessels in 2D images and curves in
challenging 3D synthetic volumes. The results show that 1) sub-Riemannian
geometry is essential in achieving top performance and 2) that grouping via the
fast analytic approximations performs almost equally, or better, than
data-adaptive fast marching approaches on and .Comment: 18 pages, 9 figures, 3 tables, in review at JMI
Self-duality and associated parallel or cocalibrated structures
We find a remarkable family of structures defined on certain
principal -bundles associated with any
given oriented Riemannian 4-manifold . Such structures are always
cocalibrated. The study starts with a recast of the Singer-Thorpe equations of
4-dimensional geometry. These are applied to the Bryant-Salamon cons\-truction
of complete -holonomy metrics on the vector bundle of self- or
anti-self-dual 2-forms on . We then discover new examples of that special
holonomy on disk bundles over and ,
respectively, the real and complex hyperbolic space. Only in the end we present
the new structures on principal bundles.Comment: 20 pages; final version, to appear in Annales Academi{\ae}
Scientiarum Fennic{\ae
A tensor theory of space-time as a strained material continuum
The classical theory of strain in material continua is reviewed and
generalized to space-time. Strain is attributed to "external" (matter/energy
fields) and intrinsic sources fixing the global symmetry of the universe
(defects in the continuum). A Lagrangian for space-time is worked out, adding
to the usual Hilbert term an "elastic" contribution from intrinsic strain. This
approach is equivalent to a peculiar tensor field, which is indeed part of the
metric tensor. The theory gives a configuration of space-time accounting both
for the initial inflation and for the late acceleration. Considering also the
contribution from matter the theory is used to fit the luminosity data of type
Ia supernovae, giving satisfactory results.Comment: Revised to match the version accepted for publication in Class.
Quantum Gra
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