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, $P(X)$ equipped with the Wasserstein metric $d^W$. We show that this metric space is isometric to a totally convex subset of a Hilbert space, $L^2[0,1]$, which allows for concrete calculations, contrary to the usual state of affairs in the theory of optimal transport. We prove explicitly that $(P(X),d^W)$ 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 $SE(2)$ and $SE(3)$. In our distance approximations we consider homogeneous norms on nilpotent groups that locally approximate $SE(n)$, and which are obtained via the exponential and logarithmic map on $SE(n)$. 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 $SE(n)$. 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 $\mathbb{R}^n$ and $SE(n)$.Comment: 18 pages, 9 figures, 3 tables, in review at JMI

Self-duality and associated parallel or cocalibrated G2{\mathrm{G}}_2 structures

We find a remarkable family of $\mathrm{G}_2$ structures defined on certain principal $\mathrm{SO}(3)$-bundles $P_\pm\longrightarrow M$ associated with any given oriented Riemannian 4-manifold $M$. 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 $\mathrm{G}_2$-holonomy metrics on the vector bundle of self- or anti-self-dual 2-forms on $M$. We then discover new examples of that special holonomy on disk bundles over ${\cal H}^4$ and ${\cal H}^2_{\mathbb{C}}$, respectively, the real and complex hyperbolic space. Only in the end we present the new $\mathrm{G}_2$ 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