Evolution equations for a wide range of Einstein-matter systems

We use an orthonormal frame approach to provide a general framework for the first order hyperbolic reduction of the Einstein equations coupled to a fairly generic class of matter models. Our analysis covers the special cases of dust and perfect fluid. We also provide a discussion of self-gravitating elastic matter. The frame is Fermi–Walker propagated and coordinates are chosen such as to satisfy the Lagrange condition. We show the propagation of the constraints of the Einstein-matter system.

Brownian motion on Perelman’s almost Ricci-flat manifold

We study Brownian motion and stochastic parallel transport on Perelman’s almost Ricci flat manifold {\mathcal{M}=M\times\mathbb{S}^{N}\times I}, whose dimension depends on a parameter N unbounded from above. We construct sequences of projected Brownian motions and stochastic parallel transports which for {N\to\infty} converge to the corresponding objects for the Ricci flow. In order to make precise this process of passing to the limit, we study the martingale problems for the Laplace operator on {\mathcal{M}} and for the horizontal Laplacian on the orthonormal frame bundle {\mathcal{OM}}. As an application, we see how the characterizations of two-sided bounds on the Ricci curvature established by A. Naber applied to Perelman’s manifold lead to the inequalities that characterize solutions of the Ricci flow discovered by Naber and the second author.

Nonstandard Action of Diffeomorphisms and Gravity’s Anti-Newtonian Limit

A tensor calculus adapted to the Anti-Newtonian limit of Einstein gravity is developed. The limit is defined in terms of a global conformal rescaling of the spatial metric. This enhances spacelike distances compared to timelike ones and in the limit effectively squeezes the lightcones to lines. Conventional tensors admit an analogous Anti-Newtonian limit, which however transforms according to a non-standard realization of the spacetime Diffeomorphism group. In addition to the type of the tensor the transformation law depends on, a set of integer-valued weights is needed to ensure the existence of a nontrivial limit. Examples are limiting counterparts of the metric, Einstein, and Riemann tensors. An adapted purely temporal notion of parallel transport is presented. By introducing a generalized Ehresmann connection and an associated orthonormal frame compatible with an invertible Carroll metric, the weight-dependent transformation laws can be mapped into a universal one that can be read off from the index structure. Utilizing this ‘decoupling map’ and a realization of the generalized Ehresmann connection in terms of scalar field, the limiting gravity theory can be endowed with an intrinsic Levi–Civita type notion of spatio-temporal parallel transport.

Soliton propagation of electromagnetic field vectors of polarized light ray traveling along with coiled optical fiber on the unit 2-sphereS²

In this paper, we relate the evolution equation of the electric field and magnetic field vectors of the polarized light ray traveling along with a coiled optical fiber on the unit 2-sphere S² into the nonlinear Schrödinger's equation by proposing new kinds of binormal motions and new kinds of Hasimoto functions in addition to commonly known formula of the binormal motion and Hasimoto function. All these operations have been conducted by using the orthonormal frame of spherical equations that is defined along with the coiled optical fiber lying on the unit 2-sphere S². We also propose perturbed solutions of the nonlinear Schrödinger's evolution equation that governs the propagation of solitons through electric field (E) and magnetic field (M) vectors. Finally, we provide some numerical simulations to supplement the analytical outcomes.

Soliton propagation of electromagnetic field vectors of polarized light ray traveling in a coiled optical fiber in the ordinary space

In this paper, we relate the evolution equations of the electric field and magnetic field vectors of the polarized light ray traveling in a coiled optical fiber in the ordinary space into the nonlinear Schrödinger’s equation by proposing new kinds of binormal motions and new kinds of Hasimoto functions in addition to commonly known formula of the binormal motion and Hasimoto function. All these operations have been conducted by using the orthonormal frame of Bishop equations that is defined along with the coiled optical fiber. We also propose perturbed solutions of the nonlinear Schrödinger’s evolution equation that governs the propagation of solitons through the electric field [Formula: see text] and magnetic field [Formula: see text] vectors. Finally, we provide some numerical simulations to supplement the analytical outcomes.

Scale Transformations in Metric-Affine Geometry

This article presents an exhaustive classification of metric-affine theories according to their scale symmetries. First it is clarified that there are three relevant definitions of a scale transformation. These correspond to a projective transformation of the connection, a rescaling of the orthonormal frame, and a combination of the two. The most general second order quadratic metric-affine action, including the parity-violating terms, is constructed in each of the three cases. The results can be straightforwardly generalised by including higher derivatives, and implemented in the general metric-affine, teleparallel, and symmetric teleparallel geometries.

A Highlight-Generation Method for Rendering Translucent Objects

The acquisition of translucent objects has become a very common task thanks to the progress of 3D scanning technology. Since the characteristic soft appearance of translucent objects is due to subsurface scattering, the details are naturally left out in this appearance. For objects that have complex shapes, this lack of detail is obviously more prominent. In this paper, we propose a method to preserve the details of surface geometry by adding highlight effects. In generating highlight effects, our method employs a local orthonormal frame and combines, in a novel way, the incoming and outgoing light in approximating the subsurface scattering process. When the incident illuminant direction changes from nearly overhead to nearly horizontal, our method effectively preserves complex surface geometry details in the appearance of translucent materials. Through experiments, we show that our method can store surface features as well as maintain the translucency of the original materials and even enhance the perception of translucency. By numerically comparing the generated highlight effects with those generated by the traditional Bidirectional Reflectance Distribution Function (BRDF) models with different bandwidth parameters, we demonstrate the validity of our proposed method.