We propose several different types of construction principles for new classes Toda field theories based on root systems defined Lorentzian lattices. In analogy to conformal and affine semi-simple Lie algebras, also their extensions come about in massive variants. carry out the Painlevé integrability test proposed theories, finding general only one integer valued resonance corresponding energy-m...

We review both the kinematics and dynamics of non-lorentzian theories their associated geometries. First, we introduce kinematical spacetimes symmetry algebras. Next, construct actions describing particle in some these spaces using method nonlinear realisations. explain relation with coadjoint orbit method. continue discussing three types gravity theories: Galilei gravity, Newton-Cartan Carroll...

It has been speculated that Lorentzian wormholes of the Morris-Thorne type might be allowed by the laws of physics at submicroscopic, e.g. Planck, scales and that a sufficiently advanced civilization might be able to enlarge them to classical size. The purpose of this paper is to explore the possibility that inflation might provide a natural mechanism for the enlargement of such wormholes to ma...

Just as for non-abelian gauge theories at strong coupling, discrete lattice methods are a natural tool in the study of non-perturbative quantum gravity. They have to reflect the fact that the geometric degrees of freedom are dynamical, and that therefore also the lattice theory must be formulated in a background-independent way. After summarizing the status quo of discrete covariant lattice mod...

v. dannon showed that spherical curves in e4 can be given by frenet-like equations, and he thengave an integral characterization for spherical curves in e4 . in this paper, lorentzian spherical timelike andspacelike curves in the space time 41 r are shown to be given by frenet-like equations of timelike andspacelike curves in the euclidean space e3 and the minkowski 3-space 31 r . thus, finding...

we study connected orientable spacelike hypersurfaces $x:m^{n}rightarrowm_q^{n+1}(c)$, isometrically immersed into the riemannian or lorentzian space form of curvature $c=-1,0,1$, and index $q=0,1$, satisfying the condition $~l_kx=ax+b$,~ where $l_k$ is the $textit{linearized operator}$ of the $(k+1)$-th mean curvature $h_{k+1}$ of the hypersurface for a fixed integer $0leq k

This paper summarizes the findings and recommendations of a recent panel on Future Directions in Control, Dynamics, and Systems, sponsored by the US Air Force Office of Scientific Research. A set of grand challenges that illustrate some of the recommendations and opportunities are provided. Finally, the paper describes two new courses being developed at Caltech that are aligned with the recomme...

We summarize the main ideas of General Relativity and Lorentzian geometry, leading to a proof of the simplest of the celebrated Hawking-Penrose singularity theorems. The reader is assumed to be familiar with Riemannian geometry and point set topology. Introduction Historically, much of the development of Riemannian geometry has been driven by General Relativity. This theory models spacetime as ...

In this paper using the Clifford bundle formalism a Lagrangian theory of the Yang-Mills type (with a gauge fixing term and an auto interacting term) for the gravitational field in Minkowski spacetime is presented. It is shown how two simple hypothesis permits the interpretation of the formalism in terms of effective Lorentzian or teleparallel geometries. In the case of a Lorentzian geometry int...