Piecewise Flat Manifold Research Articles

Spectral analysis of causal dynamical triangulations via finite element method

We examine the dual graph representation of simplicial manifolds in causal dynamical triangulations (CDT) as a means to build observables and propose a new representation based on the finite element methods (FEM). In particular, with the application of FEM techniques, we extract the (low-lying) spectrum of the Laplace-Beltrami (LB) operator on the Sobolev space ${H}^{1}$ of scalar functions on piecewise flat manifolds and compare them with corresponding results obtained by using the dual graph representation. We show that, except for nonpathological cases in two dimensions, the dual graph spectrum and spectral dimension do not generally agree, neither quantitatively nor qualitatively, with the ones obtained from the LB operator on the continuous space. We analyze the reasons for this discrepancy and discuss its possible implications on the definition of generic observables built from the dual graph representation.

Computing discrete harmonic differential forms in a given cohomology class using finite element exterior calculus

Computational topology research of the past two decades has emphasized combinatorial techniques while numerical methods such as numerical linear algebra remain underutilized. While the combinatorial techniques have been very successful in diverse areas, for some applications, it is worth considering the numerical counterparts. We discuss one such application. Harmonic forms are elements of the kernel of the Hodge Laplacian operator and contain information about the topology of the manifold. If a particular cohomology class is chosen, the closed differential form with the smallest norm in that class is a harmonic form. We use these well-known facts to give an algorithm for solving the following problem: given a piecewise flat manifold simplicial complex (with or without boundary) and a closed piecewise polynomial differential form representing a cohomology class, find the discrete harmonic form in that cohomology class. We give a least squares based algorithm to solve this problem and show that the computed form satisfies the finite element exterior calculus (FEEC) equations for being a harmonic form. The piecewise polynomial spaces used are the spaces of trimmed polynomial forms, that is, arbitrary degree polynomial generalizations of Whitney forms which are used in FEEC. We also survey other methods for finding harmonic forms.

Discrete conformal variations and scalar curvature on piecewise flat two- and three-dimensional manifolds

A piecewise flat manifold is a triangulated manifold given a geometry by specifying edge lengths (lengths of 1-simplices) and specifying that all simplices are Euclidean. We consider the variation of angles of piecewise flat manifolds as the geometry varies in a particular way, which we call a conformal variation. This variation generalizes variations within the class of circles with fixed intersection angles (such as circle packings) as well as other formulations of conformal variation of piecewise flat manifolds previously suggested. We describe the angle derivatives of the angles in two- and three-dimensional piecewise flat manifolds, giving rise to formulas for the derivatives of curvatures. The formulas for derivatives of curvature resemble the formulas for the change of scalar curvature under a conformal variation of Riemannian metric. They allow us to explicitly describe the variation of certain curvature functionals, including Regge’s formulation of the Einstein-Hilbert functional (total scalar curvature), and to consider convexity of these functionals. They also allow us to prove rigidity theorems for certain analogues of constant curvature and Einstein manifolds in the piecewise flat setting.

Reggeʼs Einstein–Hilbert functional on the double tetrahedron

The double tetrahedron is the triangulation of the three-sphere gotten by gluing together two congruent tetrahedra along their boundaries. As a piecewise flat manifold, its geometry is determined by its six edge lengths, giving a notion of a metric on the double tetrahedron. We study notions of Einstein metrics, constant scalar curvature metrics, and the Yamabe problem on the double tetrahedron, with some reference to the possibilities on a general piecewise flat manifold. The main tool is analysis of Reggeʼs Einstein–Hilbert functional, a piecewise flat analogue of the Einstein–Hilbert (or total scalar curvature) functional on Riemannian manifolds. We study the Einstein–Hilbert–Regge functional on the space of metrics and on discrete conformal classes of metrics.

Collisions in piecewise flat gravity in 3+1 dimensions

We consider the (3 + 1)-dimensional locally finite gravity model proposed by 't Hooft (2008 Found. Phys. 38 733–57). In particular we revisit the problem of resolving collisions of string defects. We provide a new geometric description of the configurations of strings using piecewise flat manifolds and use it to resolve a more general class of collisions. We argue that beyond certain bounds for the deficiency/surplus angles no resolutions may be found that satisfy the imposed causality conditions.

A version of the connection representation of Regge action

We define for any 4-tetrahedron (4-simplex) the simplest finite closed piecewise flat manifold consisting of this 4-tetrahedron and of one other 4-tetrahedron identical up to reflection to the present one (call it a bisimplex built on the given 4-simplex, or a two-sided 4-simplex). We consider an arbitrary piecewise flat manifold. The gravity action for it can be expressed in terms of the sum of the actions for the bisimplices built on the 4-simplices constituting this manifold. We use a representation of each bisimplex action in terms of rotation matrices (connections) and area tensors. This gives some representation of any piecewise flat gravity action in terms of connections. The action is a sum of terms each depending on the connection variables referring to a single 4-tetrahedron. Application of this representation to the path integral formalism is considered. Integrations over connections in the path integral reduce to independent integrations over finite sets of connections on separate 4-simplices. One of the consequences is exponential suppression of the result at large areas or lengths (compared to the Plank scale). This is important for the consistency of the simplicial description of spacetime.

Path integral in the simplest Regge calculus model

The simplest (3+1)D Regge calculus model (with three-dimensional discrete space and continuous time) is considered which describes evolution of the simplest closed two-tetrahedron piecewise flat manifold in the continuous time. The measure in the path integral which describes canonical quantisation of the model in terms of area bivectors and connections as independent variables is found. It is shown that selfdual-antiselfdual splitting of the variables simplifies the integral although does not admit complete separation of (anti-)selfdual sector.