Gauged Gravity via Spectral Asymptotics of non-Laplace type Operators

We construct invariant differential operators acting on sections of vector bundles of densities over a smooth manifold without using a Riemannian metric. The spectral invariants of such operators are invariant under both the diffeomorphisms and the gauge transformations and can be used to induce a new theory of gravitation. It can be viewed as a matrix generalization of Einstein general relativity that reproduces the standard Einstein theory in the weak deformation limit. Relations with various mathematical constructions such as Finsler geometry and Hodge-de Rham theory are discussed.Comment: Version accepted by J. High Energy Phys. Introduction and Discussion significantly expanded. References added and updated. (41 pages, LaTeX: JHEP3 class, no figures

A structure-preserving split finite element discretization of the split wave equations

We introduce a new finite element (FE) discretization framework applicable for covariant split equations. The introduction of additional differential forms (DF) that form pairs with the original ones permits the splitting of the equations into topological momentum and continuity equations and metric-dependent closure equations that apply the Hodge-star operator. Our discretization framework conserves this geometrical structure and provides for all DFs proper FE spaces such that the differential operators hold in strong form. We introduce lowest possible order discretizations of the split 1D wave equations, in which the discrete momentum and continuity equations follow by trivial projections onto piecewise constant FE spaces, omitting partial integrations. Approximating the Hodge-star by nontrivial Galerkin projections (GP), the two discrete metric equations follow by projections onto either the piecewise constant (GP0) or piecewise linear (GP1) space. Our framework gives us three schemes with significantly different behavior. The split scheme using twice GP1 is unstable and shares the dispersion relation with the P1-P1 FE scheme that approximates both variables by piecewise linear spaces (P1). The split schemes that apply a mixture of GP1 and GP0 share the dispersion relation with the stable P1-P0 FE scheme that applies piecewise linear and piecewise constant (P0) spaces. However, the split schemes exhibit second order convergence for both quantities of interest. For the split scheme applying twice GP0, we are not aware of a corresponding standard formulation to compare with. Though it does not provide a satisfactory approximation of the dispersion relation as short waves are propagated much too fast, the discovery of the new scheme illustrates the potential of our discretization framework as a toolbox to study and find FE schemes by new combinations of FE spaces

Finite element exterior calculus: from Hodge theory to numerical stability

This article reports on the confluence of two streams of research, one emanating from the fields of numerical analysis and scientific computation, the other from topology and geometry. In it we consider the numerical discretization of partial differential equations that are related to differential complexes so that de Rham cohomology and Hodge theory are key tools for the continuous problem. After a brief introduction to finite element methods, the discretization methods we consider, we develop an abstract Hilbert space framework for analyzing stability and convergence. In this framework, the differential complex is represented by a complex of Hilbert spaces and stability is obtained by transferring Hodge theoretic structures from the continuous level to the discrete. We show stable discretization is achieved if the finite element spaces satisfy two hypotheses: they form a subcomplex and there exists a bounded cochain projection from the full complex to the subcomplex. Next, we consider the most canonical example of the abstract theory, in which the Hilbert complex is the de Rham complex of a domain in Euclidean space. We use the Koszul complex to construct two families of finite element differential forms, show that these can be arranged in subcomplexes of the de Rham complex in numerous ways, and for each construct a bounded cochain projection. The abstract theory therefore applies to give the stability and convergence of finite element approximations of the Hodge Laplacian. Other applications are considered as well, especially to the equations of elasticity. Background material is included to make the presentation self-contained for a variety of readers.Comment: 74 pages, 8 figures; reorganized introductory material, added additional example and references; final version accepted by Bulletin of the AMS, added references to codes and adjusted some diagrams. Bulletin of the American Mathematical Society, to appear 201

Numerische Approximation in Riemannschen Mannigfaltigkeiten mithilfe des Karcher'schen Schwerpunktes

(1) Let $(M,g)$ be a compact Riemannian manifold without boundary, and let $\Delta$ be the $n$-dimensional standard simplex. Following Karcher (1977), we consider, for $n+1$ given points $p_i \in M$, the function \ E: M \times \Delta \to \R, (a,\lambda) \mapsto \lambda^0 d^2(a,p_0) + \dots + \lambda^n d^2(a,p_n), \ where $d$ is the geodesic distance in $M$. If all $p_i$ lie in a sufficiently small geodesic ball, then $x: \lambda \mapsto argmin_a E(a,\lambda)$ is a well-defined mapping $\Delta \to M$ (5.3). We call $s := x(\Delta)$ the Karcher simplex with vertices $p_i$. Suppose $\Delta$ carries a flat Riemannian metric $g^e$ induced by edge lengths $d(p_i,p_j)$. If all edge lengths are small than $h$ and $vol(\Delta,g^e) \geq \alpha h^n$ for some $\alpha > 0$, then we show in 6.17 and 6.23 that \begin{equation} %\tag{A.1a} |(x^*g - g^e)(v,w)| \leq c h^2 |v| |w|, \qquad % |(\nabla^{x^*g} - \nabla^{g^e})_v w| \leq c h |v| |w| \end{equation} with some constant $c$ depending only on the curvature tensor $R$ of $(M,g)$ and $\theta$. With little effort, this gives interpolation estimates for functions $u: s \to \R$ (7.4) and $y: s \to N$ for a second Riemannian manifold $N$ (7.14). Also, following Leibon und Letscher (2000), this simplex construction allows for the definition of a Voronoi decomposition (8.7). Thus we can consider $(M,g)$ to be triangulated in the following. On each simplex, $g$ is approximated by a metric $g^e$ with (A1.a), and weakly differentiable functions $u \in H^1(M,g)$ can be approximated by polynomials $u_h \in P^1(M)$. Via the standard method of surface finite element methods (Dziuk 1988, Holst and Stern 2012), variational problems such as the Poisson problem (10.13, 10.17, 13.14) or the Hodge decomposition (10.15) in $H^1(M,g)$ can be compare to those in $H^1(M,g^e)$ and their Galerkin approximations in $P^1(M)$. Corresponding to the standard surface finite element theory for problems on submanifolds of $\R^m$, also submanifolds $S \subset M$ may be approximated by Karcher simplices. The geometry error is the same as for submanifolds of $\R^m$ plus an additional term $ch^2$ (11.18). (2) Let $M$ be the geometric realisation of a simplicial complex $K$. The simplicial cohomology $(C^k(K), \partial^*)$ has been interpreted as discrete outer calculus (DEC) by Desbrun and Hirani (2003, 2005). We define spaces $P^{-1}\Omega^k \subset L^\infty\Omega^k$ and outer differentials and give an isometric cochain map $C^k \to P^{-1}\Omega^k$ (9.11). This reduces the computation of variational problems in discrete outer calculus to variational problems in a trial space of non-conforming differential forms. We investigate the approximation properties of $P^{-1}\Omega^k$ in $H^1\Omega^k$ (9.19, 9.20) and compare the solutions to variational problems in both spaces (10.26--28).(1) Sei $(M,g)$ eine unberandete, kompakte Riemannsche Mannigfaltigkeit und $\Delta$ das $n$-dimensionale Standardsimplex. Für $n+1$ gegebene Punkte $p_i \in M$ betrachten wir mit Karcher (1977) die Funktion \ E: M \times \Delta \to \R, (a,\lambda) \mapsto \lambda^0 d^2(a,p_0) + \dots + \lambda^n d^2(a,p_n), \ worin $d$ der geodätische Abstand in $M$ sei. Liegen alle $p_i$ in einem hinreichend kleinen geodätischen Ball, so ist $x: \lambda \mapsto argmin_a E(a,\lambda)$ eine wohldefinierte Funktion $\Delta \to M$ (5.3). Wir nennen $s := x(\Delta)$ das Karcher-Simplex mit Ecken $p_i$. Auf $\Delta$ sei eine flache Riemannsche Metrik $g^e$ durch Vorgabe von Seitenlängen $d(p_i,p_j)$ definiert. Wenn alle Seitenlängen kleiner als $h$ sind und $vol(\Delta,g^e) \geq \alpha h^n$ für ein $\alpha > 0$ ist, so zeigen wir in 6.17 und 6.23 \begin{equation} %\tag{A.1a} |(x^*g - g^e)(v,w)| \leq c h^2 |v| |w|, \qquad % |(\nabla^{x^*g} - \nabla^{g^e})_v w| \leq c h |v| |w| \end{equation} mit einer nur vom Krümmungstensor $R$ von $(M,g)$ und $\theta$ abhängigen Konstanten $c$. Daraus folgen mit wenig Aufwand Interpolationsabschätzungen für Funktionen $u: s \to \R$ (7.4) und $y: s \to N$ für eine zweite Riemannsche Mannigfaltigkeit $N$ (7.14). Auch erlaubt diese Simplexdefinition, auf Grundlage der Voronoi-Zerlegung von Leibon und Letscher (2000) eine Karcher-Delaunay-Triangulierung zu definieren (8.7). Daher können wir im folgenden ganz $(M,g)$ als trianguliert annehmen. Auf jedem Simplex ist $g$ durch eine Metrik $g^e$ mit (A1.a) approximiert, und schwach differenzierbare Funktion $u \in H^1(M,g)$ lassen sich durch stückweise polynomielle $u_h \in P^1(M)$ approximieren. In der üblichen Weise (Dziuk 1988, Holst und Stern 2012) lassen sich daher Variationsprobleme wie das Poissonproblem (10.13, 10.17, 13.14) oder die Hodge-Zerlegung (10.15) in $H^1(M,g)$ mit denjenigen in $H^1(M,g^e)$ und ihren Galerkin-Approximationen in $P^1(M)$ vergleichen. Anknüpfend an die gängige Finite-Elemente-Theorie für Probleme auf Untermannigfaltigkeiten des $\R^m$ lassen sich auch Untermannigfaltigkeiten $S \subset M$ durch Karcher-Simplexe approximieren. Der dabei auftretende Geometriefehler ist gleich dem für Untermannigfaltigkeiten des $\R^m$ zuzüglich eines Terms $ch^2$ (11.18). (2) Sei $M$ die geometrische Realisierung eines simplizialen Komplexes $K$. Die simpliziale Kohomologie $(C^k(K), \partial^*)$ ist von Desbrun und Hirani (2003, 2005) als diskretes äußeres Kalkül (DEC) interpretiert worden. Wir definieren Räume $P^{-1}\Omega^k \subset L^\infty\Omega^k$ und äußere Differentiale und geben eine isometrische Kokettenabbildung $C^k \to P^{-1}\Omega^k$ an (9.11). Damit ist die Berechnung von Variationsproblemen im diskreten äußeren Kalkül auf Variationsprobleme in einem Raum von nicht- konformen Ansatz-Differentialformen zurückgeführt. Wir untersuchen die Approximationseigenschaften von $P^{-1}\Omega^k$ in $H^1\Omega^k$ (9.19, 9.20) und vergleichen die Lösungen von Variationsproblemen in ihnen (10.26--28)

Generalised Sorkin-Johnston and Brum-Fredenhagen States for Quantum Fields on Curved Spacetimes

The presented work contains a new construction of a class of distinguished quasifree states for the scalar field and Proca field on globally hyperbolic spacetimes. Our idea is based on the axiomatic construction of the Sorkin-Johnston (SJ) state \cite{Sorkin:2017fcp}; we call these states \emph{generalised SJ states}. We give a concrete application of this framework with the construction of the `thermal' SJ state. By slightly modifying the construction of generalised SJ states, we also introduce a new class of Hadamard states, which we call \emph{generalised SJ states with softened boundaries}. We show when these states satisfy the Hadamard condition and compute the Wick polynomials. Finally we construct the SJ and Brum-Fredenhagen (BF) states for the Proca field on ultrastatic slabs with compact spatial sections. We show that the SJ state construction fails for the Proca field, yet the BF state is well defined and, moreover, satisfies the Hadamard condition

Local coderivatives and approximation of Hodge Laplace problems

The standard mixed finite element approximations of Hodge Laplace problems associated with the de Rham complex are based on proper discrete subcomplexes. As a consequence, the exterior derivatives, which are local operators, are computed exactly. However, the approximations of the associated coderivatives are nonlocal. In fact, this nonlocal property is an inherent consequence of the mixed formulation of these methods, and can be argued to be an undesired effect of these schemes. As a consequence, it has been argued, at least in special settings, that more local methods may have improved properties. In the present paper, we construct such methods by relying on a careful balance between the choice of finite element spaces, degrees of freedom, and numerical integration rules. Furthermore, we establish key convergence estimates based on a standard approach of variational crimes