The null distance of Sormani and Vega encodes the manifold topology as well as the causality structure of a (smooth) spacetime. We extend this concept to Lorentzian length spaces, the analog of (metric) length spaces, which generalize Lorentzian caus
ality theory beyond the manifold level. We then study Gromov-Hausdorff convergence based on the null distance in warped product Lorentzian length spaces and prove first results on its compatibility with synthetic curvature bounds.
The main aim of the present work is to arrive at a mathematical theory close to the historically original conception of generalized functions, i.e. set theoretical functions defined on, and with values in, a suitable ring of scalars and sharing a num
ber of fundamental properties with smooth functions, in particular with respect to composition and nonlinear operations. This is how they are still used in informal calculations in Physics. We introduce a category of generalized functions as smooth set-theoretical maps on (multidimensional) points of a ring of scalars containing infinitesimals and infinities. This category extends Schwartz distributions. The calculus of these generalized functions is closely related to classical analysis, with point values, composition, non-linear operations and the generalization of several classical theorems of calculus. Finally, we extend this category of generalized functions into a Grothendieck topos of sheaves over a concrete site. This topos hence provides a suitable framework for the study of spaces and functions with singularities. In this first paper, we present the basic theory; subsequent ones will be devoted to the resulting theory of ODE and PDE.
We prove existence and uniqueness of a solution to the Cauchy problem corresponding to the equation begin{equation*} begin{cases} partial_t u_{varepsilon,delta} +mathrm{div} {mathfrak f}_{varepsilon,delta}({bf x}, u_{varepsilon,delta})=varepsilon Del
ta u_{varepsilon,delta}+delta(varepsilon) partial_t Delta u_{varepsilon,delta}, {bf x} in M, tgeq 0 u|_{t=0}=u_0({bf x}). end{cases} end{equation*} Here, ${mathfrak f}_{varepsilon,delta}$ and $u_0$ are smooth functions while $varepsilon$ and $delta=delta(varepsilon)$ are fixed constants. Assuming ${mathfrak f}_{varepsilon,delta} to {mathfrak f} in L^p( mathbb{R}^dtimes mathbb{R};mathbb{R}^d)$ for some $1<p<infty$, strongly as $varepsilonto 0$, we prove that, under an appropriate relationship between $varepsilon$ and $delta(varepsilon)$ depending on the regularity of the flux ${mathfrak f}$, the sequence of solutions $(u_{varepsilon,delta})$ strongly converges in $L^1_{loc}(mathbb{R}^+times mathbb{R}^d)$ towards a solution to the conservation law $$ partial_t u +mathrm{div} {mathfrak f}({bf x}, u)=0. $$ The main tools employed in the proof are the Leray-Schauder fixed point theorem for the first part and reduction to the kinetic formulation combined with recent results in the velocity averaging theory for the second.
We introduce an analogue of the theory of length spaces into the setting of Lorentzian geometry and causality theory. The r^ole of the metric is taken over by the time separation function, in terms of which all basic notions are formulated. In this w
ay we recover many fundamental results in greater generality, while at the same time clarifying the minimal requirements for and the interdependence of the basic building blocks of the theory. A main focus of this work is the introduction of synthetic curvature bounds, akin to the theory of Alexandrov and CAT$(k)$-spaces, based on triangle comparison. Applications include Lorentzian manifolds with metrics of low regularity, closed cone structures, and certain approaches to quantum gravity.
We show that the Hawking--Penrose singularity theorem, and the generalisation of this theorem due to Galloway and Senovilla, continue to hold for Lorentzian metrics that are of $C^{1, 1}$-regularity. We formulate appropriate wea
Generalized smooth functions are a possible formalization of the original historical approach followed by Cauchy, Poisson, Kirchhoff, Helmholtz, Kelvin, Heaviside, and Dirac to deal with generalized functions. They are set-theoretical functions defin
ed on a natural non-Archimedean ring, and include Colombeau generalized functions (and hence also Schwartz distributions) as a particular case. One of their key property is the closure with respect to composition. We review the theory of generalized smooth functions and prove both the local and some global inverse function theorems.
We extend the validity of the Penrose singularity theorem to spacetime metrics of regularity $C^{1,1}$. The proof is based on regularisation techniques, combined with recent results in low regularity causality theory.
We introduce the notion of functionally compact sets into the theory of nonlinear generalized functions in the sense of Colombeau. The motivation behind our construction is to transfer, as far as possible, properties enjoyed by standard smooth functi
ons on compact sets into the framework of generalized functions. Based on this concept, we introduce spaces of compactly supported generalized smooth functions that are close analogues to the test function spaces of distribution theory. We then develop the topological and functional analytic foundations of these spaces.
We provide a detailed proof of Hawkings singularity theorem in the regularity class $C^{1,1}$, i.e., for spacetime metrics possessing locally Lipschitz continuous first derivatives. The proof uses recent results in $C^{1,1}$-causality theory and is b
ased on regularisation techniques adapted to the causal structure.
We show that many standard results of Lorentzian causality theory remain valid if the regularity of the metric is reduced to $C^{1,1}$. Our approach is based on regularisations of the metric adapted to the causal structure.
