Related papers: The singularity theorems of General Relativity and…

Given a recollement of three proper dg algebras over a noetherian commutative ring, e.g. three algebras which are finitely generated over the base ring, which extends one step downwards, it is shown that there is a short exact sequence of…

The nature of gravity is fundamental to understand the scaffolding of the Universe and its evolution. Einstein's general theory of relativity has been scrutinized for over ninety five years and shown to describe accurately all phenomena…

General relativity successfully describes space-times at scales that we can observe and probe today, but it cannot be complete as a consequence of singularity theorems. For a long time there have been indications that quantum gravity will…

We give an explicit form for the Lorentzian vertices recently introduced for possibly defining the dynamics of loop quantum gravity. As a result of so doing, a natural regularization of the vertices is suggested. The regularized vertices…

In this paper, it is shown why Lorentz Transformation implies the general case where observed events are not necessarily in the inertia frame of any observer but assumes a special scenario when determining the length contraction and time…

In the early 80's S.-T. Yau posed the problem of establishing the rigidity of the Hawking-Penrose singularity theorems. Approaches to this problem have involved the introduction of Lorentzian Busemann functions and the study of the geometry…

In this paper, a geometric resolution of singularities algorithm is developed. This method is elementary in its statement and proof, using explicit coordinate systems as much as possible. Each coordinate change used in the resolution…

Einstein's general theory of relativity is the standard theory of gravity, especially where the needs of astronomy, astrophysics, cosmology and fundamental physics are concerned. As such, this theory is used for many practical purposes…

This paper serves as an introduction to $C^0$ causal theory. We focus on those parts of the theory which have proven useful for establishing spacetime inextendibility results in low regularity -- a question which is motivated by the strong…

Several aspects of regularity theory for parabolic systems are investigated under the effect of random perturbations. The deterministic theory, when strict parabolicity is assumed, presents both classes of systems where all weak solutions…

We provide a detailed proof of Hawking's 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…

In 1945 Einstein concluded that [1]: 'The present theory of relativity is based on a division of physical reality into a metric field (gravitation) on the one hand, and into an electromagnetic field and matter on the other hand. In reality…

We briefly review some results concerning the problem of classical singularities in general relativity, obtained with the help of the theory of differential spaces. In this theory one studies a given space in terms of functional algebras…

We point out that spacetime singularities play a useful role in gravitational theories by eliminating unphysical solutions. In particular, we argue that any modification of general relativity which is completely nonsingular cannot have a…

Our assumption that spacetime is a continuum leads to many challenges in mathematical physics. Singularities, divergent integrals and the like threaten many of our favorite theories, from Newtonian gravity to classical electrodynamics,…

The theories of quantum mechanics and relativity dramatically altered our understanding of the universe ushering in the era of modern physics. Quantum theory deals with objects probabilistically at small scales, whereas relativity deals…

The purpose of this article is to investigate the structure of singular measures from a microlocal perspective. Motivated by the result of De Philippis-Rindler [10] and the notions of wave cone of Murat-Tartar [19,20,26,27] and of…

This paper, which is meant to be a tribute to Minkowski's geometrical insight, rests on the idea that the basic observed symmetries of spacetime homogeneity and of isotropy of space, which are displayed by the spacetime manifold in the…

We present a systematic study of causality theory on Lorentzian manifolds with continuous metrics. Examples are given which show that some standard facts in smooth Lorentzian geometry, such as light-cones being hypersurfaces, are wrong when…

We investigate suitable, physically motivated conditions on spacetimes containing certain submanifolds - the so-called {weakly trapped submanifolds} - that ensure, in a set of neighboring metrics with respect to a convenient topology, that…