Related papers: Interpreting Mass via Riemannian Polyhedra
In this article, we classify the set of asymptotic mass-like invariants for asymptotically hyperbolic metrics. It turns out that the standard mass is just one example (but probably the most important one) among the two families of…
Some geometric structures with associated Riemannian metrics have been considered in the book.
The correct characterization of the concept of mass point in general relativity is a straightforward consequence of the original form of solution given by Schwarzschild to the problem of the Einstein field of a material point.
The notion of center of mass, which is very useful in kinematics, proves to be very handy in geometry (see [1]-[2]). Countless applications of center of mass to geometry go back to Archimedes. Unfortunately, the center of mass cannot be…
The general relativistic notion of gravitational and inertial mass is discussed from the general viewpoint of the tidal forces implicit in the curvature and the Einstein field equations within ponderable matter. A simple yet rigorously…
In this paper, we present a series of techniques to describe General Relativity using Geometric Algebra (GA). We emphasize the physical interpretation of quantities and provide a step-by-step guide for performing calculations. In doing so,…
An explicit lower bound for the mass of an asymptotically flat Riemannian 3-manifold is given in terms of linear growth harmonic functions and scalar curvature. As a consequence, a new proof of the positive mass theorem is achieved in…
A notion of general manifolds is introduced. It covers all usual manifolds in mathematics. Essentially, it is a way how to get a bigger 'fibration' over a site which locally coincides with a given one. An enrichment with generalized…
We define a geometric quantity for asymptotically hyperbolic manifolds, which we call the volume-renormalized mass. It is essentially a linear combination of the ADM mass surface integral and a renormalization of the volume. We show that…
These notes on Riemannian geometry use the bases bundle and frame bundle, as in Geometry of Manifolds, to express the geometric structures. It has more problems and omits the background material. It starts with the definition of Riemannian…
The four-dimensional gauge group of general relativity corresponds to arbitrary coordinate transformations on a four-manifold. Theories of gravity with a dynamical structure remarkably like Einstein's theory can be obtained on the basis of…
In this article we show how holomorphic Riemannian geometry can be used to relate certain submanifolds in one pseudo-Riemannian space to submanifolds with corresponding geometric properties in other spaces. In order to do so, we shall first…
We extend Witten's spinor proof of the positive mass theorem to large classes of complete asymptotically flat non-spin manifolds, including all manifolds of dimension less than or equal to 11 and all manifolds of dimension less than 26…
The set of all metrics that can be placed on a given manifold defines an infinite-dimensional `superspace' that can itself be imbued with the structure of a Riemannian manifold. Geodesic distances between points on Met$(M)$ measure how…
General results on asymptotic expansions of Feynman diagrams in momenta and/or masses are reviewed. It is shown how they are applied for calculation of massive diagrams.
In this report I discuss the relations between systoles and volumes of hyperbolic manifolds and a conjecture of Lehmer about the Mahler measure of non-cyclotomic polynomials.
We define a physically reasonable mass for an asymptotically Robertson-Walker (ARW) manifold which is uniquely defined in the case of a normalized representation.
In this paper, we investigate the geometry of Einstein-type equation on a Riemannian manifold, unifying various particular geometric structures recently studied in the literature, such as critical point equation and vacuum static equation.…
Modern machine learning increasingly leverages the insight that high-dimensional data often lie near low-dimensional, non-linear manifolds, an idea known as the manifold hypothesis. By explicitly modeling the geometric structure of data…
An original way of presentation of the Schwarzschild black hole in the form of a point-like mass with making the use of the Dirac $\delta$-function, including a description of a continuous collapse to such a point mass, is given. A…