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This brief review is intended to introduce gravitational physicists to recent developments in which general relativity is being used to describe certain aspects of condensed matter systems, e.g., superconductivity.
We prove that the Einstein equations can be solved in a very general form for arbitrary spacetime dimensions and various types of vacuum and non-vacuum cases following a geometric method of anholonomic frame deformations for constructing…
Gaussian graphical models are used throughout the natural sciences, social sciences, and economics to model the statistical relationships between variables of interest in the form of a graph. We here provide a pedagogic introduction to…
This document gathers the notes of a 30-hour review course on gravitation, which I delivered in January 2018 and January 2019 at the African Institute for Mathematical Sciences of Cameroon (AIMS-Cameroon). Its main goal is to propose a big…
The aim of this work is to use the notions of Riemann's geometry introduced in Part I, to analyze the foundations of Einstein's theory of general relativity.
We review recent progress in taking the large dimension limit of Einstein's equations. Most of our analysis is classical in nature and concerns situations where there is a black hole horizon although we briefly discuss various extensions…
There are various types of global and local spacetime invariant in general relativity. Here I focus on the local invariants obtainable from the curvature tensor and its derivatives. The number of such invariants at each order of…
This paper presents a brief but comprehensive introduction to certain mathematical techniques in General Relativity. Familiar mathematical procedures are investigated taking into account the complications of introducing a non trivial…
Using ideas from continuum mechanics we construct a theory of gravity. We show that this theory is equivalent to Einstein's theory of general relativity; it is also a much faster way of reaching general relativity than the conventional…
After a brief summary of the foundations of general relativity, we will concentrate on the stationary exact solutions of the Einstein and Einstein-Maxwell equations. A number of these solutions can be interpreted as black holes,…
In this article, we discuss the idea of gravitational lensing, from a systematic, historical and didactic point of view. We show how the basic lensing equation together with the concepts of geometrical optics opens a space of implications…
In this manuscript we review the theoretical foundations of gravitational waves in the framework of Albert Einstein's theory of general relativity. Following Einstein's early efforts we first derive the linearised Einstein field equations…
We consider the interaction of gravity, as expressed by Einstein's Equations of General Relativity, to other force fields. We describe some recent results, discussing both the mathematics, and the physical interpretations. These results…
We present in the most natural way, that is, in the context of the theory of vector and principal bundles and connections in them, fundamental geometrical concepts related to General Relativity and one of its extensions, the Einstein-Cartan…
We provide an introduction to selected recent advances in the mathematical understanding of Einstein's theory of gravitation.
It is well known that Einstein's General Relativity (GR) achieved a great success and overcame lots of experimental tests. On the other hand, GR also showed some shortcomings and flaws which today advise theorists to ask if it is the…
We are entering an era where the numerical construction of generic spacetimes is becoming a reality. The use of computer simulations, in principle, allows us to solve Einstein equations in their full generality and unravel important…
Based on the distinction between the covariant and contravariant metric tensor components in the framework of the affine geometry approach and also on the choice of the contravariant components, it was shown that a wide variety of third,…
The study of general two dimensional models of gravity allows to tackle basic questions of quantum gravity, bypassing important technical complications which make the treatment in higher dimensions difficult. As the physically important…
A general framework for the solutions of the constraints of pure gravity is constructed. It provides with well defined mathematical criteria to classify their solutions in four classes. Complete families of solutions are obtained in some…