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Extended Theories of Gravity can be considered a new paradigm to cure shortcomings of General Relativity at infrared and ultraviolet scales. They are an approach that, by preserving the undoubtedly positive results of Einstein's Theory, is…
Modifications of General Relativity usually include extra dynamical degrees of freedom, which to date remain undetected. Here we explore the possibility of modifying Einstein's theory by adding solely nondynamical fields. With the minimal…
It is a little known fact that while he was developing his theory of general relativity, Einstein's initial idea was a variable speed of light theory. Indeed space-time curvature can be mimicked by a speed of light $c(r)$ that depends on…
We show that Einstein's $R^{\hat{0} \hat{0}}$ equation for nonrelativistic matter and strong gravitational fields is identical with Newton's equation for relative radial acceleration of neighbouring freefalling particles, spherically…
Spacetime curvature plays the primary role in general relativity but Einstein later considered a theory where torsion was the central quantity. Just as the Einstein-Hilbert action in the Ricci curvature scalar R can be generalized to f(R)…
Attempts to merge Einsteinian gravity with Newtonian run into inconsistencies because in Newton's gravity time is absolute and the speed of gravity is infinite. Such an assumption was in a focus of attention of scientists in 19th century…
The existence of gravitational radiation is a natural prediction of any relativistic description of the gravitational interaction. In this chapter, we focus on gravitational waves, as predicted by Einstein's general theory of relativity.…
The mantra about gravitation as curvature is a misnomer. The curvature tensor for a standard of rest does not describe acceleration in a gravitational field but the \underline{gradient} of the acceleration (e.g. geodesic deviation). The…
Einstein Gravity in 2+1 dimensions arises as a consequence of the equations of motion of a gauge model in an external metric. Newton's constant appears as an order parameter of a spontaneously broken discrete symmetry. Matter is coupled in…
In relativistic mechanics the energy-momentum of a free point mass moving without acceleration forms a four-vector. Einstein's celebrated energy-mass relation E=mc^2 is commonly derived from that fact. By contrast, in Newtonian mechanics…
The confrontation between Einstein's theory of gravitation and experiment is summarized. Although all current experimental data are compatible with general relativity, the importance of pursuing the quest for possible deviations from…
Einstein said that gravity is an acceleration like any other acceleration. But gravity causes relativistic effects at non-relativistic speeds; so gravity could have relativistic origins. And since the strong force is thought to cause most…
We present three natural but distinct formalisations of Einstein's special principle of relativity, and demonstrate the relationships between them. In particular, we prove that they are logically distinct, but that they can be made…
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…
One of the highlight of this note is that the author presents the relativistic gravity field that Einstein was looking for. The field is a byproduct of the matter in motion. This field can include both the discrete and continuous…
General Relativity is today the best theory of gravity addressing a wide range of phenomena. Our understanding of physical laws, from cosmology to local scales, cannot be properly formulated without taking into account it. It is based on…
We review the foundations of Einstein's general theory of relativity, discuss recent progress in the tests of relativistic gravity, and present motivations for new generation of high-accuracy gravitational experiments. We discuss the…
The accepted idea that the expansion of the universe is accelerating needs, for compatibility to general relativity, the introduction of some unusual forms of matter. However, several authors have proposed that instead of making weird…
It is very likely that the quantum description of spacetime is quite different from what we perceive at large scales, $l\gg (G\hbar/c^3)^{1/2}$. The long wave length description of spacetime, based on Einstein's equations, is similar to the…
This article reviews the concept of Lorentz invariant relative velocity that is often misunderstood or unknown in high energy physics literature. The properties of the relative velocity allow to formulate the invariant flux and cross…