Related papers: Einstein on Involutions in Projective Geometry
Enumerative geometry, the art and science of counting geometric objects satisfying geometric conditions, has seen a resurgence of activity in recent years due to an influx of new techniques that allow for enriched computations. This paper…
Hilbert's paper on ``The Foundations of Physics (First Communication),'' is now primarily known for its parallel publication of essentially the same gravitational field equations of general relativity which Einstein published in a note on…
Euclidean geometry is among the earliest forms of mathematical thinking. While the geometric primitives underlying its constructions, such as perfect lines and circles, do not often occur in the natural world, humans rarely struggle to…
These are expanded notes from lectures given at the \'{E}tats de la Recherche workshop on "Derived algebraic geometry and interactions". These notes serve as an introduction to the emerging theory of Poisson structures on derived stacks.
Albert Einstein postulated the equivalence of energy and mass, developed the theory of special relativity, explained the photoelectric effect, and described Brownian motion in five papers, all published in 1905, 100 years ago. With these…
Einstein's theory of relativity is largely thought of as one of the most important discoveries of the 20$^{th}$ century and continues to pass observational tests over 100 years later. Yet, it is Newtonian gravity, a 350 year old formalism…
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.
This work analyzes the difficulties in learning and teaching Einstein's theory of special relativity. An extensive bibliographic review has been performed, considering articles published in the most relevant journals on science education,…
These notes represent approximately one semester's worth of lectures on introductory general relativity for beginning graduate students in physics. Topics include manifolds, Riemannian geometry, Einstein's equations, and three applications:…
We investigate connections between the geometry of linear subspaces and the convergence of the alternating projection method for linear projections. The aim of this article is twofold: in the first part, we show that even in Euclidean…
In this expository paper, we present a survey about the history of the geometrization conjecture and the background material on the classification of Thurston's eight geometries. We also discuss recent techniques for immersive visualization…
Patterson discussed the curvature identities on Riemannian manifolds in [14], and a curvature identity for any 6-dimensional Riemannian manifold was independently derived from the Chern-Gauss-Bonnet Theorem [8]. In this paper, we provide…
After a historical discussion of Einstein's 1907 principle of equivalence, a homogeneous gravitational field in Minkowski spacetime is constructed. It is pointed out that the reference frames in gravitational theory can be understood as…
In the class of metrics of a generic conformal structure there exists a distinguishing metric. This was noticed by Albert Einstein in a lesser-known paper of 1921 (Berl. Ber., 1921, pp. 261-264). We explore this finding from a geometrical…
The starting point of this work is the original Einstein action, sometimes called the Gamma squared action. Continuing from our previous results, we study various modified theories of gravity following the Palatini approach. The metric and…
In his 1916's first paper on gravitational waves Einstein began to speculate on interactions between the principles of the old quantum theory and his theory of gravitation. With this contribution Einstein has stimulated a lot of similar…
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…
We establish an explicit correspondence between two--dimensional projective structures admitting a projective vector field, and a class of solutions to the $SU(\infty)$ Toda equation. We give several examples of new, explicit solutions of…
In 1640's, Blaise Pascal discovered a remarkable property of a hexagon inscribed in a conic - Pascal Theorem, which gave birth of the projective geometry. In this paper, a new geometric invariant of algebraic curves is discovered by a…
Einstein Equivalence Principle (EEP) requires all matter components to universally couple to gravity via a single common geometry: that of spacetime. This relates quantum theory with geometry as soon as interactions with gravity are…