Myers and Hawking theorems: Geometry for the limits of the Universe

Abstract

It is known that the celebrated theorem by Hawking which assures the existence of a Big-Bang under physically motivated hypotheses, uses geometric ideas inspired in classical Myers theorem. Our aim here is to go a step further: first, a result which can be interpreted as the exact analogy in pure Riemannian geometry to Hawking theorem will be proven and, then, the isomorphic role of the hypotheses in both theorems will be analyzed. This will provide some interesting links between Riemannian and Lorentzian geometries, as well as an introduction to the latter. The reader interested only in Riemannian Geometry can regard this new result as a simple application of Myers theorem combined with the properties of focal points. However, readers with broader perspectives will learn that when a geometer thinks in our space as a complete Riemannian manifold, a relativist may think in our spacetime as predictable, or that suitable bounds on the Ricci tensor will force geodesics either to converge in the space or to join in the time. Moreover, the limitation of the distance from any point to a hypersurface in a Riemannian manifold, may turn out into a catastrophic relativistic limit for the duration of our physical Universe.