Ultra-relativistic scaling limits of multi-metric theories

Abstract

Lie algebra expansion is an elegant method to obtain higher dimensional algebras and using this method one can write some interesting ultra-relativistic gravitational theories beginning from the Poincar\'e algebra. This method has been used in many other studies, for which we will provide some explanations and references for these studies in the following sections. In this work, we will first give a brief introduction to the gauge theories, which are necessary in order to realize the gravitational theories as a gauge theory, especially the algebraic structure of gravitational theories. This is crucial for many gravity theories, such as supergravity. After that, we will review the general aspects of differential geometry shortly. This will give us the main mathematical framework to study gravity as a gauge theory. Even though we will not be able to review all the related topics, we will go through all the necessary topics in order to gather all the necessary information. Thirdly, we will try to understand the theories of gravity, especially general relativity, as a gauge theory. After a simple introduction to the second-order formalism of GR, we will define the first-order formalism and its action. After that, we will obtain GR beginning from the Poincar\'e algebra and by gauging this algebra. After that, we attempt to gain a few knowledge about bi-metric/bi-gravity theories by examining some specific examples. Of course these examination will not be enough to understand the subjects such as Massive Gravity and Bi-Gravity however, we hope that readers can follow easily. As we move on to spoken chapter, we will give a detailed reference for the people who are encouraged to learn the subject. As a last section, we will focus on scaling limit, specifically algebraic aspects of it. We will first show that gravity theories can be obtained by using Lie Algebra Expansion. In this section we start with Poincar\'e algebra then divide its generators to two subspaces. After that, we will obtain Carroll Gravity as a gauge theory. Then, we will not stop in that point and push forward to a generalization of it by giving the method to obtain the same results by contraction of a multi-metric theory by illustrating this procedure with a few examples.