Beyond Trinity: The Many Faces of General Relativity

The general theory of relativity gives a geometric description of gravity within the framework of Riemannian geometry. Two other descriptions of the theory have been found based on teleparallel geometries. These three dynamically equivalent formalisms are dubbed as the geometric trinity of gravity. In this paper, we go beyond this so-called trinity and present several new theories that are dynamically equivalent to general relativity. All these theories employ a non-Riemannian geometrical framework. We present a unified treatment in which all equivalent formalisms are treated as subcases of a single general theory. We show that the equivalence between these non-Riemannian theories and general relativity is restricted to only the metric field equations. We extend this treatment to modified gravity theories and underline the dynamical nature of the boundary term. Finally, we analyse the kinematics of curves within these theories and show that the focussing theorem is violated.