General relativity from five dimensional Chern-Simons theory

We derive general relativity (GR) from a five dimensional Chern-Simons theory defined on a manifold with a boundary. The underlying mechanism is not a Kaluza-Klein reduction, rather, GR appears as an effective theory at the (four dimensional) boundary. It has been suggested many times that general relativity (GR) may not be a fundamental theory but only an effective theory arising from a still unknown well-defined quantum field theory. In this paper we explore in this direction. We show that Einstein equations in four dimensions arise in a natural way from a five dimensional Chern-Simons theory. It is a well known fact that reducing the fields in a given action before varying it, does not necessarily give the same equations of motion if one first vary and then make the reduction. Normally, one would say that the correct equations of motion are those following from the variation of the full action, that is before imposing any condition on the fields. There are cases, however, in which one does not know which is the correct or physically relevant action; it may happen that imposing conditions on the fields before varying the action may give rise to an interesting theory. We consider in this paper a five dimensional Chern-Simons theory for the group ISO(3, 2) or ISO(4, 1), depending on the sign of the cosmological constant. The action is defined on a five dimensional manifold M which has a boundary denoted by ∂M . We shall impose