Matter couplings in theories of gravity beyond General Relativity

In Einstein's theory of General Relativity it is well understood how to couple matter fields to gravity. The standard approach is often referred to as the principle of minimal coupling which contains two steps. First, the equations describing the matter are formulated in Minkowski space which means that time and space are no longer distinct concepts but become part of a 4-dimensional Lorentzian manifold, flat initially. Second, one replaces partial derivatives with covariant derivatives and the Minkowski metric with an arbitrary spacetime metric.

This process, however, can yield different gravitational theories when one considers the metric and connection as independent variables. When applied to Dirac fields one naturally arrives at Einstein-Cartan theory, a theory well known since the 1920s with much renewed interesting from the 1960s onwards. It is well known that the electromagnetic field cannot be coupled minimally in Einstein-Cartan theory as the torsion field would couple to the electromagnetic potential, thereby breaking its gauge invariance. The also applied to non-Abelian gauge fields.

The project will investigate the minimal coupling procedure in various modified theories of gravity and study the resulting field equations. Various recently proposed theories of gravity have unusual properties, for instance, some models are no longer locally Lorentz invariant but are invariant under global Lorentz transformations. The matter coupling procedure in such models is known to be subtle but it is possible to prescribe a consistent coupling procedure which return the general relativistic equations in the appropriate limit. Less is know about models which can loose diffeomorphism invariance on small scales.

In addition to minimal couplings one can also consider non-minimal couplings. By this one means any interaction term involving the matter field and a geometrical quantity, like the curvature scalar. However, one can also consider a variety of other non-minimal couplings where the matter interacts with a boundary term, for example, which may not necessarily have a geometrical interpretation. This is a relatively new field of research allowing for a large amount of possible models to be investigated.

Research Areas: mathematical sciences -> mathematical physics