Metric transformations: disformal invariance and mimetic symmetries

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I will talk about transformations of the metric within Scalar-tensor theories of gravity. Quite often they are seen just as a mathematical trick. However, in fact that they have helped to clarify many aspects of scalar-tensor theories.

The simplest and very well-known metric transformation is a local rescaling of the metric, a.k.a. conformal transformations. Bekenstein generalised them including a preferred direction given by the gradient of a scalar field. In this talk, I will focus on such a generalisation, so-called disformal transformation. I will show some examples in which disformal transformations naturally appear. In particular, they are of interest in Horndeski theory and its extensions. I will further discuss how the structure of the theory changes (or not) under such transformations. First, I will show that, as in the case of a conformal transformation, observables are invariant under a disformal transformation as well. Second, by means of Hamiltonian analysis, I will give some insight into the structure of generalised scalar-tensor theories.

There is one special case though, called Mimetic gravity, in which the transformation is “singular”. In that case, the transformation bears some inner symmetry which gives rise to an extra degree of freedom in the theory.