Abstract
We discuss field theories appearing as a result of applying field transformations with derivatives (differential field transformations, DFTs) to a known theory. We begin with some simple examples of DFTs to see the basic properties of the procedure. In this process, the dynamics of the theory might either change or be conserved. After that, we concentrate on the theories of gravity which appear as a result of various DFTs applied to general relativity, namely the mimetic gravity and Regge–Teitelboim embedding theory. We review the main results related to the extension of dynamics in these theories, as well as the possibility to write down the action of a theory after DFTs as the action of the original theory before DFTs plus an additional term. Such a term usually contains some constraints with Lagrange multipliers and can be interpreted as an action of additional matter, which might be of use in cosmological applications, e.g., for the explanation of the effects of dark matter.
Highlights
The choice of a proper parameterization is arguably one of the most crucial to solving almost all problems in the theory of gravity
It must be stressed that here and hereafter we use the concept of conservation of dynamics in the following sense: The dynamics of the theory after the field transformation is conserved if the amount of initial values necessary to solve the equations of motion (EoM) of the theory in the new variables coincide with the amount of initial values in the original ones
It is worth noting that, if the connection in the original action in Equation (24) is not a priori symmetric, the situation drastically changes: the original theory is not equivalent to general relativity (GR) anymore and contains additional degrees of freedom [4], whereas differential field transformations (DFTs) transforms the theory into GR, i.e., we arrive at a case of the restricted dynamics
Summary
The choice of a proper parameterization is arguably one of the most crucial to solving almost all problems in the theory of gravity. It allows one to lower the order of derivatives in the gravitational action, which often turns out very helpful, and addresses many other issues in the theory of gravity, e.g., its quantization [3] and unification with other theories (for a historical survey, see [4] and the references therein; see [5]). Another early attempt to reformulate GR in terms of alternate variables was made by Cartan [6]. We briefly discuss the generalization of the mimetic transformation called disformal transformation
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