Abstract
The idea of “asymptotically free” gravity is implemented using a constrained mimetic scalar field. The effective gravitational constant is assumed to vanish at some limiting curvature. As a result singularities in spatially flat Friedmann and Kasner universes are avoided. Instead, the solutions in both cases approach a de Sitter metric with limiting curvature. We show that quantum metric fluctuations vanish when this limiting curvature is approached.
Highlights
In [1] mimetic matter was introduced utilizing reparametrization of the physical metric gμν in terms of an auxiliary metric hμν and a scalar field φ in the form gμν = hμν hαβ φ,αφ,β (1)This definition implies that φ identically satisfies gμν φ,μφ,ν = 1. (2)Because the physical metric is invariant under Weyl transformations of hμν, the trace of the equations obtained by variation of the Einstein action with respect to the metric vanishes identically
The constrained scalar field allows us to build invariants which in synchronous coordinates can be expressed exclusively in terms of first order time derivatives of the metric. This opens the possibility to modify General Relativity in a simple way avoiding problematic higher order time derivative terms which generically lead to ghost degrees of freedom
Such a generalization of Einstein theory happens to be very interesting and allows us for example to implement the idea of limiting curvature and resolve spacelike singularities in Friedmann and Kasner universes as well as in black holes
Summary
In [1] mimetic matter was introduced utilizing reparametrization of the physical metric gμν in terms of an auxiliary metric hμν and a scalar field φ in the form gμν = hμν hαβ φ,αφ,β (1). This definition implies that φ identically satisfies gμν φ,μφ,ν = 1. This quantity is the only measure of curvature G can depend on without introducing higher time derivatives in the modified Einstein equation. Assuming that G vanishes at some limiting curvature characterized by ( φ)2L we will implement in this way the idea of “asymptotic freedom” for gravity and investigate its possible consequences
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