Abstract
We prove the equivalence between non-local gravity with an arbitrary form factor and a non-local gravitational system with an extra rank-2 symmetric tensor. Thanks to this reformulation, we use the diffusion-equation method to transform the dynamics of renormalizable non-local gravity with exponential operators into a higher-dimensional system local in spacetime coordinates. This method, first illustrated with a scalar field theory and then applied to gravity, allows one to solve the Cauchy problem and count the number of initial conditions and of non-perturbative degrees of freedom, which is finite. In particular, the non-local scalar and gravitational theories with exponential operators are characterized by, respectively, two and four initial conditions in any dimension and, respectively, by one and eight degrees of freedom in four dimensions. The fully covariant equations of motion are written in a form convenient to find analytic non-perturbative solutions.
Highlights
The authors did not realize that some of the initial conditions indicated to solve the Cauchy problem in nonlocal systems are not independent, so that the number of initial conditions is reduced from 4 to 2 in the case of the scalar field
Section 1, page 4: “The counting of initial conditions and degrees of freedom is carried out in section 2.3, where we find that this number is, respectively, 4 and 1 for the real non-local scalar” should be “The counting of initial conditions and degrees of freedom is carried out in section 2.3, where we find that this number is, respectively, 2 and 1 for the real non-local scalar”
Given a non-local action with exponential non-locality for tensorial field φμν··· representing n physical degrees of freedom, the diffusion-equation method relies on a second-order localized system for a field Φμν··· and an auxiliary field χμν··· with the same symmetry properties as φ, leading to 2n initial conditions.” should be “In particular, these conditions are valid at r = βr∗, where, χ is fully determined once φ is known
Summary
Erratum: Initial conditions and degrees of freedom of non-local gravity
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