New representation of the nonlocal ghost-free gravity theory
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ew Representation of the Nonlocal GhostFree Gravity Theory1 A. O. Barvinskya and Yu. V. Gusevb, c a
Theory Department, Lebedev Physics Institute, Leninsky Prospect 53, Moscow, 119991 Russia b Lebedev Research Center in Physics, Leninsky Prospect 53, Moscow, 119991 Russia cIRMACS Centre, Simon Fraser University, 8888 University Drive, Burnaby, B.C. V5A 1S6 Canada Abstract—A new representation is found for the action of the recently suggested ghostfree nonlocal gravity models generating de Sitter or Antide Sitter background with an arbitrary value of the effective cosmological constant. This representation allows one to extend applications of these models from maximally symmetric to generic Einstein spaces and black hole solutions, but clearly indicates violation of the general relativistic limit in this class of theories, induced by their infrared behavior. It is shown that this limit can be recovered in a special conformal frame of these theories, and their relation to critical gravity models is also briefly dis cussed. DOI: 10.1134/S1063779613020068 1
1. INTRODUCTION
A new approach to the dark energy problem, that has recently been suggested in [1], is inspired by the necessity to avoid the fine tuning problem. This approach suggests the theory in which the de Sitter or antide Sitter evolution can occur at any value of the effective cosmological constant Λ⎯the antithesis to the dark energy scale encoded in the action of the model and fine tuned to the observational data. A con crete observable value of Λ in this theory is supposed to be selected by the mechanism analogous to symmetry breaking [1]. Interestingly, the realization of this approach quite unexpectedly has also led to the ana logue of the dark matter phenomenon characterized at large distances by gravitational attraction stronger than in general relativity or Newton theory. The action of this theory was shown to generate vacuum equations of motion which have as a solution the de Sitter or antide Sitter background. This back ground bears only transversetraceless gravitons as propagating physical modes and is free from ghost instabilities. The stability property was proven in [1] by very extensive calculations for a maximally symmetric background and then extended in [2] to generic Ein stein spaces Rμν = Λgμν with a vanishing traceless part of the Ricci tensor E μν ≡ R μν – 1 g μν R = 0, 4
(1.1)
Thus, this model could be regarded as one of the first modifications of the Einstein theory made by Einstein himself, who for reasons of unification with electro magnetism suggested to replace the Einstein tensor
Gμν = Rμν – 1/2gμνR in the left hand side of Einstein equations by Eμν [3]. The action with these properties is the following 2
nonlocal functional of the spacetime metric gμν, 2 ⎫ 1/2 ⎧ μν M 1 S = dxg ⎨ –R + αR G μν ⎬, ˆ 2 䊐+P ⎩ ⎭
∫
ˆ ≡ P μν = aR ( μν ) + b ( g R μν + g μν R ) P αβ ( αβ ) αβ αβ (μ
ν)
+ cR ( α δ β ) + dRg αβ g
μν
μν
(1.2)
(1.3)
+ eRδ αβ ,
where the hat denotes matrices acting on symmetric tenso
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