Abstract
Quasi-topological terms in gravity can be viewed as those that give no contribution to the equations of motion for a special subclass of metric ansätze. They therefore play no rôle in constructing these solutions, but can affect the general perturbations. We consider Einstein gravity extended with Ricci tensor polynomial invariants, which admits Einstein metrics with appropriate effective cosmological constants as its vacuum solutions. We construct three types of quasi-topological gravities. The first type is for the most general static metrics with spherical, toroidal or hyperbolic isometries. The second type is for the special static metrics where gttgrr is constant. The third type is the linearized quasitopological gravities on the Einstein metrics. We construct and classify results that are either dependent on or independent of dimensions, up to the tenth order. We then consider a subset of these three types and obtain Lovelock-like quasi-topological gravities, that are independent of the dimensions. The linearized gravities on Einstein metrics on all dimensions are simply Einstein and hence ghost free. The theories become quasi-topological on static metrics in one specific dimension, but non-trivial in others. We also focus on the quasi-topological Ricci cubic invariant in four dimensions as a specific example to study its effect on holography, including shear viscosity, thermoelectric DC conductivities and butterfly velocity. In particular, we find that the holographic diffusivity bounds can be violated by the quasi-topological terms, which can induce an extra massive mode that yields a butterfly velocity unbound above.
Highlights
Einstein gravity extended with higher-order Riemann curvature polynomial invariants in general admits multiple vacua that are maximally symmetric spacetimes, such as the Minkowski, de Siter or anti-de Sitter (AdS) spacetimes
We focus on the quasi-topological Ricci cubic invariant in four dimensions as a specific example to study its effect on holography, including shear viscosity, thermoelectric DC conductivities and butterfly velocity
Our theories are all ghost free at the linear order of perturbations around maximally-symmetric spacetimes and Einstein metrics. (If the theory depends on D, the ghost-free condition is satisfied in D dimensions.) In other words, the decoupling of the massive scalar and spin-2 modes yields a general class of linearized quasi-topological Ricci gravities which we call the W series
Summary
We begin to consider Einstein gravity extended with cubic Ricci-tensor polynomial invariants. Note that we introduced κ0, the inverse of Newton’s constant, the bare cosmological constant Λ0 and three coupling constants (e1, e2, e3) associated with the three cubic invariants. 1 + 2 e2R (gμρRνσ − gμσRνρ − gνρRμσ + gνσRμρ) , Pμ3νρσ = 34 e3 gμρRνγ Rσγ − gμσRνγ Rργ − gνρRμγ Rσγ + gνσRμγ Rργ (2.5). It is clear that the theory admits Einstein metrics as vacuum solutions. It is convenient to define an effective cosmological constant Λeff , namely. In this convention, for negative Λeff , the radius of the AdS vacuum is (D − 1)(D − 2).
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