Notes on hyperscaling violating Lifshitz and shear diffusion

Abstract

We explore in greater detail our investigations of shear diffusion in hyperscaling violating Lifshitz theories in arXiv:1604.05092 [hep-th]. This adapts and generalizes the membrane-paradigm-like analysis of Kovtun, Son and Starinets for shear gravitational perturbations in the near horizon region given certain self-consistent approximations, leading to the shear diffusion constant on an appropriately defined stretched horizon. In theories containing a gauge field, some of the metric perturbations mix with some of the gauge field perturbations and the above analysis is somewhat more complicated. We find a similar near-horizon analysis can be obtained in terms of new field variables involving a linear combination of the metric and the gauge field perturbation resulting in a corresponding diffusion equation. Thereby as before, for theories with Lifshitz and hyperscaling violating exponents z, θ satisfying z < 4 − θ in four bulk dimensions, our analysis here results in a similar expression for the shear diffusion constant with power-law scaling with temperature suggesting universal behaviour in relation to the viscosity bound. For z = 4 − θ, we find logarithmic behaviour.

Hyperscaling violation, quasinormal modes and shear diffusion

“…This is consistent with D found in [25,27], using a membrane-paradigm-like near horizon analysis (generalizing [32]), and the corresponding guess…”

“…However the correlation functions obtained above, in the Kubo limit, continue to reveal universal behaviour for the viscosity bound with η s = 1 4π as we discuss in section 3.1. In appendix A and B, we provide some technical details, and in appendix C, we review the membrane paradigm approach to the shear diffusion constant studied in [25,27].…”

“…We turn on perturbations of the form e −iωt+iqx h µν (r), e −iωt+iqx a µ (r), and restrict ourselves to radial gauge (h µr = a r = 0). Then, as in [25,27], shear diffusion can be studied by mapping it to charge diffusion in a theory in one lower dimension obtained by compactifying one of the spatial directions x i enjoying translation invariance, say y. Motivated by this, we define the variables…”

“…The first three equations, namely (2.4), (2.5) and (2.6) are the three relevant components of the Einstein equations while (2.7) is the linearized Maxwell's equation: we refer to [27] for details (in part reviewed in the appendix). Now, following [28,29], we note that there is a residual gauge invariance in these variables representing fluctuations of the form above: the metric fluctuations transform under infinitesimal diffeomorphisms as h µν → h µν − ∇ µ ξ ν − ∇ ν ξ µ , with the gauge functions ξ µ (t, x, r) ≡ ξ µ (r)e −iωt+iqx .…”

“…Some previous investigations appear in e.g. [15][16][17][18][19][20][21][22][23][24][25][26][27], in part reviewed in [5]. In the context of gauge/gravity duality [1][2][3][4] for relativistic theories, various transport properties are encoded in the quasinormal modes of the dual gravitational black branes, see e.g.…”