Abstract
We clarify the relation between the Noether charge associated to an arbitrary vector field and the equations of motions by revisiting Wald formalism. For a time-like Killing vector, aspects of the Noether charge suggest that it is dual to the heat current in the boundary for general holographic theories. For a space-like Killing vector, we interpret the Noether charge (at the transverse direction) as shear stress of the dual fluid so we can compute the ratio of shear viscosity to entropy density by simply using the infrared data on the black hole event horizon. We test the new method for Einstein gravity and Gauss-Bonnet gravity and find that it produces correct results for both cases even in the presence of additional matter fields.
Highlights
By carefully examining Wald formalism, we clarify the relation between the Noether charge associated to an arbitrary vector field and the Einstein equations of motion
By analyzing aspects of the Noether charge associated to a timelike Killing vector field, we argue that it provides an alternatively reasonable definition for the heat current in the boundary for general holographic theories
For a timelike Killing vector field, we study aspects of the Noether charge and argue that it is dual to the heat current of the boundary theory
Summary
The AdS/CFT correspondence provides a new powerful method to compute the transport coefficients of strongly coupled systems that live on the boundary of asymptotically anti–de Sitter (AdS) space-times, including the ratio of shear viscosity to entropy density [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18] and thermalelectric conductivities (for nice reviews, see [19,20,21]). For Einstein-MaxwellDilaton theories, the authors in [28] presented a nice result by constructing a radially conserved charge independent of the holographic stress tensor but for general cases the situation is not clear Quite recently, it was argued in [29] that the Noether charge associated to a timelike Killing vector field is dual to the holographic heat current. We interpret the Noether charge Qry at the transverse direction as shear stress of the boundary fluid such that we can compute shear viscosity by using the IR data on the black hole event horizon This reproduces the celebrated result η=S 1⁄4 1=ð4πÞ for Einstein gravity even in the presence of matter fields.
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