Abstract
Abstract We show that holographic RG flow can be defined precisely such that it corresponds to emergence of spacetime. We consider the case of pure Einstein’s gravity with a negative cosmological constant in the dual hydrodynamic regime. The holographic RG flow is a system of first order differential equations for radial evolution of the energy-momentum tensor and the variables which parametrize it’s phenomenological form on hypersurfaces in a foliation. The RG flow can be constructed without explicit knowledge of the bulk metric provided the hypersurface foliation is of a special kind. The bulk metric can be reconstructed once the RG flow equations are solved. We show that the full spacetime can be determined from the RG flow by requiring that the horizon fluid is a fixed point in a certain scaling limit leading to the non-relativistic incompressible Navier-Stokes dynamics. This restricts the near-horizon forms of all transport coefficients, which are thus determined independently of their asymptotic values and the RG flow can be solved uniquely. We are therefore able to recover the known boundary values of almost all transport coefficients at the first and second orders in the derivative expansion. We conjecture that the complete characterisation of the general holographic RG flow, including the choice of counterterms, might be determined from the hydrodynamic regime.
Highlights
Which has been instrumental in making the holographic correspondence precise
We show that holographic renormalization group (RG) flow can be defined precisely such that it corresponds to emergence of spacetime
It is not clear how to define the rules for constructing counterterms generally, and what is the precise role of multi-trace operators in the holographic renormalization group (RG) flow
Summary
We will first briefly review fluid mechanics in an arbitrary weakly curved background metric. We will denote the number of independent scalars, transverse vectors, and symmetric, traceless and transverse tensors at n-th order in derivative expansion as m(sn), m(vn) and m(tn) respectively. All scalar and tensor transport coefficients are functions of the entropy density s or alternatively the temperature T , which like the equation of state can be obtained from the underlying quantum field theory. At each order in derivative expansion, only a certain combination of symmetric, traceless and transverse tensors are Weyl covariant. Only five linear combinations of the eight independent symmetric, traceless and transverse tensors are Weyl covariant. Where lAdS is the AdS radius and Tb is the Hawking temperature
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