Abstract
We consider relativistic hydrodynamics in the limit where the number of spatial dimensions is very large. We show that under certain restrictions, the resulting equations of motion simplify significantly. Holographic theories in a large number of dimensions satisfy the aforementioned restrictions and their dynamics are captured by hydrodynamics with a naturally truncated derivative expansion. Using analytic and numerical techniques we analyze two and three-dimensional turbulent flow of such fluids in various regimes and its relation to geometric data.
Highlights
We consider the turbulent behavior of event horizons of asymptotically AdS spaces, in the limit where the number of dimensions, d, is very large
To obtain a large d limit, one considers a metric ansatz in d = n + p + 1 spacetime dimensions, where the dynamics depend on p + 1 dimensions
The holographic transport coefficients satisfy precisely the relations required for the aforementioned simplification of the large d hydrodynamic equations
Summary
Following a Wilsonian type of coarse graining, the equations of motion of relativistic hydrodynamics may be characterized by a handful of fields: a unit normalized velocity field uμ(x), a temperature field T (x) and, in the presence of charged matter, a chemical potential μ(x). When working in the limit where gradients of the hydrodynamic fields are small compared to the inverse mean free path, all observables may be expressed as local functions of the hydrodynamic fields. These expressions are referred to as constitutive relations. In a d spacetime dimensional conformal theory in flat space (and in the absence of charge, anomalies or parity breaking terms) the constitutive relations for the stress tensor take the form
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