Abstract
Over the past few decades, a host of theoretical evidence has surfaced that suggest a connection between theories of gravity and the Navier-Stokes (NS) equation of fluid dynamics. It emerges out that a theory of gravity can be treated as some kind of fluid on a particular surface. Motivated by the work carried out by Bredberg et al. [6], our paper focuses on including certain modes to the vacuum solution which are consistent with the so called hydrodynamic scaling and discuss the consequences, one of which appear in the form of Damour Navier Stokes (DNS) equation with the incompressibility condition. We also present an alternative route to the results by considering the metric as a perturbative expansion in the hydrodynamic scaling parameter ϵ and with a specific gauge choice, thus modifying the metric. It is observed that the inclusion of certain modes in the metric corresponds to the solution of Einstein’s equations in presence of a particular type of matter in the spacetime. This analysis reveals that gravity has both the NS and DNS description not only on a null surface, but also on a timelike surface. So far we are aware of, this analysis is the first attempt to illuminate the possibility of presenting the gravity dual of DNS equation on a timelike surface. In addition, an equivalence between the hydrodynamic expansion and the near-horizon expansion has also been studied in the present context.
Highlights
We describe an alternative framework in which the geometry described in [6], is viewed as a perturbative expansion of the leading order metric and leading to NS or Damour Navier Stokes (DNS) dynamics depending on the manner in which the metric is modified at a particular order
It was well known that the DNS equation governs the geometric data on any null surface in the spacetime
In our paper we have shown that a restricted version of the DNS equation arises even when the geometry under consideration is a timelike hypersurface
Summary
We shall briefly review the scaling symmetry of the incompressible NS equation, which forms the basis of the work [6] and lays the foundation of our analysis. If the amplitudes of the solution space (vi, P ) of the incompressible NS equation is scaled down by the parameter : vi (xi, τ ) = vi( xi, 2τ ) ; P (xi, τ ) = 2P ( xi, 2τ ) ,. A term proportional to ∂iv arises in our analysis later in the paper, as an addition to the incompressible NS equation. This resulting equation is shown to obey the same scaling laws (2.1) (the explicit proof is shown in appendix A). In our analysis in the subsequent sections, we focus on including certain additional modes consistent with the above hydrodynamic scaling sourced by an appropriately defined bulk matter tensor and working with a modified metric than in [6]. We shall see that it includes the NS equation, and gives birth to a restricted type of DNS equation which preserves its structure under the scaling (2.1) of the fluid parameters
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