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Title: Entropy density of spacetime and the Navier-Stokes fluid dynamics of null surfaces

Abstract

It has been known for several decades that Einstein's field equations, when projected onto a null surface, exhibit a structure very similar to the nonrelativistic Navier-Stokes equation. I show that this result arises quite naturally when gravitational dynamics is viewed as an emergent phenomenon. Extremizing the spacetime entropy density associated with the null surfaces leads to a set of equations which, when viewed in the local inertial frame, becomes identical to the Navier-Stokes equation. This is in contrast to the usual description of the Damour-Navier-Stokes equation in a general coordinate system, in which there appears a Lie derivative rather than a convective derivative. I discuss this difference, its importance, and why it is more appropriate to view the equation in a local inertial frame. The viscous force on fluid, arising from the gradient of the viscous stress-tensor, involves the second derivatives of the metric and does not vanish in the local inertial frame, while the viscous stress-tensor itself vanishes so that inertial observers detect no dissipation. We thus provide an entropy extremization principle that leads to the Damour-Navier-Stokes equation, which makes the hydrodynamical analogy with gravity completely natural and obvious. Several implications of these results are discussed.

Authors:
 [1]
  1. Inter-University Centre for Astronomy and Astrophysics (IUCAA), Pune, Maharashtra 411 007 (India)
Publication Date:
OSTI Identifier:
21505006
Resource Type:
Journal Article
Journal Name:
Physical Review. D, Particles Fields
Additional Journal Information:
Journal Volume: 83; Journal Issue: 4; Other Information: DOI: 10.1103/PhysRevD.83.044048; (c) 2011 American Institute of Physics; Journal ID: ISSN 0556-2821
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; DENSITY; EINSTEIN FIELD EQUATIONS; ENTROPY; FLUID MECHANICS; GRAVITATION; METRICS; NAVIER-STOKES EQUATIONS; SPACE-TIME; SURFACES; DIFFERENTIAL EQUATIONS; EQUATIONS; FIELD EQUATIONS; MECHANICS; PARTIAL DIFFERENTIAL EQUATIONS; PHYSICAL PROPERTIES; THERMODYNAMIC PROPERTIES

Citation Formats

Padmanabhan, T. Entropy density of spacetime and the Navier-Stokes fluid dynamics of null surfaces. United States: N. p., 2011. Web. doi:10.1103/PHYSREVD.83.044048.
Padmanabhan, T. Entropy density of spacetime and the Navier-Stokes fluid dynamics of null surfaces. United States. https://doi.org/10.1103/PHYSREVD.83.044048
Padmanabhan, T. 2011. "Entropy density of spacetime and the Navier-Stokes fluid dynamics of null surfaces". United States. https://doi.org/10.1103/PHYSREVD.83.044048.
@article{osti_21505006,
title = {Entropy density of spacetime and the Navier-Stokes fluid dynamics of null surfaces},
author = {Padmanabhan, T},
abstractNote = {It has been known for several decades that Einstein's field equations, when projected onto a null surface, exhibit a structure very similar to the nonrelativistic Navier-Stokes equation. I show that this result arises quite naturally when gravitational dynamics is viewed as an emergent phenomenon. Extremizing the spacetime entropy density associated with the null surfaces leads to a set of equations which, when viewed in the local inertial frame, becomes identical to the Navier-Stokes equation. This is in contrast to the usual description of the Damour-Navier-Stokes equation in a general coordinate system, in which there appears a Lie derivative rather than a convective derivative. I discuss this difference, its importance, and why it is more appropriate to view the equation in a local inertial frame. The viscous force on fluid, arising from the gradient of the viscous stress-tensor, involves the second derivatives of the metric and does not vanish in the local inertial frame, while the viscous stress-tensor itself vanishes so that inertial observers detect no dissipation. We thus provide an entropy extremization principle that leads to the Damour-Navier-Stokes equation, which makes the hydrodynamical analogy with gravity completely natural and obvious. Several implications of these results are discussed.},
doi = {10.1103/PHYSREVD.83.044048},
url = {https://www.osti.gov/biblio/21505006}, journal = {Physical Review. D, Particles Fields},
issn = {0556-2821},
number = 4,
volume = 83,
place = {United States},
year = {Tue Feb 15 00:00:00 EST 2011},
month = {Tue Feb 15 00:00:00 EST 2011}
}