Abstract
We give a brief review of the recently-formulated fluid/gravity correspondence. Originating from the AdS/CFT correspondence, this constitutes a one-to-one map between configurations of a conformal fluid dynamics in \(d\) dimensions and solutions to Einstein’s equations in \(d+1\) dimensions. The map is fully constructive; for a given fluid configuration, it is completely algorithmic to write down the spacetime geometry and deduce its causal properties. In particular, the bulk solutions describe a regular generic, non-uniform and dynamical, black hole which at late times settles down to a stationary planar black hole. We briefly indicate the iterative construction of such solutions, extract the key physical properties, and discuss further generalizations and open questions.
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Notes
- 1.
AdS is a space of constant negative curvature, which introduces a length scale, called the AdS scale \(R_{\rm AdS}\), corresponding to the radius of curvature. The black hole size is then measured in terms of this AdS scale; large black holes have horizon radius \(r_+ > R_{\rm AdS}\).
- 2.
In what follows, the five-sphere of \(\hbox{AdS}_5 \times S^5\) will play a passive role, so we will ignore it and consider the five-dimensional (asymptotically) AdS bulk spacetime only.
- 3.
Note, however, that there are regular solutions which do not fall into the long-wavelength category, such as small black holes in AdS, which correspondingly are not described by fluid configurations.
- 4.
Note that (6.7) does not have any \(\partial_{\mu} T\) terms appearing explicitly, since by implementing the zeroth order stress tensor conservation, we have expressed the temperature derivatives in terms of the velocity derivatives.
- 5.
We use the standard prescription (cf. [3]),
$$ T^\mu_\nu = -2 \lim_{r \to \infty} r^4 (K^\mu_\nu - \delta^\mu_\nu) $$where \(K_{\mu\nu }\) is the extrinsic curvature tensor on a constant-\(r\) surface.
- 6.
Here we quote the stress tensor as written originally in [5]; subsequently, slightly simpler and more general expressions were found; cf. e.g. [19] for a review. (Also, for simplicity we have absorbed an overall factor of \(16\pi G_N^{(5)}\) which would appear in the conventionally defined stress tensor as conjugate to boundary time translations.)
- 7.
Although including higher orders may remedy this discrepancy [21].
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Acknowledgements
It is a pleasure to thank my collaborators, Sayantani Bhattacharyya, R. Loganayagam, Gautam Mandal, Shiraz Minwalla, Takeshi Morita, Harvey Reall, Mukund Rangamani, and Mark Van Raamsdonk for marvelous collaborations on various aspects of fluid dynamics. I would also like to thank the participants of the Fifth Aegean summer school for their enthusiasm and questions, and the organizers, especially Lefteris Papantonopoulos, for putting together an excellent summer school. This work was supported in part by STFC Rolling grant and by the National Science Foundation under the Grant No. NSF PHY05-51164.
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Hubeny, V.E. (2011). Fluid Dynamics from Gravity. In: Papantonopoulos, E. (eds) From Gravity to Thermal Gauge Theories: The AdS/CFT Correspondence. Lecture Notes in Physics, vol 828. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04864-7_6
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