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Register a new userScale generation in ~R^2 gravity from super-critical dynamics
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K. Odagiri
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This paper has been withdrawn by the author, due to obsolete reference [4], insufficient discussion in sec. 4, and major conceptual error in sec. 5.
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Galactic rotation curves and lack of direct observations of Dark Matter may indicate that General Relativity is not valid (on galactic scale) and should be replaced with another theory. There is the only variant of Absolute Parallelism which solutions are free of arising singularities, if D=5 (there is no room for changes). This variant does not have a Lagrangian, nor match GR: an equation of `plain R^2-gravity (ie without R-term) is in sight instead. Arranging an expanding O_4-symmetrical solution as the basis of 5D cosmological model, and probing a universal_function of mass distribution (along very-very long the extra dimension) to place into bi-Laplace equation (R^2 gravity), one can derive the Law of Gravitation: 1/r^2 transforms to 1/r with distance (not with acceleration).
We present a detailed analysis of the construction of $z=2$ and $z eq2$ scale invariant Hov{r}ava-Lifshitz gravity. The construction procedure is based on the realization of Hov{r}ava-Lifshitz gravity as the dynamical Newton-Cartan geometry as well as a non-relativistic tensor calculus in the presence of the scale symmetry. An important consequence of this method is that it provides us the necessary mechanism to distinguish the local scale invariance from the local Schrodinger invariance. Based on this result we discuss the $z=2$ scale invariant Hov{r}ava-Lifshitz gravity and the symmetry enhancement to the full Schrodinger group.
We generalise the computations of arXiv:0712.2456 to generate long wavelength, asymptotically locally AdS_5 solutions to the Einstein-dilaton system with a slowly varying boundary dilaton field and a weakly curved boundary metric. Upon demanding regularity, our solutions are dual, under the AdS/CFT correspondence, to arbitrary fluid flows in the boundary theory formulated on a weakly curved manifold with a prescribed slowly varying coupling constant. These solutions turn out to be parametrised by four-velocity and temperature fields that are constrained to obey the boundary covariant Navier Stokes equations with a dilaton dependent forcing term. We explicitly evaluate the stress tensor and Lagrangian as a function of the velocity, temperature, coupling constant and curvature fields, to second order in the derivative expansion and demonstrate the Weyl covariance of these expressions. We also construct the event horizon of the dual solutions to second order in the derivative expansion, and use the area form on this event horizon to construct an entropy current for the dual fluid. As a check of our constructions we expand the exactly known solutions for rotating black holes in global AdS_5 in a boundary derivative expansion and find perfect agreement with all our results upto second order. We also find other simple solutions of the forced fluid mechanics equations and discuss their bulk interpretation. Our results may aid in determining a bulk dual to forced flows exhibiting steady state turbulence.
Black branes in AdS5 appear in a four parameter family labeled by their velocity and temperature. Promoting these parameters to Goldstone modes or collective coordinate fields -- arbitrary functions of the coordinates on the boundary of AdS5 -- we use Einsteins equations together with regularity requirements and boundary conditions to determine their dynamics. The resultant equations turn out to be those of boundary fluid dynamics, with specific values for fluid parameters. Our analysis is perturbative in the boundary derivative expansion but is valid for arbitrary amplitudes. Our work may be regarded as a derivation of the nonlinear equations of boundary fluid dynamics from gravity. As a concrete application we find an explicit expression for the expansion of this fluid stress tensor including terms up to second order in the derivative expansion.
We generalize recent work to construct a map from the conformal Navier Stokes equations with holographically determined transport coefficients, in d spacetime dimensions, to the set of asymptotically locally AdS_{d+1} long wavelength solutions of Einsteins equations with a negative cosmological constant, for all d>2. We find simple explicit expressions for the stress tensor (slightly generalizing the recent result by Haack and Yarom (arXiv:0806.4602)), the full dual bulk metric and an entropy current of this strongly coupled conformal fluid, to second order in the derivative expansion, for arbitrary d>2. We also rewrite the well known exact solutions for rotating black holes in AdS_{d+1} space in a manifestly fluid dynamical form, generalizing earlier work in d=4. To second order in the derivative expansion, this metric agrees with our general construction of the metric dual to fluid flows.
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