Algebraically special solutions in AdS/CFT

TL;DR

This work analyzes the AdS/CFT interpretation of Robinson-Trautman (RT) spacetimes, a class of algebraically special solutions in AdS gravity, showing that the dual CFT in 2+1 dimensions experiences a time-dependent, essentially local boundary geometry. By developing a scalar, Weyl-covariant formalism for conformal fluids in 2+1 dimensions and applying it to RT and Kerr–AdS geometries, the authors extract higher-order transport coefficients and constrain entropy production terms, linking exact bulk solutions to derivative expansions in the boundary theory. The results reveal that, under RT evolution, certain four- and six-derivative transport coefficients are uniquely fixed while three-derivative terms remain unconstrained, and the entropy current divergence aligns with the expected eight-derivative shear-squared structure; Kerr–AdS serves as a check where the boundary behaves as a perfect conformal fluid. Overall, the paper demonstrates a powerful, scalar-based framework for deriving higher-order hydrodynamic data from algebraically special AdS bulk solutions and outlines extensions to broader bulk families and higher dimensions.

Abstract

We investigate the AdS/CFT interpretation of the class of algebraically special solutions of Einstein gravity with a negative cosmological constant. Such solutions describe a CFT living in a 2+1 dimensional time-dependent geometry that, generically, has no isometries. The algebraically special condition implies that the expectation value of the CFT energy-momentum tensor is a local function of the boundary metric. When such a spacetime is slowly varying, the fluid/gravity approximation is valid and one can read off the values of certain higher order transport coefficients. To do this, we introduce a formalism for studying conformal, relativistic fluids in 2+1 dimensions that reduces everything to the manipulation of scalar quantities.

Figures (2)

• Figure 1: Conformal structure of a RT spacetime with negative cosmological constant. The solution exists to the future of the null hypersurface $u = u_0$, has a timelike infinity, and approaches the Schwarzschild-AdS solution as $u \rightarrow \infty$. However, for large $m$, the hypersurface $u = \infty$ is actually a null singularity. There is also a curvature singularity at $r = 0$.
• Figure 2: Penrose diagram for the time-reversed (extended) RT solution. The shaded region represents the extension to the future of the hypersurface $t = t_0$, where the spacetime is no longer RT. The location of the event horizon $\mathscr{H}^+$ is also shown. The solution approaches Schwarzschild-AdS as $t\rightarrow -\infty$ but (for large $m$) there exists no $C^1$ extension across this null surface.