Numerical solution of gravitational dynamics in asymptotically anti-de Sitter spacetimes

TL;DR

<3-5 sentence high-level summary>This work develops a robust numerical framework for solving gravitational dynamics in asymptotically AdS spacetimes using a characteristic, null-slicing approach with spectral discretization. By exploiting residual diffeomorphisms and an apparent-horizon IR cutoff, the method turns Einstein's equations into a nested set of radial ODEs that can be evolved in time, enabling stable simulations of far-from-equilibrium dynamics. The authors demonstrate the approach on three test problems—homogeneous isotropization, planar shock collisions, and 2D turbulence—observing rapid isotropization times, locally hydrodynamic behavior post-collision, and a turbulent cascade with horizon-geometry signatures that align with the fluid/gravity correspondence. This framework significantly extends the range of holographic, strongly coupled systems accessible to numerical study and lays groundwork for exploring more complex, less symmetric dynamics in gauge/gravity duality.

Abstract

A variety of gravitational dynamics problems in asymptotically anti-de Sitter (AdS) spacetime are amenable to efficient numerical solution using a common approach involving a null slicing of spacetime based on infalling geodesics, convenient exploitation of the residual diffeomorphism freedom, and use of spectral methods for discretizing and solving the resulting differential equations. Relevant issues and choices leading to this approach are discussed in detail. Three examples, motivated by applications to non-equilibrium dynamics in strongly coupled gauge theories, are discussed as instructive test cases. These are gravitational descriptions of homogeneous isotropization, collisions of planar shocks, and turbulent fluid flows in two spatial dimensions.

Figures (18)

• Figure 1: Focusing of null infalling radial geodesics and consequent formation of caustics. Only the radial direction and one spatial direction are shown. The grey shaded "blob" represents some perturbation in the geometry causing focusing of infalling geodesics. The shaded area at the bottom of each figure represents events behind the apparent horizon. Left panel: caustic formation outside the apparent horizon. Right panel: caustic hidden behind apparent horizon.
• Figure 2: Possible forms of apparent horizon evolution induced by gravitational infall. Only the radial and one spatial direction are shown. The solid, dashed, and dotted lines, bounding progressively lighter shaded regions, show the position of the apparent horizon at three times $t_0$, $t_1$, and $t_2$, respectively, with $t_0 < t_1 < t_2$. Right panel: planar horizon topology at all times, to which our methods apply. Left panel: non-planar horizon topology (at times $t_1$ and $t_2$), requiring different computational methods.
• Figure 3: Homogeneous isotropization results. Left panel: Anisotropy function $B(t,u)/u^3$. The anisotropy function rapidly attenuates, with exponentially damped oscillations. Right panel: Pressure anisotropy $\delta p = T_{zz} - \tfrac{1}{2} (T_{xx}+T_{yy})$, relative to the equilibrium pressure $p_{\rm eq} = \frac{1}{8} N_\text{c}^2 (\pi T)^4$, as a function of time. At early times the pressures anisotropy is very large. However, just as the anisotropy function vanishes exponentially fast, so does the pressure anisotropy.
• Figure 4: A plot of $e^{|{\rm Re\, \lambda_1}| t}\delta p/ p_{\rm eq}$ as well as the lowest quasinormal mode (also multiplied by a factor of $e^{|{\rm Re \lambda_1}| t}$). The fit to the lowest quasinormal mode agrees with the numerics at the 1 part in $10^4$ level or better after time $t = 10.$
• Figure 5: Plots of $b_+$ (left) and $\lambda_+$ (right) for a single narrow shock of width $w = 0.075$ moving in the $+z$ direction. The choice of gauge parameter $\lambda_+$ is such that $u = 1$ corresponds with Fefferman-Graham coordinate $\tilde{\rho} = 8$. On the boundary, $u = 0$, the shock is centered at $z = 0$ at the time shown, $t = 0$. However, in Eddington-Finkelstein coordinates the shock increasingly extends into the $+z$ direction as one goes deeper into the bulk. This also manifests itself in the gauge parameter $\lambda$, which differs significantly from its background value in front of the shock. In regions where $b_+ = 0$ the geometry is that of AdS$_5$.