Simulation of Black Hole Collisions in Asymptotically Anti–de Sitter Spacetimes

We present results from the evolution of spacetimes that describe the merger of asymptotically global anti–de Sitter black holes in 5D with an SO(3) symmetry. Prompt scalar field collapse provides us with a mechanism for producing distinct trapped regions on the initial slice, associated with black holes initially at rest. We evolve these black holes towards a merger, and follow the subsequent ring down. The boundary stress tensor of the dual conformal field theory is conformally related to a stress tensor in Minkowski space that inherits an axial symmetry from the bulk SO(3). We compare this boundary stress tensor to its hydrodynamic counterpart with viscous corrections of up to second order, and compare the conformally related stress tensor to ideal hydrodynamic simulations in Minkowski space, initialized at various time slices of the boundary data. Our findings reveal far-from-hydrodynamic behavior at early times, with a transition to ideal hydrodynamics at late times.

Figure 1

The effect of changing the size of the regulator BH as seen from the $x_1=x_2=0$ spatial slice of Minkowski space at $t^{\prime }=0.32$. Here, the mass of the regulator BH is expressed as a percentage of the total spacetime mass. Left: the energy density distribution along $x_3$. Right: the $x_3$ component of the three-velocity along $x_3$.

Figure 2

Left: comparison on $\mathbb{R}\times S^3$, showing the normalized mismatch between the boundary stress tensor and its hydrodynamic counterpart, with viscous corrections of up to second order. Here, $|.|$ denotes an $L^2$ norm over $\chi \in [0,\pi ]$; for more details, see Ref. [21], where a similar analysis was performed on distorted ${\mathrm{AdS}}_5$ black holes. Data from the highest resolution run (dashed black line) are displayed with an estimated uncertainty (gray shaded region). Right: comparison on ${\mathbb{R}}^{3,1}$, showing the time evolution of the energy density at the origin, with data from the boundary stress tensor of the gravity solution (“GR”) and from ideal $3+1$ dimensional relativistic hydrodynamics (“Ideal Hydro”). For computational reasons, a small but nonvanishing viscosity was used in the hydrodynamic simulation. The hydrodynamics is initialized with gravity data at $t_i^{\prime }=0.3,0.9,2.1$.

Figure 3

Spacetime diagram of the bulk solution for a BH-BH simulation with a 1.9% mass regulator BH. The vertical is parametrized by $\rho \cos \chi$ for $\rho \in [0,1]$ and $\chi \in [0,\pi ]$, and the horizontal is parametrized by bulk global time $t$. The apparent horizon (solid black line) bounds each trapped region (gray shaded region). At early times $t<\pi /16$, there are three distinct trapped regions corresponding to two physical BHs and a regulator BH at the origin. The physical BHs move towards each other from rest and eventually merge with the regulator BH around $t\approx \pi /16$. The resulting postmerger BH rings down and eventually becomes quiescent at late times $t\gtrsim \pi$. The Poincaré patch (blue shaded region) is the wedge shaped region that tapers up towards $t=\pi$. The boundary of the Poincaré patch is where we extract all CFT quantities.

Figure 4

Snapshots of the energy density ${\epsilon }_{{\mathbb{R}}^{3,1}}$ in Minkowski space for a BH-BH simulation with a 1.9% mass regulator BH, linearly rescaled such that all panels attain the same numerical maximum, to emphasize qualitative features of the flow. The four different panels contain the spatial profile of the energy density in the $x_1\text{−}{x}_3$ plane at four different constant $t^{\prime }$ slices. One can recover the full spatial dependence of the flow by simply rotating each of these panels about the $x_3$ axis.