Holographic dual of a Hawking pair has a wormhole

In previous work, two of us constructed the holographic dual of a single Einstein-Podolsky-Rosen (EPR) pair in maximally supersymmetric Yang-Mills theory. Here we extend that work in two directions. First, we use a Randall-Sundrum brane to couple the entangled pair to dynamical gravity. Second, we consider the entangled pair in the background of a black brane with the particles on opposite sides of the horizon, turning the EPR pair into a Hawking pair. Along the way, we use the Hawking pair to exhibit the confinement of the super Yang-Mills theory on a black brane background. We give general arguments that $\mathrm{ER}=\mathrm{EPR}$ should be understood as a duality between two equivalent descriptions of the same physical reality.

Figure 1

The Poincaré patch of ${\mathrm{AdS}}_{d+1}$, in both flat space and ${\mathrm{AdS}}_d$ slicing. The green dash-dotted lines are the standard constant $z$ slices in flat slicing, where each slice is $d$-dimensional Minkowski space. The solid red slices are constant $z$ slices when the metric on the slice is taken to be ${\mathrm{AdS}}_d$ with radius $L$. The dashed black lines correspond to fixed radial distance on the ${\mathrm{AdS}}_d$ slice but arbitrary $z$ in the ${\mathrm{AdS}}_d$ slicing. The solid blue line at the top of the plot is the conformal boundary of ${\mathrm{AdS}}_{d+1}$. In flat slicing, it is naturally equipped with the $d$-dimensional Minkwoski metric. In ${\mathrm{AdS}}_d$ slicing, it is the union of the two ${\mathrm{AdS}}_d$ slices at $z/L=\pm \pi /2$, glued together along their common boundary. These boundary metrics are of course the same up to a Weyl transformation.

Figure 2

String dual to a quark-antiquark pair in mSYM coupled to dynamical gravity, as realized by a critical RS brane. The string ends on the RS brane, and the endpoints are dual to a heavy dynamical quark and antiquark, which uniformly accelerate away from each other for all time. When the initial separation $2b$ is large compared to the inverse cutoff scale $L$, there is a wormhole on the world sheet of the dual string. However, the wormhole disappears when the initial separation is of order the inverse cutoff ($b<L$).

Figure 3

Strings dual to two different entangled Hawking pairs in the AdS black string geometry for two different values of $r_q=r_{\overline{q}}$. The circles are quark and antiquark pair, respectively; dashed and solid line are two possible strings connecting the two. The solid line is the analytic solution (28) with $r_q=r_{\overline{q}}=\infty$. The figures show the $r$ and $z$ coordinates of (6) with $g_d$ given by the eternal AdS black brane (18). The time and transverse $\overrightarrow{x}$ directions are suppressed.

Figure 4

Figure from Ref. [40] which depicts the causal structure of the Kruskal ${\mathrm{AdS}}_d$ black hole present on each slice. Here, $r_0$ is the horizon $r_h$. The coordinate axes are the radial and time coordinate of the Kruskal coordinate system, which covers the entire space. The solid lines are the $r\to \infty$ boundaries of the extended black hole geometry. The dashed lines are the black hole singularities and the dotted lines are past and future horizons.

Figure 5

The string configurations for a range of values of $r_q$ where the green, dashed vertical line is the location of the horizon $r_h=1$ and the red, dashed horizontal line represents the boundary $z=\pi /2$.

Figure 6

The numerically obtained potential (blue, curved line), $V(r_q)$, plotted against the location of the quark $r_q$. The dashed lines are the analytic solutions in the asymptotic regions $r_q\gg r_h$ (red, horizontal) and $r_q\sim r_h$ (green, vertical) given in the following section.