Gravity dual of Connes cocycle flow

We define the “kink transform” as a one-sided boost of bulk initial data about the Ryu-Takayanagi surface of a boundary cut. For a flat cut, we conjecture that the resulting Wheeler-DeWitt patch is the bulk dual to the boundary state obtained by the Connes cocycle (CC) flow across the cut. The bulk patch is glued to a precursor slice related to the original boundary slice by a one-sided boost. This evades ultraviolet divergences and distinguishes our construction from a one-sided modular flow. We verify that the kink transform is consistent with known properties of operator expectation values and subregion entropies under CC flow. CC flow generates a stress tensor shock at the cut, controlled by a shape derivative of the entropy; the kink transform reproduces this shock holographically by creating a bulk Weyl tensor shock. We also go beyond known properties of CC flow by deriving novel shock components from the kink transform.

Figure 1

Kink transform. Left: a Cauchy surface $\mathrm{\Sigma }$ of the original bulk $\mathcal{M}$. An extremal surface $\mathcal{R}$ is shown in red. The orthonormal vector fields $t^a$ and $x^a$ span the normal bundle to $\mathcal{R}$; $x^a$ is tangent to $\mathrm{\Sigma }$. Right: the kink transformed Cauchy surface ${\mathrm{\Sigma }}_s$. As an initial data set, ${\mathrm{\Sigma }}_s$ differs from $\mathrm{\Sigma }$ only in the extrinsic curvature at $\mathcal{R}$ through Eq. (3.4). Equivalently, the kink transform is a relative boost in the normal bundle to $\mathcal{R}$, Eq. (3.21).

Figure 2

The kink-transformed spacetime ${\mathcal{M}}_s$ is generated by the Cauchy evolution of the kinked slice ${\mathrm{\Sigma }}_s$. This reproduces the left and right entanglement wedges $D(a)$ and $D(a^{\prime })$ of the original spacetime $\mathcal{M}$. The future and past of the extremal surface $\mathcal{R}$ are in general not related to the original spacetime.

Figure 3

Straight slices $\mathrm{\Sigma }$ (red) in a maximally extended Schwarzschild (left) and Rindler (right) spacetime get mapped to kinked slices ${\mathrm{\Sigma }}_s$ (blue) under the kink transform about $\mathcal{R}$.

Figure 4

On a fixed background with boost symmetry, the kink transform changes the initial data of the matter fields. In this example, $\mathcal{M}$ is Minkowski space with two balls relatively at rest (red).The kink transform is still Minkowski space, but the balls collide in the future of $\mathcal{R}$ (blue).

Figure 5

A boundary subregion $A_0$ (pink) has a quantum extremal surface denoted $\mathcal{R}$ (brown) and an entanglement wedge denoted $a$. The complementary region $A_0^{\prime }$ (light blue) has the entanglement wedge $a^{\prime }$. CC flow generates valid states, but one-sided modular flow is only defined with a UV cutoff. For example, one can consider regulated subregions $A^{(\epsilon )}$ (deep blue) and ${A^{\prime }}^{(\epsilon )}$ (red). In the bulk, this amounts to excising an infrared region (gray) from the joint entanglement wedge resulting in a regulated entanglement wedge $D({\mathrm{\Sigma }}_{\epsilon })$ (yellow).

Figure 6

An arbitrary spacetime $\mathcal{M}$ with two asymptotic boundaries is transformed to a physically different spacetime ${\mathcal{M}}_s$ by performing a kink transform on the Cauchy slice $\mathrm{\Sigma }$. A piecewise geodesic (dashed gray line) in $\mathcal{M}$ connecting $x$ and $y$ with boost angle $2\pi s$ at $\mathcal{R}$ becomes a geodesic between $x_s$ and $y$ in ${\mathcal{M}}_s$.

Figure 7

Holographic proofs. Left: Boundary causality is respected by the red curve that goes through the bulk in a spacetime $\mathcal{M}$; this is used in proving the ANEC. The RT surfaces ${\mathcal{R}}_1$ and ${\mathcal{R}}_2$ must be spacelike separated; this is used in proving the QNEC. Right: In the kink transformed spacetime ${\mathcal{M}}_s$ as $s\to \infty$, the QNEC follows from causality of the red curve, which only gets contributions from the Weyl shocks (blue) at ${\mathcal{R}}_1$ and ${\mathcal{R}}_2$, and the metric perturbation in the region between them.