On Spacetime Entanglement

Abstract

We examine the idea that in quantum gravity, the entanglement entropy of a general region should be finite and the leading contribution is given by the Bekenstein-Hawking area law. Using holographic entanglement entropy calculations, we show that this idea is realized in the Randall-Sundrum II braneworld for sufficiently large regions in smoothly curved backgrounds. Extending the induced gravity action on the brane to include the curvature-squared interactions, we show that the Wald entropy closely matches the expression describing the entanglement entropy. The difference is that for a general region, the latter includes terms involving the extrinsic curvature of the entangling surface, which do not appear in the Wald entropy. We also consider various limitations on the validity of these results.

Figures (2)

• Figure 1: (Colour online) $S_\textrm{\tiny EE}$, $C_2$ and $C'_2=\delta\partial_{\tilde{\ell}}C_2$ as a function of $\tilde{\ell}$ for $d=3, 4, 5, 6$. The vertical axes are normalized with $S_0=\frac{H^{d-2}}{2G_d}$. The first plot confirms that for $\tilde{\ell}\gg\delta$, the dominant contribution in entanglement entropy is the BH term, i.e.,$S_0$. Also the last plot reveals that $C'_2$ becomes positive for $\tilde{\ell}\lesssim\delta$, indicating a limitation with this model.
• Figure 2: (Colour online) $S_\textrm{\tiny EE}$, $C_3$ and $C'_3=\delta\partial_{\tilde{R}}C_3$ as a function of $\tilde{R}$ for $d=3, 4, 5, 6$. The vertical axes are normalized with $S_0=\mathcal{A}(\tilde{\Sigma})/(4 G_d)$. The plot of $S_\textrm{\tiny EE}$ confirms that for $\tilde{R}\gg\delta$, the dominant contribution is the BH term, i.e.,$S_0$. The last plot reveals that for $d=4, 5, 6$, $C'_3$ becomes positive for $\tilde{R}\lesssim\delta$. Also note that for $d=3$, $C'_3$ is positive for all $\tilde{R}$.