Emergent times in holographic duality

In holographic duality an eternal Anti-de Sitter black hole is described by two copies of the boundary Conformal Field Theory in the thermal field double state. In this paper we provide explicit constructions in the boundary theory of infalling time evolutions which can take bulk observers behind the horizon. The constructions also help to illuminate the boundary emergence of the black hole horizons, the interiors, and the associated causal structure. A key element is the emergence, in the large $N$ limit of the boundary theory, of a type ${\mathrm{III}}_1$ von Neumann algebraic structure from the type I boundary operator algebra and the half-sided modular translation structure associated with it.

Figure 1

The Penrose diagram of an eternal black hole. The dashed lines are event horizons, and the wavy lines are the singularities.

Figure 2

Left: evolution of the $t=0$ bulk slice under $H_R-H_L$, where $H_{R,L}$ denote the Hamiltonians of the boundary theories. Center: evolution of the $t=0$ slice under $H_R+H_L$, the action of which is singular at the horizon. In fact any linear combination of $H_R$ and $H_L$ other than $H_R-H_L$ is expected to have a singular action at the horizon. Right: a smooth Kruskal-like evolution. If such an evolution can be described in a diffeomorphism invariant way, it must be emergent in the boundary theory.

Figure 3

Kruskal diagram for an eternal black hole. The dashed lines are event horizons, the solid red lines are the singularities, and the solid black lines are the boundaries.

Figure 4

Below $T_{HP}$ the bulk theory is two separate global AdS spacetimes whose small excitations are entangled in the thermal field double state.

Figure 5

The single-trace algebras ${\mathcal{A}}_1$ and ${\mathcal{A}}_2$ associated with the two different Cauchy slices shown are inequivalent, even though they share a causal diamond, since single-trace operators do not obey any equation of motion among themselves (standard Heisenberg evolution takes a single-trace operator outside of the algebra). The same statements apply to algebras generated by generalized free fields (e.g. subalgebras of $\mathcal{Y}$) which do not obey any equations of motion.

Figure 6

Left: Rindler regions of Minkowski spacetime. Right: AdS Rindler regions of the bulk spacetime. The vertical lines denote the boundary, and the dashed lines are Rindler horizons. Each AdS Rindler region has the corresponding Minkowski Rindler region as its boundary.

Figure 7

RT surface, ${\gamma }_R$, for a boundary spatial region $R$. $E_R$ denotes the entanglement wedge. Here we only draw a spatial section of the bulk. The bulk IR divergence of the area of ${\gamma }_R$ comes from the part near the boundary (circled red regions) and reflects the type ${\mathrm{III}}_1$ nature of boundary algebra ${\mathcal{B}}_R$. The type ${\mathrm{III}}_1$ nature of ${\widehat{\mathcal{Y}}}_R$ is reflected in the UV divergences of ${\mathcal{S}}_{E_R}$, which comes from UV degrees of freedom near ${\gamma }_R$ in the bulk (highlighted by orange wavy lines).

Figure 8

Left: operators inserted at the bifurcating horizon do not appear to be factorizable. Right: a Wilson line in the eternal black hole geometry and its factorization into left and right operators via the bifurcation surface.

Figure 9

Slightly separated Rindler regions on a spatial slice. The split property implies that there is a tensor factorization of the Hilbert space with respect to an operator algebra $\mathcal{N}$ contained in the union of the green and red regions above, even though no such tensor factorization exists for the red and blue regions alone when ${\epsilon }_{\mathrm{b}}=0$.

Figure 10

Left: the algebra of the subregion $\mathcal{N}$ that leads to the half-sided modular inclusion structure for $x^{-}$ translation. Right: the algebra of the subregion $\mathcal{N}$ that leads to the half-sided modular inclusion structure for $x^{+}$ translation.

Figure 11

Left: $\mathcal{N}$ denotes the spacetime subregion with $t\le 0$. The vertical axis is time, and for simplicity we have taken the spatial manifold to be a circle (vertical boundaries in the figure are identified). Right: $\mathcal{N}$ denotes the spacetime subregion with $t\ge 0$.

Figure 12

For both plots in Fig. 11 instead of letting the region describing $\mathcal{N}$ be bounded by the $t=0$ slice, we can choose a slice $t=f(\chi )$ where $\chi$ is the boundary spatial coordinate and $f$ an arbitrary periodic function.

Figure 13

Left: support of a bulk field operator ${\varphi }_R(X)$ in the right region on the boundary in the Penrose diagram. The supported region is highlighted with blue color. Right: boundary support of a bulk field operator ${\varphi }_F$ in the future region.

Figure 14

The respective proposed bulk duals for the boundary subregions indicated in Fig. 11.

Figure 15

Left: when a bulk field ${\varphi }_R(X_1)$ with $X_1\in R$ is transported by a null Kruskal coordinate distance $-U_1+U_2$ (since $U_1<0$), it enters the light cone of ${\varphi }_L(X_2)$. The shaded region is a cartoon for the spread of $\mathrm{\Phi }(X_1;s)$. The orange dashed lines are event horizons, and the purple dashed lines give the light cones of $X_2$. Right: the commutator between the evolved right operator and fixed left bulk field is equal to the commutator of the evolved left and right fields with the same difference of evolution parameters. We use this to evolve the left operator almost all the way to the past horizon. The commutator can only be nonzero then if the evolved right operator remains supported on left oscillators after now also applying the evolution that brought the left field to the horizon. The blue shaded region is a cartoon for the spread of $\mathrm{\Phi }(X_1;s+v)$, and the purple shaded region is a cartoon for the spread of $\mathrm{\Phi }(X_2;v)$. The boundaries and singularities are suppressed in each figure.

Figure 16

The left plot gives trajectories of (11.10). The right plot gives constant $s$ surfaces evolved from the $\eta =0$ slice. The orange dashed lines are the event horizons and black solid lines are the boundaries, while the red solid lines are the singularities.

Figure 17

The counterparts of Fig. 16 when using $\mathcal{N}$ as in the right plot of Fig. 11.