# [1911.12333] [Almheiri, Hartman, Maldacena, Shaghoulian, Tajdini] Replica Wormholes & Information Paradox ## Refs - JT - Harlow & Jafferis (H&J), 1804.01081 - Gábor Sárosi, 1711.08482 - Island & Wormholes: 2006.06872 - Calculations: - 1911.12333 ## $\mathrm{AdS}_2$ Basics - Euclidean Poincaré: $$ \dd{s}^2 = \frac{\dd{x} \dd{\bar{x}}}{z^2}, \quad z = \frac{x + \bar{x}}{2}, \quad z < 0 $$ Note that we choose to look at the $z < 0$ region for future convenience. Poincaré horizon: $$ 0> z \to -\infty, \quad \mcal{I}^\pm \colon\ x^\mp \to -\infty $$ - Rindler: $$ x = \tanh ky, \quad x \to -1, \quad y = \sigma + i\tau \to -\infty $$ $y = \sigma + i\tau$ is similar to some cylindrical coordinate. $\tau\colon$ Euclidean time, $\tau = it$, Lorentzian $y^\pm = \sigma \pm t$. Note that $y^+ = \bar{y},\ y^- = y$. $k$ is related to the temperature. To see this explicitly, compute the induced metric in Euclidean signature, and we get: $$ \dd{s}^2 = \frac{ k^2 \dd{y} \dd{\bar{y}} }{ \pqty{ \tfrac{1}{2}\, \sinh 2k\sigma }^2 }, \quad \sigma = \frac{y + \bar{y}}{2} $$ - $\sigma \to -\epsilon$, $\dd{s}^2 = \frac{1}{\epsilon} \dd{y} \dd{\bar{y}}$, we see a cutoff boundary with flat metric. - $\sigma \to -\infty$, i.e. near the Euclidean "horizon", after some trial and error, we see that if we set $\rho = e^{2k\sigma} \to 0$, then $\dd{\rho} = 2k \rho \dd{\sigma}$, $$ \dd{s}^2 \sim \frac{ (2k)^2 }{ \pqty{ \tfrac{1}{2}\, e^{-2k\sigma} }^2 } \pqty{ \frac{\dd{\rho}^2}{(2k)^2\rho^2} + \dd{\tau}^2 } = 4\,( \dd{\rho}^2 + \rho^2 (2k)^2 \dd{\tau}^2 ) $$ i.e. the Euclidean geometry is smooth around $\rho \to 0$ if $2k\beta = 2\pi$, or $k = \frac{\pi}{\beta}$. In fact, one can get the standard Poincaré disk with: $$ w = e^{\frac{2\pi}{\beta} y}, \quad \dd{s}^2 = \dd{w}\dd{\bar{w}} \bigg/ \pqty{ \frac{1 - \abs{w}^2}{2} }^2 $$ - The Rindler patch is in fact exactly the "right half" of a two sided "black hole". To see this, consider: $$ x = \tan u, \quad \begin{aligned} x &\to -1, \quad u \to -\tfrac{\pi}{4},\\ x &\to -\infty, \quad u \to -\tfrac{\pi}{2}, \end{aligned} $$ $$ \dd{s}^2 = \frac{ \dd{u} \dd{\bar{u}} }{ \pqty{ \tfrac{1}{2}\, \sin \,(u + \bar{u}) }^2 }, \quad \sigma = \frac{y + \bar{y}}{2} $$ This is, in fact, global $\mathrm{AdS_2}$, with two asymptotic boundaries at $\frac{u + \bar{u}}{2} = 0, -\frac{\pi}{2}$. ## JT Gravity AdS/CFT: too strong, not realistic! $\leadsto$ holographic systems. How do we see that some theory of quantum gravity is holographic or not? Case study: JT gravity. Different UV completions of JT gravity (H&J): ### Canonical quantization JT gravity is renormalizable! $$ S \sim \frac{1}{16\pi G_N} \int \dd[2]{x} \Phi\,(R + 2) + \cdots $$ Classical power counting: mass dim $ [g] = 0, [x] = -1 $, $ [\Phi] = 0, \ [R] = +2, \ [G_N] = 0 $. $$ 0 = \var{S} = \int_M \dd[2]{x} \sqrt{-g} \ (\mathrm{EOM}) + \int_{\pd M} \dd{x} \sqrt{\abs{\gamma}} \ (\mathrm{BC}) $$ Boundary conditions (BC) at the cutoff $r_c$ --- asymptotic $\mathrm{AdS}_ 2$. Classical solution: two-sided black hole. Singularities emerge at the $\infty$-ly blue-shifted surface. Given such BC, the solution to the EOM is _unique!_ With only one free parameter. Details: H&J, Section 2. Conformal compactification: $x = \tan z$, $$ \dd{s}^2 = (\sec{z})^2 \pqty{ -\dd{\tau}^2 + \dd{z}^2 }, \quad z\in (-\tfrac{\pi}{2}, +\tfrac{\pi}{2}), \\ \Phi = \Phi_h \sec z \cos \tau $$ Alternatively, use coordinates in the right patch: horizon $r_s = \Phi_h / \phi_b$, $$ T^{tt}_\mathrm{CFT} = H = \frac{2\Phi_h^2}{\phi_b} $$ **Lorentzian path integral: no sum of topologies!** Classical solution is completely fixed by $\Phi_h$, or $r_s$, or equivalently, $H$. One can thus canonically quantized such system like any QM system: find dynamical d.o.f. and promote the Poisson brackets to commutators. Here the natural conjugate variables are $(t,H)$, which can be transformed to $(L,P)$ where $L$ is the cutoff-independent "renormalized bulk size"; see H&J (2.29). We have: $$ H = \frac{P^2}{2\phi_b} + \frac{2}{\phi_b}\,e^{-L} $$ **Note:** it is kind of weird to include $t$ in the phase space. However, note that what we are actually doing is trying to assign a natural weight to each configuration (labeled by $\Phi_h$ or $H = E$). One finds the result is equivalent to **Hartle--Hawking construction / Euclidean path integral, if we include only the trivial (disk) topology!** [H&J] **Information paradox:** Couple the right patch to a reservoir to let it evaporates! $S_{\text{res}}$ can be computed by a RT surface $$ S_{\text{res}} \gg S_{BH} \sim 8\pi\Phi_0 + 8\pi^2\phi_b T_0 $$ Interpretation: there are remnants not counted by $S_{BH}$! H&S: _this renormalizable bulk theory can have an arbitrarily large number of low-energy excitations near black hole horizon._ [Details? NOT shown / cited by H&S.] ### Recommendations - 1804.01081: to better understand JT - 1905.08762: contains most of the calculations that H&S relies on - 1211.3494: Aron Wall on maximin & HRT.