Bag-of-gold spacetimes, Euclidean wormholes, and inflation from domain walls in AdS/CFT

Abstract

We use Euclidean path integrals to explore the set of bulk asymptotically AdS spacetimes with good CFT duals. We consider simple bottom-up models of bulk physics defined by Einstein-Hilbert gravity coupled to thin domain walls and restrict to solutions with spherical symmetry. The cosmological constant is allowed to change across the domain wall, modeling more complicated Einstein-scalar systems where the scalar potential has multiple minima. In particular, the cosmological constant can become positive in the interior. However, in the above context, we show that inflating bubbles are never produced by smooth Euclidean saddles to asymptotically AdS path integrals. The obstacle is a direct parallel to the well-known obstruction to creating inflating universes by tunneling from flat space. In contrast, we do find good saddles that create so-called "bag-of-gold" geometries which, in addition to their single asymptotic region, also have an additional large semi-classical region located behind both past and future event horizons. Furthermore, without fine-tuning model parameters, using multiple domain walls we find Euclidean geometries that create arbitrarily large bags-of-gold inside a black hole of fixed horizon size, and thus at fixed Bekenstein-Hawking entropy. Indeed, with our symmetries and in our class of models, such solutions provide the unique semi-classical saddle for appropriately designed (microcanonical) path integrals. This strengthens a classic tension between such spacetimes and the CFT density of states, similar to that in the black hole information problem.

Figures (15)

• Figure 1: A moment of time symmetry in a bag-of-gold spacetime. The region outside the minimal surface is precisely AdS-Schwarzschild, as is a small part of the interior. The rest of the interior is a (say, radiation dominated) FLRW universe.
• Figure 2: Both signs of $\alpha_e(r_{\min{}})$ can be found for $\lambda>0$ and for $\lambda <0$. The cases $D=3$ (left) and $D=4$ (right) are shown. The figure displays results at $\lambda=0$, but $\alpha_e(r_{\min{}})$ is continuous so the results for small positive or negative $\lambda$ are essentially identical. Regions with $\alpha_e(r_{\min{}}) >0$ are shaded red while those with $\alpha_e(r_{\min{}}) <0$ are shaded green. In the purple regions $V<0$ at all $r$ and Euclidean solutions do not exist.
• Figure 3: Left: Unshaded regions provide allowed Euclidean SAdS exteriors with $r$ increasing outward for $\lambda > - (\kappa-1)^2$ (case ($I$)) with $\mu_i=0$. The angular direction is Euclidean time. Since $\alpha_e(r_{\max{}}) <0$, one must excise the (shaded) region $r> r_{\max }$. The sign of $\alpha_e(r_{\min{}})$ is positive in the top panel and negative in the bottom panel. Center: Allowed interiors for $\lambda >0$ shown as (unshaded) regions of Euclidean dS with $r$ increasing inward. Since $\alpha_i(r_{\min{}}) >0$, one must excise the (shaded) region $r> r_{\min }$. The sign of $\alpha_i(r_{\max{}})$ is also positive at top but is negative at bottom. Right: Allowed interiors for $0 > \lambda > - (\kappa - 1)^2$ shown as (unshaded) regions of Euclidean AdS, with $r$ increasing to the right and Euclidean time running vertically. Here $\alpha_i(r_{\min{}}) >0$ requires one to excise the (shaded) region $r> r_{\min }$. The sign of $\alpha_i(r_{\max{}})$ is also positive at top but is negative at bottom. Sewing any allowed interior to any allowed exterior shows the full solution to have no asymptotic region.
• Figure 4: A degenerate case $(I)$ exterior (unshaded region) with AlAdS boundary. Two copies of Euclidean SdS are shown, with an allowed case $(I)$ domain wall trajectory shown inside the left copy. The sign of $\alpha_e$ requires us to discard the shaded region. Gluing the remaining unshaded regions together along the co-dimension 2 horizons ($H$) gives a singular (degenerate) solution analogous to those advocated in Farhi:1989yrFischler:1989seFischler:1990pk.
• Figure 5: Left: A domain wall intersection that would allow both an $r_{\max}$ and an AlAdS boundary (straight line at top of figure), but which does not satisfy conservation of the stress-energy tensor. Right: A domain wall intersection that conserves the stress-energy tensor.