Eternal traversable wormhole

TL;DR

The paper shows that an eternal traversable wormhole can be realized in nearly-$AdS_2$ gravity by a cross-boundary interaction that yields negative bulk energy, a mechanism that maps naturally onto two coupled SYK models sharing a common gravitational subsector. By deriving and solving a unified low-energy action (Schwarzian-like), it identifies the boundary graviton as the key degree of freedom with an energy gap controlled by a parameter t' and the SL(2) symmetry, and it analyzes both ground-state and finite-temperature phases. The authors demonstrate a Hawking-Page-type transition in the coupled-SYK system, study the spectrum and correlation functions, and show that the ground state remains closely related to the thermofield double of decoupled systems for small couplings, with controlled corrections calculable in large-N and large-q limits. The work provides a concrete bridge between gravitational traversable-wormhole physics and quantum-mechanical models, suggesting broader applicability to systems with emergent conformal symmetry and highlighting avenues for exploring Euclidean wormholes and topological aspects in simpler holographic setups.

Abstract

We construct a nearly-$AdS_2$ solution describing an eternal traversable wormhole. The solution contains negative null energy generated by quantum fields under the influence of an external coupling between the two boundaries. In parallel, we discuss two SYK systems coupled by a relevant interaction. The physics of the two cases is very similar. They both share a "gravitational" subsector which is identical. The solution within this subsector sets the stage for dynamics which is almost conformal invariant. We study this system in detail, both in gravity and in the SYK model. The coupled SYK models have an interesting phase diagram at finite temperature, displaying the usual Hawking-Page transition between the thermal AdS phase at low temperature and the black hole phase at high temperature. Interestingly, these two phases are continuously connected in the microcannonical ensemble.

Figures (23)

• Figure 1:
• Figure 2: (a) Trajectories of the physical boundaries (in magenta) for the Nearly-$AdS_2$ geometry with a global time isometry. These trajectories are the lines where the dilaton acquires its boundary value. It can be obtained by introducing an interaction between the two boundaries. (b) We can describe the fluctuations of the boundary trajectories in terms of a pair of functions, $t_l(u)$ and $t_r(u)$, mapping (rescaled) proper time $u$ along the trajectory to the global $AdS_2$ time coordinate $t$. The dotted lines can be viewed as insertions of the interaction Hamiltonian. They join points with the same value of $u$ on both boundaries. (c) The physical boundaries for a Nearly-$AdS_2$ geometry with thermal isometry. Here the two boundary trajectories cover only a finite range of global time and we cannot send a signal between the two trajectories.
• Figure 3: Schematic comparison of the Euclidean path integral that prepares the $|TFD\rangle$ state and the ground state of coupled system $|G\rangle$. The thermal circle is related to two parallel lines by a conformal transformation. The conformal symmetry is weakly broken. The competition of $H_L+H_R$ and $H_{\rm int}$ controls the symmetry breaking and selects the $|TFD\rangle$ with a certain temperature as the ground state.
• Figure 4: Efective potential for the single degree of freedom associated to the gravitational mode, see (\ref{['PotV']}). Here $\varphi_m$ is the minimum and $\varphi_0$ is where it crosses zero.
• Figure 5: (a) We turn off the interaction hamiltonian at $u=0$. Then the boundary trajectories look like those of the thermofield double state after that time. We see that at zero time, the ground state of the coupled system is very close to the thermofield double state of the decoupled system. We also display the diagrams that are summed over when we make the approximation in (\ref{['Expec']}). (b) some of the diagrams not included in (\ref{['Expec']}). (c) and (d) are the two diagrams that compute the leading corrections to the overlap of the two states in section \ref{['SecOver']}.