Traversable Wormholes, Couplings, and Energy Conditions

A two-sided AdS wormhole becomes traversable in a controlled window when an explicitly timed coupling between the boundaries produces negative averaged null energy on the horizon. The effect is a perturbative time advance for a probe; its sign, operator ordering, signal energy, and gravitational backreaction are essential, and the construction does not permit acausal signaling relative to the coupled boundary system.

Required background. Lorentzian Einstein–Rosen Bridges and Two-Boundary States fixes the uncoupled state. Averaged Null Energy Condition supplies the null-energy diagnostic. Shockwaves, OTOCs, and Scrambling supplies the near-horizon scattering regime.

Helpful background. Quantum Energy Inequalities in Curved Spacetime and Quantum Interest and Negative-Energy Compensation constrain negative energy. Quantum Communication and Entanglement Distribution separates a channel from shared entanglement.

Double-trace opening of the horizon

Start with two noninteracting large-$N$ theories in the TFD state and couple them briefly at boundary time $t_0$:

$$\delta H(t)=-h(t)\,O_L(t)O_R(t),\qquad \int dt\,h(t)=g .$$

The minus sign is part of the convention; changing it changes the effect. To first order,

$$\delta\langle T_{UU}(U)\rangle =i\int dt\,h(t)\, \langle[O_L(t)O_R(t),T_{UU}(U)]\rangle_{\mathrm{TFD}} .$$

For the operator, dimension, timing, and sign used by Gao, Jafferis, and Wall, the horizon integral

$$\mathcal E_U=\int_{-\infty}^{\infty}dU\, \delta\langle T_{UU}\rangle$$

is negative Gao, Jafferis, and Wall 2017. This violates the averaged null energy condition along the relevant generator in the interacting state without requiring a classical exotic source.

Application: the probe time advance

Linearized Einstein equations integrate the null stress into a Kruskal shift. With coordinates chosen so that the future horizon is $V=0$, a probe crossing the shell sees

$$V\longrightarrow V+\Delta V,\qquad \Delta V=-\mathcal K_d\,G_N\mathcal E_U,$$

where $\mathcal K_d>0$ depends on transverse geometry and coordinate normalization. Thus $\mathcal E_U<0$ yields $\Delta V>0$ in this convention, opening a causal window from one exterior to the other.

The signal must be inserted early enough to reach the shifted horizon but late enough that the coupling has acted in the required causal order. In the probe regime the signal momentum $p_U$ obeys both

$$G_N |p_U\mathcal E_U|\ll1 \quad\text{and}\quad \Delta V\ \text{larger than the signal's wave-packet width}.$$

The first inequality controls backreaction; the second makes transmission resolvable. Their overlap bounds the number and energy of messages. In the teleportation description, the double-trace coupling supplies the classical interaction needed to use preexisting entanglement Maldacena, Stanford, and Yang 2017.

Sign, ordering, and backreaction tests

Three adversarial changes diagnose the mechanism.

1. Reverse $g$. Then $\mathcal E_U$ and $\Delta V$ reverse at leading order, producing a delay rather than an opening.
2. Move the signal or coupling outside the required time ordering. The commutator vanishes or has the wrong support, so no traversable trajectory appears.
3. Increase the signal energy. Its positive-energy shock shifts the horizon oppositely and eventually closes the opening; the probe calculation then fails.

The averaged integral must use a complete affinely parameterized horizon generator. A local negative pulse alone is insufficient, because compensating positive energy and quantum inequalities constrain its magnitude and duration.

Causality and scope

The total Hamiltonian includes an explicitly nonlocal $L$–$R$ coupling. Signaling through the bulk is causal relative to that boundary interaction; it is not superluminal communication in a single local boundary theory. The calculation is perturbative in $g$ and $G_N$, assumes controlled correlators, and does not establish a macroscopic, autonomous, or asymptotically flat traversable wormhole. Euclidean connected saddles are a different category, treated next in Euclidean Wormholes and Connected Boundary Amplitudes.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Gao, P., D. L. Jafferis, and A. C. Wall. “Traversable Wormholes via a Double Trace Deformation.” Journal of High Energy Physics 2017, 12 (2017): 151. DOI.
• Maldacena, J., D. Stanford, and Z. Yang. “Diving into Traversable Wormholes.” Fortschritte der Physik 65 (2017): 1700034. DOI.