# A Study of Time Delay from Different Time Zones

Netta Engelhardt (University of California, Santa Barbara) and Sebastian Fischetti (Imperial College) gave us an insight into their communication methods whilst collaborating for their research paper recently published in CQG.

On a dark London evening and a sunny California day — January 19, 2016, to be precise — Netta sent Sebastian a Skype message:

So began a new project for this dynamic duo, published recently in CQG. Unlike our previous project, this one presented a new challenge (with which researchers are all too familiar): we were separated by an eight-hour time difference. Thus began a three-way collaboration: Netta, Sebastian, and Skype (with the third member being the least cooperative).

The process began well:

and was improved by the barely compatible time zones:

Despite the logistical challenges, we were enthusiastic. Our line of inquiry was understanding perturbative quantum gravity via fundamental principles of quantum field theory. Our approach used holography in the form of AdS/CFT, which is a statement about string theory in a “box”. This box is a type of spacetime called Anti-de Sitter space (AdS), and the AdS/CFT duality states that the dynamics of string theory in asymptotically AdS spacetimes can be described in terms of a quantum field theory. This quantum field theory “lives” on the boundary of AdS.

According to this duality, fundamental principles of quantum field theory should translate to fundamental properties of gravity (in particular, string theory) in AdS. For example, unitary evolution of quantum field theories implies that (within the framework of AdS/CFT) black hole evaporation must be unitary. Likewise, causality is a basic property of quantum field theory: nothing can travel faster than light, and two observers outside of each others’ light-cones cannot exchange signals. What, then, is the equivalent statement in the gravitational dual?

Despite the occasional lapse in communications,

we found ways of working around the other’s absence:

The goal was to derive a condition on perturbative quantum string theory (in AdS) that would guarantee boundary causality. More precisely, boundary causality is equivalent to the statement that light rays cannot travel faster through an asymptotically AdS spacetime — the “bulk” — than on its boundary. In other words, light rays must experience a time delay when travelling through the bulk, as shown in the figure.

Boundary causality requires that bulk light rays $\gamma$ (in red) arrive with a positive (left) or zero (right) time delay relative to boundary light rays (in black). The only known spacetime that behaves as the right figure is pure AdS itself; the left figure represents generic asymptotically AdS spacetimes. Figure 1 from our CQG paper. © 2016 IOP Publishing Ltd. All rights reserved.

Fortunately, a well-known theorem by Gao and Wald guarantees that any classical; bulk exhibits such a time delay, with AdS itself being the only marginal case; that is, light rays in pure AdS experience no time delay relative to the boundary. Thus quantum (or stringy) perturbative effects to pure AdS could in principle violate causality of the boundary field theory. Therefore, we needed only to consider perturbations of pure AdS.

It was easy to find a sufficient condition on the bulk to guarantee boundary causality. Intuitively, if a perturbation of AdS causes all of its light cones to close “on average” — that is, if light travels more slowly on average along an entire light ray than it does in pure AdS — then boundary causality is preserved (though locally, light can still speed up). We suspected that the converse was true, but proving it was a bit of a challenge.

Luckily, the advantage of working in well-separated time zones is that the project never sleeps: someone is working on it at all times.

We proved that this averaged light-cone closing condition was, in fact, both sufficient and necessary:

We also showed that this is an entirely new condition on perturbative quantum gravity, not implying any of the favored preexisting ones (e.g. the averaged null energy condition). We believe that more general formulations of this condition exist, and despite the difficulties of a long-distance collaboration,

we look forward to working together again:

Our collaboration has many benefits:

though we never imagined certain aspects would make it to print: