# AdS/CFT and The Gao-Wald Theorem; or why my 18 month side project might finally bite the dust.

Most of my research now a-days deals with the world in 2 spatial dimensions, colloquially referred to as 2+1 dimensions if you’re a high energy theorist like myself. But for over a year now I have been working on a side project that involves using AdS/CFT to see how angular momentum effects the formation and flow of the quark gluon plasma at heavy-ion colliders such as ALICE at CERN, or RHIC at Brookhaven.

Using AdS/CFT to model such collisions is not a novel idea, but we have some local experts on the field and so I thought I’d test to waters to see if this line of work is for me. For the non-experts reading this, AdS/CFT is a fascinating result from string theory which equates quantum gravity to a certain quantum (conformal) field theory. There are then some limits you can take where the theory of quantum gravity becomes classical and the field theory becomes strongly coupled. There is sense in which the field theory is living on the boundary of the spacetime where the gravity exists. For this reason, this field is often called holography–information for a higher dimensional object (the gravity) is encoded in a lower dimensional object (the field theory). This is really cool because we know how to do classical gravity, but we don’t know how to compute things in strongly coupled theories. We can then use our extensive knowledge of General Relativity (s/o Einstein) to compute things about a strongly coupled field theory! This is a really, really, really, really, really, really, really, really, really, really big deal and has lead to some INCREDIBLY fascinating insights between the connection of quantum entanglement and classical gravity–something I will certainly be discussing in the future.

Unfortunately this project looks as if it is leading to a dead end, mainly because of the topic I’m about to discuss–The Gao-Wald Theorem. This is a theorem proven by Robert Wald (author of my favorite-ever physics text book) and his then student Sijie Gao about “time delay”. Intuitively they aim to answer the questions “how does the curvature of spacetime effect the speed of light?”. Obviously we’ve all known since grade school that nothing can travel faster than the speed of light, which is $c=299 792 458 m/s$ (aka 1). This is true for any and every local observer no matter what spacetime you are existing in. The question they are asking is a bit more subtle: take two points $p,q$ in your spacetime and some null curve $\gamma$ connecting them. If this is the “fastest possible path”, then every other path from $p$ to $q$ (either null or time-like) should have a greater elapsed time. This all sounds very simple but I assure you the technical details are mind-numbing and incredibly difficult.

The theorem that really did us in is theorem 2 in their paper “Theorems on gravitational time delay and related issues” arXiv: qr-qc/0007021. This particular theorem deals with spacetimes which can be given a time-like boundary (the normal vector to the boundary has negative norm). In so many words, this theorem says that the fastest path between two points on the boundary lies entirely within the boundary. In other words, there can be no shortcut “through the bulk”.

So what does this have to do with AdS/CFT? As I mentioned before BRIEFLY, the CFT in AdS/CFT “lives on the boundary” of AdS. Applying this result to AdS/CFT is basically saying that your boundary CFT obeys causality–nothing can travel faster than light. For if this was not the case, you could potentially take two space-like separated points in the CFT and connect them in the bulk by a null curve. This is a big no-no since space-like separated operators in ANY well behaved field theory never ever talk to each other–for them to communicate would require superluminal propagation i.e. faster than light travel.

This all still seems very simple, but to illustrate the idea let me state the theorem in all gorey detail:

$\text{Suppose } (M,g_{ab}) \text{ can be conformally embedded in a spacetime}(\tilde{M},\tilde{g}_{ab}) \text{, so that in } M \text{ we have } \tilde{g}_{ab}=\Omega^2 g_{ab} \text{ and on }\dot{M} \text{ we have } \Omega=0 \text{, where } \Omega \text{ is smooth on } \tilde{M}. \text{ Suppose } (M,g_{ab}) \text{ satisfy the following: (1) the null energy and null generic condition. (2) } \bar{M} \text{ is strongly causal. (3) For any }p,q \in \bar{M}, \, J^+(p)\cap J^- (q) \text{ is compact. (4) } \dot{M} \text{ is a time-like hypersurface in }\tilde{M}. \text{Let } p\in \dot{M}. \text{ Then, for any } q\in \dot{A}(p), \text{ we have} q\in J^+(p)-I^+(p). \text{ Furthermore, any causal curve in } \bar{M} \text{ connecting } p \text{ to } q \text{ must lie entirely in }\dot{M} \text{ and, hence, must be a null geodesic on the boundary.}$

Holy shit this formatting is truly awful and I’m still a newbie and don’t know how to fix it. Oh well. Deal with it, nerds.

Let me comment on the technical assumptions here and try to clear up exactly what they are saying. We start by saying that we can accurately talk about a boundary by doing something called “conformal compactification”. This is the process of bringing infinity to a finite distance and including it in your spacetime. That what the multiplication by the function $\Omega$ is doing. The other words are just to say that $\Omega$ should be “nice”. (1) is a statement about the curvature of spacetime. Essentially it is saying that “gravity attracts and doesn’t repel”. (2) says that the entire spacetime (bulk AND boundary–thats what the over bar means) does not contain curves which come arbitrarily close to intersecting themselves. Otherwise, time travel might be possible and thats another big no-no. (3) this is a weird one, but I think its ultimately saying that the future of $p$ and the past of $q$ (thats what $J^+(p)$ and $J^-(q)$ are) must not contain an infinite number of pathological paths from $p$ to $q$. Compactness is sort of like finiteness, except it’s not. It’s ~like~ being finite for something infinite. For example a sphere is compact, but the cartesian plane is not. But a sphere still contains an infinite number of points and so is “infinite”. A closed interval of the real line is compact, but an open interval is not. By saying the intersection of these two objects is compact means that there are no paths which leave $p$ escape to infinity and return to $q$–all of the paths that go from $p$ to $q$ must exist in some definite region of spacetime. (4) is saying that the normal to the boundary (the over dot means boundary) must have a negative norm.

The conclusion is exactly what I said earlier–the fastest path which starts at $p$ and ends at $q$ must lie entirely within the boundary of the spacetime. There are no shortcuts through the bulk.

It is apparent that our toy model for heavy ion collisions indeed violates this theorem. The big question is: why? These technical assumptions are VERY difficult to prove in explicit cases, since they ultimately involve statements like “every $p\in M$” or “every curve from $p$ to $q$” and its very hard to check an uncountably infinite number of special cases. So for right now, this project looks dead in the water, but such is the nature of theoretical physics research.