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Knots, Energy and…Wilson Loops?

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Apologies for the delayed post. I’ve been on the east coast suffering through an intense heatwave. But I’m back, baby!

This week I want to talk about something I don’t know a lot about–knots. Knot theory is a deep and beautiful topic. And it’s nice because it is ~relatively~ intuitive, as far as I can tell–it’s all about embeddings of S^1 into \mathbb{R}^3. For the non-math minded people reading this, a knot is…well, exactly what you think it is.

Why knots? Why not? But seriously I saw this video in one of my many math facebook pages (s/o >implying we can discuss mathematics) about the equivalence of two genus 2 2-surfaces (a donut with 2 holes for the non-experts). Someone in the comments posted a truly bizzare animation demonstrating the equivalence (aside: seriously everyone watch that video because it’s very cool and very trippy). This video was apparently made for a math video festival at the 1998 International Math Conference in Berlin. Lucky for me there was a whole playlist of videos from this fest and so of course I spend the next hour watching them all.

One of those videos was about…you guessed it: knots! This particular video (and the accompanying paper which was surprisingly hard to find online–email me and I’ll send you a PDF if you want) was demonstrating the time evolution of knots with respect to their Mobius energy. Under this evolution, the knots would untangle into a configuration in the same isotopy class which minimizes the energy–sometimes landing in metastable configurations before evolving to the true ground state. For those that don’t know–an isotopy is a way to say that two knots are “the same”. That is, you can do a series of moves called Reidemeister moves which can bring one into the other (although isotopy only refers to the 2nd and 3rd moves if I’m not mistaken). I’ve described this in very physics-y language, which shouldn’t be surprising to anyone reading this. So when I say “time evolve with respect to Mobius energy” I am really talking about solving the gradient flow problem \dot{x}(t)=-\nabla E(x(t)) where E(x(t)) is the Mobius energy of the system. My language seems to imply that this energy is a Hamiltonian–which it might be. Mathematically speaking this is the problem they are solving.

The energy function in question is the following: given a knot K \subset \mathbb{R}^3, the Mobius energy is defined by E(K)=\iint_{K\times K} \frac{f(x,y)}{|x-y|^{2}} dvol_K(x) dvol_K(y) . The regulator f(x,y) is chosen to vanish as x\rightarrow y faster than the denominator so as to render the energy finite. For certain choices of f the energy is invariant under the Mobius group of \mathbb{R}^3. I am not sure why they require this other than things which have symmetry are very nice, but perhaps one of my readers can enlighten me on the technicalities involved here. For this particular video, the regulator they choose is f=1-\cos(\alpha) where I BELIEVE \alpha is supposed to be the angle between x and y, but I am not 100% sure–they never specify in the paper. They do claim that this is related to a slightly more intuitive regularization which gives \tilde{E}(K)=\iint_{K\times K} \left( \frac{1}{|x-y|^{2}}-\frac{1}{d_K(x,y)^2} \right) dvol_K(x) dvol_K(y) where d_K(x,y) is the shortest distance between the points on the knot itself, by E=\tilde{E}-4. So if you want you can just think of this regulator instead. This energy is non-negative, zero for the unknot and infinite if the knot contains self-crossings–that sounds like a good energy to me!

Another way to think of this energy is in the following: discretize the knot using your favorite triangulation, put an electric charge at each vertex and compute the mean field potential induced from all the other charges. Then sum up all of these contributions. This gives essentially the same thing and is indeed a slightly more intuitive way to think about this energy.

Physicists love minimizing energy. It’s one of our top 5 favorite things to do. This got me thinking–is there a way to formulate this problem in a more physics-y language? And will that do anything useful? One first has to wonder whether this energy can indeed be interpreted as a Hamiltonian. This does not seem very easy to do, as the gradient flow equations are not necessarily Hamiltonian–but perhaps a regulator can be chosen such that it becomes Hamiltonian. (I found a set of lecture notes that says a system x'=F(x) can be both Hamiltonian and Gradient flow iff F(x) is a harmonic function. With some work I’m sure an appropriate f(x,y) and F(x) can be found such that E(K) is the Hamiltonian of the system. Whether or not the solution to this problem will preserved the full Mobius invariance or just some subgroup is a whole different ball game).

Now, one of the most amazing connections between knot theory and physics comes from the mind of Ed Witten. In his (amazing) paper titled Quantum Field Theory and the Jones Polynomial, Witten shows that expectation values of Wilson lines in pure Chern-Simons theory on a compact three manifold give a generalization of the Jones polynomial for that line. Let’s break this statement down.

Ok, take a gauge theory with gauge group SU(N). On a three dimensional Euclidean manifold, one can define the Chern-Simons action as S=\frac{k}{4\pi} \int _{M}\text{Tr} (AdA+\frac{2}{3}A\wedge A\wedge A ) . For the mathematicians reading, A is Lie algebra valued the connection 1-form on a principle SU(N) bundle over M. It depends on the local trivialization. For the physicists reading A is the usual covariant vector potential. It is not gauge invariant. For the non-technical people reading, A tells you information about the “electric” and “magnetic” field defined on the space M–a certain combination of derivatives exactly gives you these fields (I put “electric” and “magnetic” in quotes because this is a generalized version of electricity and magnetism which is more complication (and more interesting) but nonetheless the same terms can be used). The theory I have just described is an example of a topological field theory. It does not depend on the metric on M, it’s excitations are non-local and it does weird things on manifolds with boundaries–three very important characteristics of a topological field theory.

A Wilson line is ~basically~ just a line of electric flux. But more specifically it is the holonomy of A around a curve. For simplicity let’s just consider things that are in the fundamental representation of SU(N). We then define the Wilson line as W(K)=\text{Tr}\mathcal{P}e^{i\int _K A} where K is the knot of interest, the trace is taken in the fundamental representation and \mathcal{P} is something technical called path ordering that I don’t want to get into. Consider a collection of n possibly intersecting knots K_i. The remarkable insight of Witten was that the quantity Z \langle W(K_1)W(K_2)...W(K_n)\rangle=\int \mathcal{D}Ae^{iS}W(K_1)W(K_2)...W(K_n) is equal to the Jones polynomial of the link defined by the K_i's. Z is the partition function of Chern-Simons theory and acts as the normalization. We care about the unnormalized expectation value, hence why it is on the left side and not the right. This is very, very, very cool for a LOT of reasons. Incidentally these theories are also the main object of my research. How about that!! The weird \mathcal{D}A is the measure for the path integral over the field A (look it up!) and is the crux of quantum field theory (canonical quantization gang can get bent!!). If you can compute this integral you can solve the full quantum theory. It is very hard to do, though, and doesn’t actually have a very mathematically rigorous formulation (outside of lattice gauge theory where the measure is given by the Haar measure on the gauge group).

Now is there a way to connect these ideas? More specifically, can we define a Mobius energy on the configuration space of the knots such that the expectation value of a Wilson loop defined by a specific knot will evolve to a minimum in time? I do not know. Probably knot, but maybe! There doesn’t seem to be a clear relation between the Jones polynomial and the Mobius energy. One is a topological invariant and the other is certainly not (since it depends on the configuration of the knot), and it is not clear how one should affect the other.

One thing we might be able to do here is find extremize the Wilson loop with respect to it’s path. That is, we take d \langle W(K) \rangle /dK=0 and solve for the minimum configuration K. This may possibly tell us the “minimum energy” configuration for a given Wilson loop in its isotopy class. We can do the same with d \langle W(K_1)W(K_2)...W(K_n) \rangle /d(K_1...K_n). It is far from clear if this will actually work. If we include the “unnormalization” Z it probably won’t work because the unnormalized expectation value is a topological invariant and shouldn’t care about the specific configuration–it only cares about the linking numbers and things like that. If we consider the normalized expectation value, it might work.

To make things even more difficult, if we’re going to want to talk about time evolving something with a Hamiltonian we will definitely have to move out of 3 dimensions. This is because time is its own dimension and if we want to start with a certain Wilson loop in 3 dimensions at time t=0 and evolve it forward in time using the Hamiltonian associated to the Mobius energy to some time t we need to be talking about 3+1 dimensional spacetime. The problem now is that Chern-Simons theory doesn’t exist in 3+1 d and so the connection to the knot invariants is lost. Still, we could do two things: (1) use the standard Yang-Mills Hamiltonian and time evolve a Wilson loop such that it minimizes the YM energy, or (2) define a new type of gauge theory in 3+1 such that the Hamiltonian is the Mobius energy for a given knot and time evolve it using this Hamiltonian. It is not clear if (1) will give the results we want or if (2) is even possible. If it was it would be really cool!

Thanks for listening to my incoherent rambles on knots and gauge theory. These ideas truly are half baked (I’m not admitting I was high when I watched that video and came up with these ideas, but I’m also not not admitting it). If any readers wants to talk about and possibly develop these ideas further, please reach out to me at andrew.baums@gmail.com. I would love to think about this more if it means that I could write a paper about it someday and put it on my CV.

This week’s music recommendation is Denver’s City Hunter. I saw this band a couple weeks ago at the Black Lodge in Seattle and HOLY SHIT. They rip. Black metal style vocals over hardcore/power violence-y music. Inspired by your favorite 80’s slasher flicks the lead singer wore a leather ski mask, gloves and a gigantic coat. Rumor has it sometimes he pulls a knife out too, although that wouldn’t fly at the tiny DIY venue we were at. This band is really really good and if they are coming through your town you should check them out because they put on one helluva show.

Something you (probably) didn’t know Stephen Hawking did

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Stephen Hawking is undoubtably one of the most well known theorists of the latter half of the 20th century. Well, at the very least he’s one of maybe 5 modern physicists with household name recognition. And thats a pretty big deal for us! But beyond being “the black hole dude with ALS who wrote a Brief History of Time”, most non-physicists (and even some physicists!) don’t know how deep and lasting his impact on the field of General Relativity has been. In my mind, this is what sets Hawking apart from most popularizers of science (I’m looking at you Neil deGrasse Tyson you CREEP)–Hawking can talk to the talk (be an effective communicator of science to a public audience) and walk the walk (make lasting contributions to fundamental physics that will influence the work of 1000s of other scientists). Personally, I always enjoyed his writings as a budding scientist, but it wasn’t until graduate school that I really started to admire Hawking even more than I already did. No one has done more to bring General Relativity into the modern era than Stephen Hawking–well except Roger Penrose.

Roger Penrose is another one of those names that you must utter in the same breath as Hawking, but unfortunately he is not nearly as well known. His contributions to the topic of this blog post is equivalent to that of Hawking and so I must give credit where credit is due. If Hawking is the Michael Jordan of late 20th century relativity research, then Penrose is Scottie Pippen. Both were MASSIVE leaders in the field, and their collaboration brought relativity to places no one knew it could go. For those interested in this topic beyond this brief blog post, I HIGHLY suggest reading The Nature of Space and Time by Hawking and Penrose. This is a transcriptions of a series of debates between the two giants on global properties of spacetime and quantum gravity. Its technical but not as terse as a textbook. I learned a lot from it.

This brings me to the topic of this post: the singularity theorems. These are a set of theorems developed by Hawking and Penrose in the late 60’s and early 70’s which prove that any reasonable spacetime (what I mean by this will become clear) MUST contain singularities. We now know (or rather we THINK we know) that this means that there must exist black holes. I’ll talk more about the connection between singularities and black holes more in a future post. ((One technical note: the singularity theorems don’t directly point to the existence of black holes, but in most physical examples the singularities they refer to are black holes. They could also be associated to a Cauchy horizon, which is the boundary of what can be reached from a given initial surface. In other words: its like the “end” of a spacetime–kind of. Another instance could be either the beginning or the end of the universe itself! That means the big bang or a potential big crunch. All of these will play a role in the theorem below.))

This, to me, is a remarkable result in modern GR with some beautiful mathematical underpinnings, only some of which I will be able to touch on in this post. Back when first black hole solution was discovered by Schwarzchild in 1916, the idea of a singularity in spacetime–a region of infinite curvature–was pathological. It wasn’t until much later that this idea was accepted as a physical object and black holes may actually exist in our universe. Thanks to Hawking and Penrose, we know that these things HAVE to be there and HAVE to be in any universe in which the laws of gravitation are given by General Relativity (under physically reasonable assumptions!).

To make sense of this, we have to talk about a basic concept in GR (and differential geometry in general)–a curve. A curve is a 1 dimensional object which connects two points in your spacetime. Since each point in spacetime represents an event–something with a definite location and definite time of occurrence), curves interpolate between two events. For instance, I will be in the Baltimore airport on July 19th at 9:30am, but right now I am in a cafe in Seattle. I can draw curve in spacetime which represents how I got from this moment in Seattle to that moment in Baltimore. Contained in this curve is all the kinematic information about my travels–my velocities at any given time, my acceleration, my change in acceleration etc…Since I am moving “through” time, i.e. I am going from an event at one instant in time to another at a different instant, the tangent vector to this curve must be timelike. This means it has a negative norm. I then say I follow a timelike curve through spacetime. There is another type of curve called a null or lightlike curve. This means that the tangent to the curve has zero norm. Physically this means you are traveling at the speed of light.

In general, a timelike or null curve (hereby collectively referred to as causal curves), represents a general path an observer takes–meaning they can under go acceleration or change their direction at will and so forth. A special type of curve which is supposed to represent a freely falling observer is called a geodesic . A freely falling observer is someone who undergoes no forces other than the “force” of gravity. A spaceman floating through the void of space is undergoing geodesic motion. Hell, even the astronauts on the ISS are undergoing geodesic motion. These types of curves are thought of as “the shortest possible path” between two points. In Euclidean space this is a straight line. On a sphere it is an arc between the points. A geodesic generalizes this idea for arbitrary spaces. These geodesics can also be causal.

Now, each of these curves contains its own little clock–its own way of measuring how far apart points on the curve are from each other. Mathematically, we refer to this as an affine parameter. This is an arbitrary choice and indeed one can choose any affine parameter they want (so long as it obeys some conditions). For timelike curves, a good choice would be either coordinate time or proper time–these would be the time an external or local observer experiences, respectively. But there are plenty of choices that can get the job done. It all depends on the problem you are trying to answer. Now, normally these ideas are strictly local–meaning they are only well defined in a small neighborhood about some point. At some point a curve (geodesic or not) will leave that neighborhood. If you can leave this neighborhood in a mathematically consistent way, we say the geodesic is inextendable–either in the future or past. Mathematically we say that we can extend the curve to arbitrary values of its affine parameter. If you can do this for EVERY curve, the spacetime is called (by Hawking himself!) bundle complete or b-complete (if you can just do it for geodesics, its geodescially complete or g-complete).

Now, you may ask “what do curves have to do with singularities?” Well, it turns out the most satisfying way to define a singularity is to in terms of curves. In GR it is necessary to define your spacetime as a SMOOTH manifold, meaning a surface which is infinitely differentiable at every point. If you have a singularity in the spacetime, there is infinite curvature there and you bet your bottom that your spacetime is certainly not differentiable at that point. So to make sense of the formalism, we must exclude any singularities from the definition of spacetime. We can then definite a singularity as “a point where a causal curve ends (or begins)”. Since if a causal curve ends (begins) on a singularity, it can not be extended any further because there is no more spacetime for it to extend to! The statement of singularities has now been translated into a statement of b-completeness–a spacetime is FREE from singularities if it is b-complete.

Before stating two of the theorems–one due to Penrose and one due to Hawking–let me review one more useful concept. The idea is the following: take surface of any dimension in your spacetime. A surface is spacelike if all the tangent vectors on the surface have positive norms. Same with timelike and null (negative and zero norms). Now on a spacelike surface, take a normal vector defined at each point on the surface and consider geodesics which are tangent to those normal vectors. We can now talk of the evolution of this surface along these geodesic congruences–fancy words for just seeing how the surfaces changes as each of those points follows a geodesic. If these geodesics are getting closer together, then the surface is shrinking–if they are getting farther apart the surface is expanding. In general, they will do both depending where on the surface you are. Pretty simple, right? You can do the same thing but backwards by flipping the sign of the normal vector and seeing how this surface evolves backwards in time. Of special importance are null geodesics which are normal to this surface. Physically, this represents beams of light (world lines of photons) that are traveling orthogonal to this surface.

Specialize to a four dimensional spacetime like ours and consider a closed 2-dimensional space-like surface (think sphere, but more general!). We want to see how this surface evolves along both forward and backwards geodesics. IF the surface is contracting everywhere along both the forward and backward null geodesic congruence normal to the surface, we call this a trapped surface. Physically, this means that if we shine a light forward and backward in time from this surface, the all of these light rays will come together at some point in both the past and the future. This is called a trapped surface. Another type of surface which is important for these theorems is called a Cauchy surface. This is a 3 dimensional surface from which you can reach every single point in the spacetime in either direction. Mathematically, we say the domain of dependence of this surface is the entirety of spacetime. These surfaces don’t exist in every spacetime, but luckily they do in ours!

Let’s state the theorems and comment on some things. First, a theorem from Penrose and then from Hawking. I will leave out the more-complete Penrose-Hawking theorem (the one that gives the strongest conclusion) for technical ease.

Theorem 1 (Penrose 1965). A spacetime (M,g) cannot be null geodesically complete if: (1) R_{ab}K^aK^b \ge 0 for all null vectors K^a (2) there is a non-compact Cauchy surface \mathcal{H} in M (3) there is a closed trapped surface \mathcal{T} in M.

Let’s break this down. To be not be null geodesically complete means that there exists at least one null geodesic which can not be extended to arbitrary values of its affine parameter–either forward or backwards. In other words, there is at least one null geodesic which must run into a point where it stops and ceases to continue. The point at which this happens is the singularity. This conclusion is not as strong as it could be–singularities in black holes also cause timelike geodesics to end, not just null geodesics. But this is the first of a number of theorems on the subject and the conclusions and assumptions have been refined. Condition (1) says that gravity must be an attractive, not repulsive, force. Another way to say this is that energy density must be positive for every single observer. This is the slightly more correct way to think about it, but both are valid in a number of circumstances. We refer to this as the null energy condition. (3) is a relatively weak restriction–it can be shown that these should exist somewhere in spacetime. These surfaces are important for the physical interpretation of black holes are regions where light can not escape from. (2) is a very strong restriction which actually obscures the connection of this theorem to the existence of black holes–it implies the existence of a Cauchy horizon which could also cause the geodesics to stop. So the lack of geodesic completeness could be due to a genuine singularity inside a black hole, or these weird Cauchy horizons.

Let’s move onto Hawking’s contribution.

Theorem 2 (Hawking 1967). If (1) R_{ab}K^aK^b \ge 0 for all causal vectors K^a (2) the strong causality conditions holds (3) there is some past-directed unit timelike vector W and a positive constant b such that if V is the unit tangent to the past-directed timelike geodesic at p, then on each such geodesic the expansion \theta=V^a_{;a} of these geodesics becomes less than -3c/b within a distance b/c from p, where c= -W^aV_a, then there is a past incomplete non-spacelike geodesic through p.

Woah this one is a mouthful! The reason I chose to highlight this particular theorem is because it not only establishes the existence of an incomplete causal geodesic, but also tells you where it occurs. In this case, it occurs in the past. This may seem rather strange, since we have been talking about singularities in the context of black holes and we all know that nothing can escape a black holes and so how could there be a geodesic which ends in a black hole in the past? Well, this theorem is actually telling us something else! Its telling us that at some finite time in the past, the universe was in a singular state. In other words, this theorem tells us that there was a big bang that brought us all into existence!

Let’s break it down. (1) is familiar by now. It is just the null energy condition again but, slightly stronger since we must include timelike geodesics as well. (2) might be familiar from last week’s post. The idea of strong causality is a statement that there can be no curves which come arbitrarily close to intersecting themselves in our spacetime–otherwise time travel would be possible and things get a lot more confusing. (3) is a doozy. We didn’t talk about expansions, since it is a rather technical concept, but it kind of feeds into the trapped surface discussion above. An expansion of geodesics measures how close togehter or far away a family of geodesics becomes as they go on. For the expansion to go to infinity means they go far away from each other, but if they go to negative infinity they get close together. This condition says that so long as the expansion reaches a certain threshold, in this case -3c/b, then the expansion will tend to -\infty, indicating a singularity. It also gives us a time at which this occurs too.

There are many other singularity theorems that tell us a variety of instances in which singularities can occur. The two I laid out here seem to require the simplest explanation, hence why I chose them. The one due to Penrose was the first such theorem and told us that there must be at least one null incomplete geodesic. However, this theorem is rather weak because it doesn’t say if the incompleteness is in the past or future, and leaves open the possibility of being due to a Cauchy horizon. But this theorem is still strong since it’s proof deals directly with situations which occur in gravitational collapse (thats what the trapped surface condition is!) and so gives very convincing evidence of the existence of singularities in the center of black holes.

Hawking’s theorem is a little different. It tells us of a singularity in the past, but the mere fact that it is in the past tells us that it can’t come from a black hole. This is because nothing can escape a black hole and so there can be no causal geodesics which end on one. Instead it tells us something even more interesting (arguably): that the big bang happened!! Or at least the universe was in a singular state at some point in the finite past. This is pretty cool to me. Now if you’re at a party and someone asks you about Stephen Hawking you can say “Oh did you know he proved a theorem which tells us the universe started with a big bang??” and people will definitely want to buy you a drink.

I’m going to end every blog post with a music recommendation. This week it’s Have Heart: https://www.youtube.com/watch?v=xCfpY6jLFVc. I had the pleasure of being at Have Heart’s last show back in 2009. I was 16 years old at the time and was way over my head. But it remains one of the best live music experiences I have ever witnessed. The energy was amazing and has not been matched since. They had a string of reunion shows this past week that I really wish I was able to attend.

And don’t give me the “I used to listen to hardcore in high school but now I don’t because I’m an adult” bull because hardcore music still rips and is still sick and if you abandoned it because you thought it was “uncool” or “corny” then you should really do some self reflection. Sure, the scene can be toxic. It is ripe with bros and toxic masculinity and it’s fair share of creeps. But so is every single scene associated to every single music genre. Don’t tell me that theres not some Mac Demarco looking fuck bois out there playing some soft-ass indie tunes who are in your DMs with shit like “u looked really cute at the gig lol sorry i didn’t come say hi i was too shy lol anyway he’s a picture of my weenie”. Creeps in music are not specific to one genre. Toxic masculinity in music is not specific to one genre. In other words: stop hating on hardcore because you think its bro-y and toxic. Sometimes really nice, good people like really heavy, loud music.

Picture: Hawking and Penrose (right) alongside Andrew Wiles (famous for proving Fermat’s Last Theorem). Stolen from @oxunimaths on twitter.

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.

Am I Really Doing This?

Why must you make yourself even MORE busy?

-me

So I’ve decided to start a blog. Why? Well, partly to motivate myself to stay up to date on current research in my corner of the theory world and partly because I might have something to say? Jury is still out on this last part.

Science is important to me and sometimes being a researcher REALLY takes the pizzaz out of the field. Once you get down and dirty in the nitty gritty you really tend to lose the ~WOW~ factor that made you fall in love with the field in the first place. So here I am–eating a taco salad from the HUB salad bar and setting up a blog so that I can again find my love for the beautiful field of theoretical physics.

Also–I like writing! Thats a part of it too. And maybe this thing won’t be all physics all the time. Maybe I’ll throw some of my other passions into the mix (I hope you like grindcore and black metal!!!). But at the heart of this blog will always be what means the most to me–uncovering the fascinating connections between the mathematical and physical world and elucidating the nature of our universe.

I can’t always promise the grammar will be correct. But I can promise that the physics will.