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 congruence*s–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 cannot be null geodesically complete if: (1) for all null vectors (2) there is a non-compact Cauchy surface in (3) there is a closed trapped surface in .

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) for all causal vectors (2) the strong causality conditions holds (3) there is some past-directed unit timelike vector and a positive constant such that if is the unit tangent to the past-directed timelike geodesic at , then on each such geodesic the expansion of these geodesics becomes less than within a distance from , where , then there is a past incomplete non-spacelike geodesic through .

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 , then the expansion will tend to , 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.